It is 733 p.m. So I will go ahead and call this meeting of the zoning Board of Appeals to order. Good evening. My name is Christian Klein. I'm the chair of the Arlington Zoning Board of Appeals calling this meeting of Tuesday, August 3rd, 2021 to order. I'd like to confirm that all members and anticipated officials are present. Members of the Zoning Board of Appeals. Roger Dupont. Here. Here. Patrick Hanlon. Kevin Mills. Here. Donna Rourke. Here. Aaron Ford. Here. Stephen Revillac. Here. Wonderful. On behalf of the town, our administrator, I know Rick Valarelli. I believe he's going to be joining us a little later. But Vincent Lee is here to helping us out. Vincent, good to see you. And Ms. Sullivan is joining us from the chair of the conservation, not the chair, but from the conservation commission. She's our environmental assistant. Consultants to the board. Paul Haverty. Paul, good to see you. Good evening, Mr. Chairman. Good evening. And from Beta Group. Marty Novers here. Marty, good to see you. You too, Mr. Chair. And I believe you said Tyler was joining us this evening. He's not in the meeting yet, but he is coming. Okay. And I believe Laura Krauss and Bill McGrath are not coming. That's right. Actually, Laura is not on this project, Bill. But Bill can't make me correct. Okay. And then on behalf of 1165, excuse me. I mean, no, it's wrong. I'm going to have to thorn that place. Stephanie Kiefer is here. Good to see you. And I saw Gwen Noyce and Art Clipfeller both on the call as well. Good to see you both. And then Scott Blasic is here. And I see Derek Roach is here. John Hessian is here. Are there others who I'm missing, Stephanie? I'm not certain if Bob Angler is on. I don't see him here yet. I haven't seen him either. Okay. That would be the last one. So this open meeting of the Arlington Zoning Board of Appeals is being conducted remotely consistent with an act extending certain COVID-19 measures adopted during the state of emergency signed into law on June 16th, 2021. This act includes an extension until April 1st, 2022 of the remote meeting provisions of Governor Baker's March 12, 2020 executive order suspending certain provisions of the open meeting law, which suspended the requirement to hold all meetings in a publicly accessible physical location. Further, all members of public bodies are allowed to continue to participate remotely. Public bodies may continue to meet remotely so long as reasonable public access is afforded so the public can follow along with the deliberations of the meeting. An opportunity for public participation is provided during public comment periods. For this meeting, the Arlington Zoning Board of Appeals has convened a video conference via the Zoom app with online and telephone access as listed on the agenda posted to the town's website identifying how the public may join. This meeting is being recorded and it will be broadcast by ACMI. Please be aware that attendees are participating by a variety of means. Some attendees are participating by video conference. Others are participating by computer audio or by telephone. Accordingly, please be aware that other folks may be able to see you, your screen name or another identifier. Please take care to not share personal information. Anything you broadcast may be captured by the recording. We ask you to please maintain decorum during the meeting, including displaying an appropriate background. All supporting materials that have been provided members of this body are available on the town's website unless otherwise noted. Public is encouraged to follow along using the posted agenda. As Chair, I reserve the rights to take items out of order in the interest of promoting an orderly meeting. The first item on our agenda for this evening, item number two, members' voter approval meeting minutes from May 13th, 2021. Moving on to administrative items, these items relate to the operation of the board and as such will be conducted without input from the general public. The board will not take up any new business on prior hearings nor will there be the introduction of any new information on matters previously brought before the board. After introducing the item, I'll invite board members to provide any comments, questions, or motions. Please remember to mute your phone or computer when you're not speaking. Please remember to speak clearly. If members wish to engage in a discussion with other members, please do so through the chair taking care to identify yourself. So the item is the approval of the meeting minutes from May 13th, 2021. Rick Fallorelli had distributed these to the board a couple of weeks ago and hopefully everyone has had a chance to review them. I know that I saw the comments had come back from a few members. I had sent some to Rick as well. Are there any further questions or comments on the proposed May 13th, 2021 meeting minutes? Seeing none, may I have a motion, please? Mr. Chairman, I move to approve the minutes of May 13th, 2021. Thank you. May I have a second? Second. Thank you, Mr. Dupont. And a roll call vote of the board. Mr. Dupont? Aye. Mr. Hanlon? Aye. There we go. Mr. Mills? Aye. Mr. Burke? Aye. Black? Aye. Mr. Ford? Aye. And the chair votes aye. So those minutes are approved. So that moves us on to the next item on our agenda, which is Thorndike Place. I'm now turning to the comprehensive permit hearing for Thorndike Place. Here are some ground rules for effective and clear conduct of tonight's business. So at the prior hearing, the board had requested the applicant consolidate and revise documents before the board to clarify exactly what is being requested. This was necessary to clarify what was to be included in the project after several revisions to the program and to the plans. The updated documents were to include not only the plans for the project, but also requested waivers, updated traffic information, review of the applicability of provisions in the prior draft decision, and review of the proposed terms for the final disposition of the conservation land. The information submitted to the board for its review was received within the last 24 hours, much of it late this afternoon. Since the board, its consultants, the town, the public have not had an opportunity to review these updated documents ahead of this hearing. The board is requesting that the applicant very briefly explain what has been provided to clarify what documents have been added for the record. Members of the board, the board's consultants and town staff will be asked if they need any clarification on what has been included in the submission. There will not be a discussion about the contents of the submission itself. In order for the board, its consultants, the town, the public to review and comment on the submitted materials, the board will be requesting a continuance to Thursday, September 9th. Initial comments should be received from beta and the town within two weeks to allow time for review and response from the applicant. Final documents should be submitted to the board no later than September 2nd to allow time for everything to be publicly posted well ahead of that next hearing. The focus of that hearing will be to review all aspects of the proposed projects. It will include ample time for public comment. At the close of that hearing, the board's technical consultant will be tasked with preparing a revised draft decision. The draft should be made available within two weeks to allow for a full review ahead of the following hearing. The focus of that following hearing proposed for September 28th would be the review of the proposed decision, the impacts of any requested waivers and a review of the project in relation to local needs. It's the board's intention, should the full scope of the project be discussed at that hearing, that all questions are answered and that the applicant, the board, and the town agree that all aspects of the proposed project have been reviewed, that the board would vote to close the public hearing at the end of that hearing. After that point, the board would no longer be able to accept any testimony. It would have to rely on the public record that has been compiled since the initial submission of the project application in 2016. That will then begin a 40-day period in which the board is to render its decision, and that would close on November 11th, should the vote be taken on September 28th. So I will now ask the applicant to use not more than 15 to 20 minutes to explain what is in the project and what documents have been submitted. This is not an opportunity to express opinions about the proposal, nor is it time to argue its merits. It is intended to assist the board, its consultants, the town, the public to understand what has been submitted. A discussion of the project itself will take place at the next hearing. So Ms. Kiefer. Good evening. Thank you, Mr. Chairman. Good evening, members of the board. As the chairman referenced, in the past two days, actually, there have been kind of two tranches of additional submittals that have come in. On August 2nd, there was an 18-sheet set of architectural plans. And then earlier today, there was, as the board requested, an overview narrative, an updated waiver list, responses from both VAI and ESC to the beta peer review comments for civil and traffic. There was also VAI's submittal of its traffic analysis. And there were four sheets set from BSC, grading and drainage, layout and materials, utility plan, and proposed conservation parcel. And I'll just pause there for one second to make certain that the board got that proposed conservation parcel. And if for some reason it didn't come through, I'll submit it immediately to hear after. And then the last piece of the materials that were submitted were three auto-turn analysis exhibits that had been performed by BSC. And what I would like to do right now is just to have Bruce Hamilton architects and then BSC and Vanessa just briefly walk the board through just kind of what the submittals are to help guide in your review. Mr. Chairman, at your discretion, we can either just have those put up, or if you'd prefer to just have that be verbal, our team's willing to do whatever you prefer. If we could just keep it verbal, that would be fine. Okay, that's fine. And just to respond to your earlier question, I'm not seeing specifically the document that delineates the parcel line, but it may be under a different name, and I'm just not seeing it readily. Okay, well, as an abundance of caution, I'll resubmit it once we get off there, then just to make certain that it's in everyone's pocket. I appreciate that, thank you. Okay, absolutely. With that, I'm going to briefly first turn over to Scott Flassock to explain the architectural set that was submitted and kind of to highlight, some of them are resubmissions of what you've seen, but some of them were what the board had requested, so you just have that reference point. So Scott, if you would for a couple of moments. Sure, I'll keep it very brief. Good afternoon, Mr. Chairman and members of the board. So as Stephanie had just mentioned, the recent submission is 18 pages related to the architecture. The majority of that information is resubmission of previously submitted material, so what I'll do here is just briefly indicate what the new information that's provided in this submission. So since we're not going to pull it up, I'll just describe. So page three of that submission is the garage plan. There are a few edits to that plan, but the highlights include that the garage contains 84 total parking spaces. There was previous discussion about the number of accessible spaces, so we've increased that to eight accessible parking spaces, and there are also 10 electric vehicle charging stations shown on the plan, and we just want to note that that is certainly expandable based on what the market and what this population would demand. Page four also has minor edits to it. That is the ground floor plan of the independent living building. Pages eight and nine are completely new pages, and those contain duplex floor plans for those units along Dorothy Road. Pages 10 and 11 are contained minor edits of the duplex exterior elevations to go along with those floor plans, and finally pages 15 and 16 are newly submitted exterior elevations of the south and east sides of the independent living building that were not previously submitted. So that is pretty much a summary of the new and or edited information in that 18-page submission packet. Thank you. John, if you will move from kind of architectural to the civil, the sheets that you submitted, I believe that there are a total of seven, four plan sheets, and then three of the auto-turn analysis, and your microphone's off. Thank you, Stephanie. Good evening, chairman and members of the board. Very quickly, the civil, the updated civil materials that have been submitted include a BSC response to Beta's June 5th civil and wetlands peer review comments. That letter, that response to comments letter really focuses on the new 14 comments that Beta generated based on the, I believe it was the June 8th revised site plan submittal. The first five comments are kind of general site comments. One in particular that has been discussed is a fire department access. So there is a, there has been coordination between the planning department and fire department and our team, and that response is in that first portion of the letter. And then the, so that's the first five general comments in nature, the last nine are stormwater management related comments. And with those responses to all of those comments, it required some changes to the site plan. And those changes are incorporated in the layout materials plan, the grading and drainage plan, and the utilities plan that have been updated in response to Beta's comments and provided with this submission. Additionally, Stephanie had mentioned previously, in response to both Beta's civil and wetlands peer review, there were comments in the, in the Beta traffic peer review regarding vehicle turning maneuvers. So there, there are three vehicle turning exhibits included in the new information for review. And additionally, there was a question or a comment in the Beta traffic review related to the four parallel parking spaces that were previously shown on the west side of the main drive aisle. And in response to that comment, the site plan has been revised and those have been changed from four parallel spaces to six 90 degree head in parking spaces. And, you know, to respond to that question or comment, not driven by any of the peer review comments there is also in the revised plans, a modification to the bike storage area, which is located on the north wall of the senior living building. Essentially, with the change from apartments to senior living, the number of required spaces or anticipated bike storage spaces has been reduced proportionally and the new storage area has been sized to accommodate approximately 14 bicycles. And then lastly, Stephanie mentioned the proposed conservation parcel exhibit that may have not made it into your packets tonight, but that will be provided immediately after this meeting. And that shows the proposed development parcel at approximately 5.8 acres and the remaining just under 12 acres to be contained in a proposed conservation parcel with the permanent restriction placed on that for development. And so in quick summary, that's the new material that has been provided for this prior to this meeting. Thank you. You're welcome. Just to kind of somewhat dovetail on John's comment there with the parking off of the west side of the access drive that there are six head in. What that also does is, as you heard Scott describing the architectural, the basement has 84 parking spaces. So the total site, garage and surface parking remains at 96. So the total parking hasn't changed. So just to clarify that point for the board. And then with that said, we have Derek Roche here this evening and he can briefly give you an overview of Vanessa's. They submitted to one as a response to the peer review traffic and then secondly, there was an update to their traffic assessment. So thank you, Derek. Good evening members of the board. As Stephanie just said, we provided two letters today, one in response to the June 28th beta peer review letter, and one in response to request for information from the chairman of the ZBA. So in just a quick summary of the main traffic points from there, I know John just covered a couple of the points in the beta response letter in terms of auto turn. So I won't re re go over that. But in the so the main points were where we responded to an updated trip generate generation in terms of the new development program to show that the new development program does produce less trips than the previous one four to five less peak hour trips than previously. Now, now part of that in beta's comment letter, they stated we're using land use code 252, which is senior housing attached. They stated that using 253, which congregate care facility, which Mr. Chairman. Yes, please. I with with all respect, I think that that the substance we're getting into now is really more appropriate for the ninth. And here we should be just dealing with the chapter headings and what is involved in not describing the argument since that invites a kind of argument and and gets us into something where we need to get further along than than we're ready to do tonight. Thank you. Appreciate that. So we're just trying to just a narrow focus on what what what we were provided as opposed to what it implies. Sure. And I'll let Derek finish up. I think I don't think he was intending to make an argument and just just to clarify that because just it was to kind of if there had been a confusion and you'll see it in the report, it explains. But just to let you know that the AI had continued to use the land use code that you before. So it's kind of a continuity. So it wasn't trying to make an argument, but I'll let Derek because this is above my pay grade kind of provides you with just a brief overview or a brief statement as to what was covered in that, not you know, not making any arguments at this time, but just to explain to you like to help you read the report basically. We'll be a better audience for him once we've had a chance to read it. Right. Right. And that's definitely what I was just stating that to just to just clarify. We were using the same land use code as previously and we didn't switch to what beta had said. Anyway, continuing on, based on the updated trip gen, we provided an updated level of service capacity analysis at the study area intersections comparison to the prior no build in the prior build condition. We also provided ITE parking chip generation demand calculations, which beta had also summarized in their comment letter. And we concur on the number that ITE says. And that's 76 and we're providing 96, which is a surplus of 20. And then we also provided an auto turn analysis on Little John with cars parked on either side, both parallel and staggered to show that the fire truck and ambulance can make it up and out of Little John without conflict. And that's the summary of the two letters that we provided. Great. Thank you. With that said, Mr. Chairman, I think that's pretty much a kind of a cursory overview just to help the board orient itself when it's reviewing the materials. And if I didn't say this previously, I'd also provided just a narrative memo of what the project is now and then an updated drop to the list. And so I think that is the summary of what's there for the board to review and beta as well. And we're happy to fully respond to any questions or comments that you may have at the next year. Thank you. Appreciate that. And just to let the board, the public know, so those documents that have been attached to the agenda for tonight's hearing and if there's anything that's missing, we'll make sure we get them adding a nozzle carryover to the announcement or the following hearing as well. Make sure that those links remain valid. Are there any questions from members of the board about materials and whether there's anything that they were expecting to see or wanted to make sure that was included? Lots of shaking heads. Mr. Sure. In terms of the applicant's pro forma, have they, I'd like to know if they've updated the original pro forma to reflect the current economics of this project proposal? Ms. Keeper. I think Mr. Havity had gone over this previously, but in terms of the pro forma and looking at it, it's a little bit premature in this process for the board to ask for that information. I apologize for stepping in. The question is not necessarily to provide it, but just to make sure that you guys are internally keeping that the pro forma that you're maintaining internally is tuned to the new plans and is updated as we go along. We just want to make sure that we don't come farther down the road. It isn't like, oh, we just ran the financials now and there's an issue. We want to make sure that you guys are following along with that as you go. Oh, okay. That's correct, Mr. Chairman. That was the intention. We're internally looking at it. Okay. So has that document actually been prepared internally by the applicant? Ms. Keeper. Again, I'm not quite certain that that's appropriate for discussion before the zoning board. So not to be coy, but I just, I'm trying to keep what is within the purview under a 40B application. Look, we have some comfort, little Mr. Chairman, that we don't get further down the road and then we hear from the applicant that they run their numbers and they're not accurate for what is economic. That's what I'm just trying to avoid. Yeah, I would just ask Mr. Haverty if he could comment at this point. Mr. Chairman, what I would suggest is that the applicant has probably run some preliminary numbers and is comfortable with what is currently proposed. My guess is that they haven't done a full pro forma where they actually run through all of the inputs that would be allowed in a pro forma analysis so that they wouldn't necessarily have that information. The board really isn't in a position to request the applicant to provide it, and I don't think the board is requesting that, but if it were at that point, again, we could do it with regards to the original proposal because we did provide a draft decision with all of the conditions with regard to that proposal, but with regard to the current proposal, it has not seen a draft decision and therefore can't actually put together an accurate pro forma because they don't know what particular conditions the board is going to impose and how it will affect the economic viability of the project. Okay, thank you. So right now, it's really probably just a back of the napkin calculation. If I could follow up. Anything further Mr. Haverty? Question? No, Mr. Chairman, as long as we're not surprised by any you know, retraction of the current proposal by the applicant based on the economics leader, I'm satisfied for now. Okay, thank you. Thank you, Mr. Dupont. Yeah, so I have a couple of questions actually. I'm just trying to clarify a little bit about the ground rules and the playing field, and one of the questions is, originally there would have been a project eligibility letter, is that not correct? So there was a project eligibility issued, yeah. Back in 2016. Right, so in that there was a subsidy for the project as it was described, is that also not correct? That I do not know. But I'm wondering if this particular iteration with the senior living with services qualifies for the subsidy from New England funder MHP under a project eligibility letter. I just want to know whether or not that's something that has been dealt with. Keeper, do we know if that's been confirmed? The project eligibility letter has issued. And what happens is when the project completes kind of the ZVA process, the applicant then goes back for final approval and has to submit to the mass housing that it still meets all of the project eligibility and the other materials required for final approval. So it's somewhat of a programmatic issue. Okay, but as far as I was just going to follow up, Mr. Dupont, just to confirm that so as far as you're aware, the change in the program doesn't impact the eligibility or the issued eligibility letter. And as Mr. Dupont had referenced, this is through the New England fund, so through the Federal Home Loan Bank of Boston. So that sort of raises a very obvious question about what is the actual application before the board? Because we're being asked to adjudicate as far as the application is that's been presented to us currently. And I have a concern because I think it was Ms. Keeper who back in May came in and said, you know, we have a revised concept. And if I'm off on this memory, I apologize because I didn't make a note of it, but this is sort of my recollection. And it was sort of this general question about whether or not the board would be would look favorably or at least consider a new concept, which would at that time it was proposed as senior living with services and assisted living. And I do believe that the response was favorable in general. I realized that the residents still have the same concerns about traffic. They have the same concerns about flooding. But I do believe that some people at least spoke favorably about the particular change of use. And then in June when you came back and you said, okay, well, it's not going to contain assisted living. I'm just sort of sitting here wondering what the application is that we're being asked to consider. And I think everyone is aware that should we not outright deny it, but if we approve it with conditions, and the applicant feels that those conditions make the project uneconomic, that they are within their rights to go back to HAC and appeal and say this is uneconomic. At that point, though, I want to make sure that we have a commitment from the applicant that what they will be appealing is the project with respect to the senior living with services. I realize that this is not a contract situation. But I do believe that the way that we've approached this, meaning the members of the board and our town committees, conservation committee, and everyone else, I believe we have dealt with the applicant fairly and in good faith. And so I would ask that the applicant in return sort of recognize that and deal with us. And essentially we would agree that there is a covenant of good faith and fair dealing. So I would like the applicant to state clearly for the record that the application that they are pursuing now is the senior living with services. And should they go to an appeal, that that is what it would be confined to. Thank you. Okay, so I guess I would certainly for the first part of the question as to just clarifying definitively what the project scope is and if that is included in the narrative memorandum that you had submitted earlier today. It is, Mr. Chairman. And it is the senior living with services. It is not assisted living. Correct. Okay. And does the narrative include any description of, I know we had some earlier documents that had indicated what the appeal position was for the project. I think it was very specific that it was not going back to the 219 unit that was presented back in 2016 but was related back more to the 176 unit that was presented in November. And I think Mr. DuPont's question is sort of if this project were to come to an appeal before the HAC, what do you consider sort of the baseline project at this stage? I believe, Mr. Chairman, that as we stated in the narrative that we've put forth the revised concept being the going back to what the board had requested with the home ownership duplexes and then the and then a residential rental component to it. So that would be the 124 independent living senior services. With respect to moving forward, I think that we can't put the cart before the horse and it would be working towards a decision on this. However, I think I would be remiss if I advise my clients to give up any of their legal rights. And so I think that we can't respond to that at this point. But the law is what the law is, but my clients to give the board assurances, we've put a lot of expense and energy and time and thought into the project that's before the board right now. And we very much look forward to continuing working in good faith with the board towards an approval on this project. Mr. Chairman. Mr. DuPont. I would like to accept what Ms. Kiefer is saying at face value, and I do appreciate the sentiment that's being expressed. But I'm a bit incredulous that you could actually have a situation where there is an application which is being pursued currently, and we've all heard tonight about what the additional documents are, the additional information is, and that's what the board, I believe, is being asked to consider. And at the same time, what I'm hearing is, however, should that not be to our liking if the board imposes some conditions, it sounds to me like what's actually being said is, we might not even say that those conditions with regard to the senior living are uneconomic, but that in the back pocket, the applicant is keeping alive this hope or this desire to say, well, you know what, we're going to go the other way. We're going to go back to however many 176 units. And I'm just not aware of any form I've ever dealt with in years as a lawyer where you get a second bite of the apple by essentially just switching horses in midstream in the adjudicatory process. So I guess the question is from Mr. Havardy, is that something that can be done? And are there examples of that in your experience where somebody has essentially pursued one particular application and then withheld and reserved a right to go in a different direction if they don't get the decision that they would like? Mr. Havardy? Yes, Mr. Chairman, that in fact happens quite frequently in the Housing Appeals Committee process. Really the limiting factor for the applicant in moving forward in seeking approval of the Housing Appeals Committee in a different development, one that was previously presented, but perhaps not fully vetted in front of the board, is going to be that the board would then respond with, if you want to pursue the 216 unit project or the 179 unit project, whatever it is, we didn't have the opportunity to fully review that and we would request a remand back to the board for further proceedings. So the applicant is going to not want to get into further proceedings before the board if it can avoid it, so that is something of a limiting factor. To the extent the board hasn't actually had the opportunity to fully review a certain design and set of plans, the risk the applicant would face if they want to pursue an appeal on that version of the project would be that they might be subject to a remand back to the board for further proceedings. But they do have the right to go back to up to as many units as they originally presented at the Housing Appeals Committee and the only issue is whether or not the board has had the opportunity to fully review that plan. Thank you for that. Well, thank you. Is there anything further, Mr. DuBont? No, that's it for now. Thanks. Thank you, Mr. Revillac. Through you, Mr. Chair, there is a question I would like to ask Mr. Haverty. We've extended the 180 day period several times. Would it be reasonable to assume that I guess this is a half of a question and a half of a statement, but my take is that if the applicant were serious about considering an earlier iteration of the project, they could have at some point just declined to do a continuance. Is that correct, Mr. Haverty? Of course. Okay, thank you for that. He could have pushed forward on a different version of the plan and forced the board to issue a decision based upon that version if it so chose. And then it would have, you know, if it got a decision it couldn't accept, it would have availed itself of its right to appeal to the Housing Appeals Committee. I mean, I think that the fact that the applicant is continuing to provide continuance as it is continuing to make changes to the design in response to feedback as receiving from the board is indicative of their current commitments to move forward on the project as currently designed. I can't guarantee that, you know, that if the board issues a decision based upon a particular design that the applicant won't come back at some point before commencing construction and say, you know, circumstances have changed and this is actually no longer an economic development. It does happen. So I can't guarantee that to the board that they won't come back and ask for a modification of the approval. But as of right now, the indications you're receiving from the applicant is that this is something that they are willing to proceed on and that they would, you know, presuming everything goes according to plan once the board issues an approval and they go through all of their post permitting work that needs to be done. They're going to move forward on constructing the project as approved by the board. Thank you. Mr. Everlock, anything further? Nothing further, Mr. Chair. Thank you. Thank you. Just again for the board, are there any other questions that relate specifically to confirming what has been submitted to the board this evening? Seeing none, I would ask Ms. Sullivan is on the call. Emily, if you don't mind, is there any, was there any documentation that Conservation Commission was specifically looking for to be submitted at this time? Well, Christian, I think, so I know it's been discussed that we haven't received the conservation parcel information. So we look forward to reviewing that. And unfortunately, I haven't gotten a chance to review the materials that were submitted this afternoon, but the commission, you know, since the design has changed, will be reviewing the civil plans and specifically just the floodplain impacts and then other resource area impacts like into the 100 foot wetlands buffer. But I think resource area impacts and conservation parcel are probably the two biggest topics the commission will review. Thank you. Mr. Chairman. Mr. Hanlon. Just to sort of say what the obvious, but, you know, most of us have only just looked to see what the title pages of these documents are, and we've heard the discussion that we've all heard. Beta hasn't had a chance to look at them. The Conservation Commission hasn't. There are other people from the town who are not here. And there's going, I think that I'm hoping that apart from this hearing that the people who need to look at these things will identify early on where there's information that they had expected to see that wasn't there and identify it so that the applicant can react sooner rather than later to get whatever missing pieces there are if there are any in so that we don't have a problem at the end of the process where there's a kind of congestion as questions that could have been answered in August or being answered in September. So I'd just like to encourage everybody to make it a priority to survey what we've got and identify any like Kunai and then get them filled early on so that we make maximum use of our time over the next month. Thank you Mr. Hanlon. That's well taken. Just to ask Ms. Nova if there's anything in particular that Beta Group is looking for that may or may not be involved. Well I did quickly read through their response and I did note that there's a drainage report that's forthcoming by August 24th I believe. So we will need that. Hopefully they'll be able to get it to us by August 24th as stated. There are some other items in their comment letter that they stated that they would provide. So we'll go through the letter and if there's anything that they stated that they would provide that we need for our next set of comments we will let the board know. I appreciate that. Thank you. Thank you. Is there anything from the Avalanche perspective that they need from the board at this time? I don't believe so Mr. Chairman. Thank you. Mr. Chairman. Yes please Mr. Hanlon. I was just wondering given that it's the summer and so forth if Ms. Sullivan could maybe address whether the schedule that was laid out by the chair earlier on is one that will dovetail with a meeting of the Conservation Commission so that they'll have the opportunity to view this collectively and give us their collective views on it. I'm not quite sure what the meeting schedule for that commission will be during the course of the next few weeks. Ms. Sullivan? To clarify did you said September 9th? Would be the continued date yes. So the commission will meet three times between today and then so plenty of time hopefully to review everything. So what we have before us now so the intention had been for this evening to try to keep the discussion really sort of limited to what we were you know what we've received and where we're going and then really sort of fold substantive discussion of topics to the following hearing. And so with with that in mind it was my intention to hold off on public comment on this hearing and to extend it farther at the following hearing to make sure that everyone has an opportunity to fully review all the documents that have been submitted and to make sure that we have a full review of all those comments available to everyone before we we really hold a substantive discussion on the revised documentation. So with that I'm going to go ahead and wave public comment specifically just for this one hearing date but I will make sure that there is ample time at the following hearing to make sure to make up for that. Yes. This is Steve Moore on Piedmont Street and Arlington Tree Committee. My comment isn't substantive other than to ask about documentation. And in relation to the tree committee? Yes. Then Mr. Moore please as a member of the next committee. Yeah Mr. Chairman thank you. I just wanted to ask I saw on the list the waiver list that a tree plan is planning to be submitted prior to the issue of a building permit. I have quickly been going through the documentation that was submitted within the past two days. I don't yet I have not yet found if there has been a tree plan as part of that documentation set. My guess is no but I am asking the question. Thank you. I don't know if that's a question for Mr. Hessian or Ms. Geeker if you know the answer to that if a tree plan if a tree plan is planning will be provided. Within the submittal package there was not a tree planting submission and I believe that the it was referenced that it would be submitted prior to the it was requested as a condition be required submitted prior to a building permit. So that would be like a condition of approval with the suggestion that was referenced. So it's prior to the building permit but not prior to the decision of the board. Yes. Okay. Mr. Moore does that clarify? It does. I will reserve the rest of the comments to September 9th. Thank you. Thank you Mr. Moore. Okay. So the schedule that we've discussed there that I had laid out preliminarily at the start of the hearing. So we had previously discussed a possible continuance to September 9th. So I believe that should hold for everyone at this time. Go ahead and find why. So September 9th it's a Thursday night. And then to allow time to after that hearing for Mr. Haverty to work on the proposed decision and then to have that reviewed I had proposed that the following hearing on this that we continue to September 28th which is also a Tuesday evening also at 7.30 to continue that discussion and then the hope would be on that evening we would have an opportunity if we had if we felt we were prepared at that hearing we could close the public hearing on that evening if not the following Tuesday excuse me which is October 5th would be a further continuance and so I would we would request that the review period be extended out from the current day that scheduled which I believe is September 14th is September 17th we would request that that be continued out to October 8th to allow time for those hearings so the dates we would be proposing is Thursday September 9th Tuesday September 28th and a possibility of Tuesday October 6th. I want to confirm that those dates work for the board and for Mr. Haverty and for the applicant. Mr. Haverty? I'm free on those dates. Perfect let's give it to those dates work for you. The the dates work with me I was just going to suggest an interim step Mr. Chairman. For the 9th and then the 28th and then at that point how that the the interim deadline for the conclusion of the public hearing and then you know I think maybe we'll have a better sense of where we are on the 9th to see if we should do that to extend that further to the 5th but perhaps keep that at the 28th of September. Agreed yeah I just wanted to make sure that we we had the we extended the final date out long enough that if we didn't finish on the 28th we needed to utilize the 5th it was available but not that we would be counting on it. Mr. Chairman? Mr. Haverty often has advice to us on when we should set the date for the expiration of the time to close the public hearing with respect to it and I was wondering if he could comment on whether the the method that Ms. Kiefer proposed is is suitably conservative. I'm a little bit concerned with setting as an expiration date the 28th when it's conceivable that for all I know the power will be out and will be terrorist attack by a mob or something like that and we won't be able to meet that night and but on the other hand the the time will have expired and and will be in a pickle. Mr. Chairman I agree with that I always like to have that expiration date have more than 48 hours because you need 48 hours posting of a meeting so if if we could you know do an extra three days beyond the 28th just in case not all unheard of you know to get a hurricane September and if you know there's a loss of power in the town you won't be able to hold your meeting and want to be able to reschedule it and have sufficient time to properly notice it. That's fine Mr. Chairman. Okay are you are you okay with October 8th? Where did you want to? I thought you were talking three days after so the 31st. So we first would be the first well the October so October 1st so then we would be continuing to September 9th with the understanding that we would then be further continuing to September 28th and we would be extending the public hearing period to October 1st okay nodding all around. If I may Mr. Chairman just one quick question just so I think because it's not a hearing date but if I understood you correctly so relative to the September 9th hearing documents are to be submitted by September 2nd and that does that apply to the town boards as well if there's any comments do it would be helpful if it does. Absolutely. Okay thank you. You're welcome okay with that you can have a motion to extend the public review period for Thorndike Place through Friday October 1st 2021. So moved. Thank you Mr. Chairman and may I have a second? Second. Thank you Mr. Revillac. Board of the Board Mr. Dupont? Aye. Mr. Hanlon? Aye. Mr. Mills? Aye. Mr. O'Rourke? I see you I see a motion. Aye. Mr. Revillac? Aye. Mr. Ford? Aye. And the chair says aye so the public review period will be extended to Friday October 1st um and then if I could have a motion unless there's any further further specific on this we will if I could have a motion to continue Thorndike Place the Thursday 2021 at 730 p.m. September 3rd. Up bigger price September 9th. So moved. I have a second on the motion to continue Thorndike Place the Thursday September 9th 2021 at 730 p.m. Second. Thank you Mr. Dupont. Voted the board Mr. Dupont? Aye. Mr. Hanlon? Aye. Mr. Mills? Aye. Mr. O'Rourke? Aye. Mr. Revillac? Aye. Mr. Ford? Aye. And the chair votes aye so we are continued on Thorndike Place um and I will make sure that we um issue an email to uh to the to the applicant the board the town committees and to the um to our consultants with the revised schedule dates so everyone has those and as you say the interim dates as well so when we're looking to receive documentation um so with that in mind um the upcoming dates for um for this board so the next meeting date we have is Tuesday August 24th at 730 p.m. It's a public meeting to discuss the decision for 1165 Armistice Avenue um and then we have a further placeholder of Thursday September 2nd at 730 p.m. for further discussion of the decision for 1165 Armistice Avenue um because that do that is due um at the end of that week. Then uh Thursday September 9th we have just voted uh the continuance of Thorndike Place uh Tuesday September 14th we have a hearing for a project at 20 21a Lafayette Street um and then Tuesday September 28th at 730 p.m. would be our continuance of Thorndike Place and then Friday October 1st it says uh it's the final day to close the public hearing for Thorndike Place so those are the board's upcoming dates um and so with that um that is the end of the business we have scheduled for this evening so I thank you all for your participation in tonight's meeting of the Arlington Zoning Board of Appeals. I appreciate everyone's patience throughout the meeting especially wish to thank Revelle Reilly and Lee Kelly-Lanema for all their continued assistance and preparing for and hosting these online meetings. Please note the purpose of the board's reporting of the meeting is to ensure the creation of an accurate record of the proceedings and it's our understanding the reporting made by ACMI will be available on demand at acmi.tv within the coming days. If anyone has comments or recommendations please send them via email to zba-town.arlington.ma.us that email address is also listed on the Zoning Board of Appeals website and to conclude tonight's meeting I would ask for a motion to adjourn. So moved. Thank you Mr. Hanlon. You may have a second. Second. Thank you Mr. Revelle. Mr. Dupont how do you vote? Aye. Mr. Hanlon. Aye. Mr. Mills. Aye. Mr. O'Rourke. Aye. Mr. Revelle. Aye. Mr. Ford. Aye. Mr. Chair votes aye. We are adjourned. Thank you all very much. Good night. Thank you. Good night. Good night everyone. Thank you.


That's right. So welcome to the last lecture by Professor Wiesken. Right. So I want to use the last lecture to give you some more further results and an overview in what directions these mean curvature flows could be useful. And one theme I thought might be interesting to many of you is how I can use mean curvature flow to prove some isoparametric inequalities. So this lecture is about applications and the isoparametric inequalities. And to fix ideas, let's do first the one dimensional case and show you the most complicated proof of the isoparametric inequality you've ever seen. It's not complicated if you know mean curvature flow, of course. So given an initial curve, gamma 0 S1 into R2. So gamma 0 of S1 is the boundary of our initial domain that we are interested in. And then we know by Grayson's theorem that there exists a solution gamma on some maximal time interval into R2, shortening flow. And this takes this complicated curve into it remains embedded. And it takes it into a small circle and then to a point. So gamma t converges to convex round circle, shrinking round circle as t goes to t max. And if you want to study isoparametric problem, we need to know how the length and area evolve. So l of t be the length of the curve and the a of t, the area of the domain enclosed at time t. And then we get d dt of the length. It's our old formula about the volume element. And d dt of the area is the integral of the speed along the curve. And this is, of course, the integral of the geodesic curvature in our closed embedded curve. We know that this is minus 2 pi. So in fact, we know in the one-dimensional case that t max, because the area is decreasing linearly. And then at the end, it is going to be 0. So the maximal time must be the initial area divided by 2 pi. And we know at the end, both of them tend to 0. So let's look at what the isoparametric distance, the isoparametric defect, if you like, does under the flow. This is what is supposed to be positive on the initial surface. But let's pretend we don't know that. And then let's just compute under the flow this is minus 2l times integral k squared because of the formula above. And from here, we get plus 8 pi squared. Now just use Schwarz inequality. And then you get that this is less than minus 2 times the integral of k ds squared plus 8 pi squared. But integral k ds is 2 pi. If you square that, you get 4 pi squared. So this is 0. In other words, this isoparametric defect decreases under the curve-shortening flow. And since l of t and a of t by Grayson's theorem, so here I use heavily Grayson, tend to 0 as t tends to t max. It must be, since it's been decreasing, that initially it was positive. So we conclude that at the beginning, we must have had this inequality, which is the isoparametric inequality. In other words, the isoparametric inequality in the plane is a trivial consequence of Grayson's theorem. Now it turns out you can try to do that now in higher dimensions. So let's just do the same thing. So we know that because of scaling, we should take power n plus 1 over n of the area of the surface at time t. Then we know there is this optimal constant that we expect from the sphere, from the round sphere. We know how much this should be times the volume enclosed by the surface. And now we just do the same computation. We just get here minus n plus 1 over n times the area to the 1 over n times integral. Now it's the mean curvature squared d mu. And from here, we get plus c0 times the integral of the mean curvature d mu over the surface. You see, we want to compare the positive term with the negative term. We hope that the negative term wins. So what can you do? You do Schwarz inequality just like before. So we estimate this from below by minus n plus 1 over n times the measure to the 1 on n. And from the Schwarz inequality, I get a minus 1 half. And here I get the integral of h squared d mu to the 1 half.  The square root of this, estimated from below by integral h and the other square root stays here. Now you see, everything is fine. So it turns out that the x case, if you have convex surfaces, there's some kind of Minkowski inequality. Well known inequality that says that the integral of h d mu over the boundary of a domain is bigger than and the constant in the Minkowski inequality relates to the constant c0 by n over n plus 1 c0 times mn to the n minus 1 over n. So this is well known for convex surfaces. And then, of course, this means we can, using a gain holder, we can estimate integral h squared d mu to the 1 half from below by n over n plus 1 times this constant times mn to the, and now here I have to subtract 1 half. So I get 1 half minus 1 on n as the power. And therefore, these constants here obviously perfectly fit what stands here, and I get again that this is less or equal to 0. So I get again the isoparametric inequality because there's this theorem that I proved a long time ago that the convex surfaces shrink to a point just like in the one-dimensional case. So this gives you a proof, an immediate proof of the isoparametric inequality for convex regions in rn plus 1 since convex surfaces, convex surfaces movely contract to a point. Of course, these inequalities are well known. You don't need this particular proof, but you have now a new method, and you can hope to apply this method maybe in more general situations, maybe in Riemannian manifolds, assuming some curvature conditions. The first step, of course, would be to try in the high-dimensional case to extend this proof beyond surfaces that are convex. And also in Riemannian manifolds, convexity is such a strong assumption you want to do this in a more general setting. So the question is, can one extend this from convex to some other class, for example, to the class of positive mean curvature surfaces? That would be great because usually it's enough to prove the isoparametric inequality for positive mean curvature surfaces because if you have a general domain, you can always take the mean convex hull, which has non-negative mean curvature, and you could start, and that is then better in terms of the isoparametric inequality anyway, because it has the same area and has the, or even less area, and it has more volume. So you can usually start with the mean convex hull to prove the isoparametric inequality. And therefore the question is, can you extend this argument to surfaces of a non-negative mean curvature? Now, we know that in that case, however, we have possibly singularities. We could have this neck pinch. In the neck pinch, you have this long cylinder. So certainly if you were just to cut out this neck, if you could just cut out this neck and replace it with some cap, this would certainly improve the isoparametric situation. So if we could show somehow that only this or essentially this thing happens and we could extend by cutting out this neck, then we have a chance to extend this mean curvature flow proof of isoparametric inequalities to a more general setting. So the task will be try to extend the mean curvature flow beyond singularities. Very high dimensions. It is known that the singularities can be terrible. So if n is greater or equal than 3, singularities on sheets, which locally look like, say, S1 cross R2 or more general SN minus k cross Rk. k is bigger than 1. In other words, if you have more than one flat direction and you have really whole sheet, two-dimensional or higher dimensional sheet of spheres shrinking off, which can happen. I just can't draw it. We just don't know what to do, how to do the surgery. There is still a concept of weak solution that, for example, Professor Tony Gava told you about with level set approach. But certainly, the smooth approach that I'm choosing here and doing explicit surgery is completely hopeless in that case, at least for the time being. No technology for surgery available yet. So we have to somehow concentrate on the point where you have just one such neck and where this cross section here is like SN minus 1 cross R. And Carlo will tell you a little about how this works in dimensions greater or equal than 3. If you find a way to restrict to this case, what I want to do is concentrate on the case n equals 2. So the plan now is what can be done for two surfaces in R3 or in a general many-fold in 3G bar if the mean curvature is positive and the surface is embedded. And the initial surface, Carlo will tell you what you can do in higher dimensions if you have a stronger positivity condition to convexity for the surfaces, but there you don't need embeddedness. So here, I'm not going to speak about some. And this is joint work with Simon Brandlip. I think we finished this in 2016. And this is appearing as partially has appeared in Inventiones. And another part is going to appear in the Journal of the European Master Society. But this is. So we want to look at two-dimensional embedded surfaces in the Riemannian-3 many-fold and assume positive mean curvature. Now, this allows, of course, these singularities. And the first thing we have to do is we have to classify the singularities. We have to know exactly what singularities are possible. We could do this in the one-dimensional case. Remember, Grayson's theorem, we said the only possible singularity for embedded curves in R2 is the shrinking sphere. So now here, because of this picture, and because, remember, this degenerate problem, I was drawing this situation where the surface would develop a cusp. And here, you expect somehow this in the limit to be sort of this shrinking circle. And I said, here, you expect it somehow to be such a convex thing which is attached to a cylinder. And then, of course, you still have the shrinking spheres. So we want to prove that, essentially, these are the only possibilities. And to do that, one tool is something I just throw at you without proof. Carlo may say more about this. And this is a general theorem. This is joined for Carlo Sinistrari and myself a while back. So we're using a convexity result. If the initial surface, so in any dimension, has positive mean curvature, for any eta, there exists a constant c eta, which just depends on the initial surface. And maybe on the time interval where the solution exists. And on the ambient remanian manifold, so this is really quite general, such that the smallest principle curvature, or any principle curvature, is bounded from below by just a tiny negative fraction of the mean curvature, and then possibly this big constant c of eta. But the scaling invariant part in the inequality only has an arbitrarily small fraction on the right-hand side. So the negative eigenvalue that may be there is tiny near the singularity compared to the mean curvature, so that in particular, when you rescale, the negative part goes completely away. Such that any rescaling of a singularity must be weakly convex, must be a weakly convex solution of mean curvature flow in Rn plus 1. Because even if you do this in a remanian manifold, during the rescaling, since the remanian manifold has bounded curvature, after rescaling, you just see the tangent space of the remanian manifold, which is Rn plus 1. So the singularity models will always live in Euclidean space, even if your surface happens to be in a remanian manifold. And of course, you see at least this is the first step towards this picture. So even though we start here with something only mean convex, if this singularity happens, after rescaling, you see the shrinking cylinder, which is weakly convex. And if we are in this situation here, we even get something which is strictly convex. And if we are in the case where we shrink to a sphere, also we are in the case of strictly convex. So this is just saying that we cannot have limits that are completely non-convex. Not going to say anything about the proof. It's based on analysis of the evolution equations for the curvature and the mean curvature and a lot of analysis at P estimates, Mosa iteration to do this. Now, what you can do with that in the two-dimensional case is the following. So suppose M2-0 is sitting in a smooth remanian-3 manifold and embedded and a positive mean curvature. Then the only rescaling of singularities that may occur, first you see S2 shrinking, the standard shrinking sphere. In R3, the second possibility is you see S1 cross R shrinking, shrinking cylinder. And the third one is a convex internal solution. I should say strictly convex here. Strictly convex internal solution, cylindrical end. So near infinity, it looks like number 2. So it's this picture. You rescale, you find the strictly convex part and the strictly convex part is attached to something which gets more and more looking like a cylinder. So that's a classification of the rescalings that you can get.  Those regions where the curvature is big. Say, if you close to a singular time, you can use this theorem to prove what is called a canonical neighborhood theorem. So from this theorem over there, I say something to the proof in a second. You prove another theorem again with Simon. I call this canonical neighborhood theorem. And this says, given the initial surface as described, embedded positive mean curvature in the remaining manifold, constant h0, some huge constant. This is just to indicate it's huge. Obviously, everything depends on the scaling, depending on the initial surface and the ambient manifold. Sorry, yes, we are in dimensions 2. The region, the mean curvature, is bigger than that number. In other words, the almost singular region. I'm still talking about a completely smooth surface. These surfaces are still below T max, but maybe very close to the singular time. And then we may have a region where the curvature is huge. And what I'm saying is you can determine a threshold for the mean curvature such that beyond this threshold, you know exactly how the manifold looks like. So it will be like this. There will be a part of the manifold which is sort of completely unknown to you. You just know it's embedded as positive mean curvature. And the mean curvature is less than h0 in this region. So this is sort of the nice region where h is less than h0. And then you have a region where maybe you have sort of such a neck. And then it may open up again to give you a nice region. So here you have another nice region. In here, use red. And here somewhere is the red. This is the region where h is bigger than h0. There may be also sort of bad regions which look like these cylindrical regions. But they end in this convex cap. And then maybe if the surface is disconnected, there may also be a region where you have a tiny shrinking sphere. And you may also have a region where sort of a cylindrical region of high curvature closes up in two caps. And you may even have the situation where such a cylindrical region closes up in itself. And you have sort of a shrinking circle. But the point is that in all these regions where h is bigger than h0, the rescaling of this region satisfies theorem. You see either the shrinking, you are either shrinking, you are either the rescaling under the microscope, this region looks like a cylinder, or a cap, or sort of. Yes. And you can make this quantitative. So I can choose this h0. So this is part of the theorem. It's a quantitative theorem given epsilon bigger than 0 and lambda less than infinity. So this is the picture of the previous theorem, in particular how close you are to a cylinder of length lambda in a huge norm. You can even prescribe the norm and see one over epsilon after rescaling. So given epsilon, you can fix the h0. So in other words, we can even introduce these parameters to make this quantitative. This will depend on the epsilon and it will depend on the lambda. They have a very quantitative theorem that tells you, you tell me how precisely you want to see these singularities. And I tell you how long you have to wait with your curvature, how huge the curvature have to be that you have that control in the red region. So that's a very quantitative precise description of the singularities. And that's what enables you to do the surgery and restart the flow. Something Carlo will tell you more about. Now let me give you at least some ideas of how we can fit our previous results together to prove this theorem, what goes in. So the ingredients of proof, first of all, of course, the convexity estimate goes in. So we restrict you very much. Then you use the non-collapsing estimate. Of course, this implies that it remains embedded, but it remains embedded in a quantitative way. The non-collapsing estimate says that after rescaling, so non-collapsing, you have to combine these two. How? So the third step, you write down these two results and then you do the rescaling. And then these two estimates, one and two. So first, let's do it in two steps. One sees that you get either strictly convex or you get weakly convex. Now, if you get something which is weakly convex, you can see by the strong maximum principle, the rescaling, actually, if the rescaling is giving something which is only weakly convex, has to be weakly convex everywhere. If it is weakly convex everywhere, the smallest eigenvalue, one of the two eigenvalues, has to be 0 everywhere. But if it is 0 everywhere, you can see that you can integrate that direction for Binius theorem. And therefore, the surface has to contain a line. So if it is weakly convex, maybe I do the weakly convex case first. So weakly convex. This implies the rescaled surface is some gamma of t cross r. It must contain a line. And gamma t solves the mean curvature flow. And mean curvature flow turns into curve-shortening flow. Curve-shortening flow in the plane perpendicular to the line. But then you use, in particular here, in the non-collapsing estimate, there's the improved non-collapsing estimate by Brindley. So here the improved non-collapsing estimate, remember, showed the inscribed radius after rescaling is less than the mean curvature. That was the last result I showed yesterday, I think. After rescaling, because of this improved non-collapsing estimate by Brindley, mean curvature dominates smooth. But this means if we are, this must mean on gamma, must mean that the inscribed radius is bigger than 1 over the geodesic curvature. But this must mean gamma is a shrinking circle. Gamma is a shrinking. Proves that if we are in the weakly convex case, we get that this gamma t cross r is, in fact, s1 cross r, just a shrinking cylinder. So the weakly convex case is just the shrinking cylinder. We see. Now, the other possibility is that the limit is strictly convex. Now, if the limit is strictly convex, there's two possibilities. It might close up. If it's uniformly convex, then an argument similar to Meijer's theorem. Meijer's theorem is not quite enough to do it, actually. You have to work harder. But sort of think of Meijer's theorem. If it was uniformly convex all the time, you expect it to close up. And using our gradient estimates and everything else we know, we can replace Meijer's theorem to show that. So either it closes up because it's uniformly convex, ratio, the principle curvature is bounded below. But then we are in the case of s2. And that's the first case in the theorem. We have identified the second case in the theorem. We've identified the first case in the theorem, s2. The other possibility is it does not close up. But the only possibility that happens if it does not close up, it cannot be uniformly convex all the time. It cannot be uniformly convex all the time. With the smallest principle curvature, lambda 1 divided by h tending to 0 along some sequence tending to infinity. If this wouldn't go down to 0, it has to close up. But if this goes to 0, then we can rescale again around these points. And then we get back into this situation. Rescale again there. And I say rescale again there. This is an important point. You see, I'm now very, since I'm going out, I'm going away from, I found this strictly convex thing. Now I have to prove that there's a cylinder at the end. So I have to take points. Here's my sequence of points. They go out to infinity. These points are far away from the point where the maximum of the curvature is attained. If I do the rescaling, and I don't have a gradient estimate, I don't get a limit. So the limiting procedure requires the gradient estimate. And if I just have the simple gradient estimate that controls the gradient of the curvature in terms of the maximum curvature, it's not good enough to rescale here. So that's where I use the Harselhofer Kleiner estimate. Remember gradient A less than a constant times A squared. I don't write down all the conditions. But there was this interior estimate, depending on the non-collapsing estimate. Harselhofer Kleiner had the interior gradient estimate because they had the non-collapsing estimate. They didn't leave the improvement by Brentlett. But the non-collapsing estimate gave you this. This allows me to rescale points that are far away from the maximum. And then, because this tends to 0, I'm back in the case, get the cylinder again. I get a point where lambda 1 must be 0 in the limit. But then, there's a strong maximum principle, it must be 0 everywhere. And then, by the improved estimate of Brentlett, it must be, again, after rescaling S1 cross R. This is the third possibility, that I find something first, which is strictly convex. But then, I can go along this strictly convex thing, and I can find a cylinder after rescaling again infinity. So this is a sketch of the first theorem. And then, you have to do this all quantitatively. And using the fact that you can rescale everywhere, you can create this picture easily. So the key thing is the theorem over there, and the fact that we are able to rescale everywhere, and then we can show that, unless we are in the region where the curvature is not so big, we can rescale wherever we want. And we get this thing. It's all contradiction arguments to sum up. It's all contradiction arguments. But given the epsilon, we produce this number H0. And then, we have quantitative control, and can do the surgery procedure, which I'm not going to explain in detail now. Rather, I'd like to show you a little bit what can then be done that eventually follows from this with surgery procedure. So the surgery procedure is the thing that either cuts off this piece. So the surgery, I'm just doing it schematically. It would just cut off here, and it would cut off there, and it would throw away this piece, because we know what it is. We know it's a tiny, arbitrarily tiny tube around some curve. We know what this is topologically. It's an S2. And we know what this is. And this one, we would cut off with another surgery. So in this picture, we would do three surgeries, one here, one there, one there, and throw away the pieces that we know. And this procedure improves the isoparametric behavior of the surface. And I claim, now, the theorem says that, again, Simon, using strongly the surgery procedure that Carlo and I developed in the higher-dimensional case, we can show that, given any initial surface, two-dimensional surface in the Riemannian 3-manifold, embedded h greater than 0, and let's assume the Riemannian 3-manifold is compact, it's sufficient if it is what they say mean convex in the infinity that the surface cannot escape at infinity. So given any such smooth initial surface, there is a solution of mean curvature flow interrupted by, finally, many surgeries, scales at a scale, are zero radius that you cut off comparable to this constant h0 to the minus 1. The area of M and T tends to 0, as T tends to some T max, which is finite. When I say M and T, M and T may consist of, finally, many disconnected components. Even if I start with something which is connected, already after the first surgery, when I cut off this neck here, I may have two disconnected components. So when I write M and T, I mean the union of all the pieces that are still there that I haven't thrown away. So either the area goes to 0 in the finite time, or what else could happen? Well, in the general 3-manifold, the thing could just get stuck on a minimal surface. Or M and T is smooth, or M and T exists on 0 infinity. And they exist sometime T1, bigger than finite T1, such that M and T is smooth. So no more surgeries after T1, Mn, I always write Mn. This is a 2 surface. And M2T converges smoothly to a smooth, weakly stable minimal surface. I draw some pictures in a second, weakly stable. So examples and remarks. So first of all, if you are in R3, T max is less than infinity, because even a complicated surface can be enclosed by some sphere which dies in finite time. So certainly the area will all go away in finite time. So in the Euclidean case, you can just have finitely many such surgeries. And the thing decomposes into these pieces. By the way, you can easily write down initial data of positive mean curvature. Take a thin, very thin circle, axially symmetric. And then you get this picture here, that you can have, in fact, in R3, a very thin torus. And if it is completely symmetric, it will shrink to a circle. So you can have this picture in R3. Now, in the Riemannian manifold, many things can happen. So for example, if the Riemannian manifold looks like this, if this is N3G bar, and your initial surface, see which color we use. Let's use yellow for the initial surface. So if the initial surface is something like this, then the mean curvature vector points in this direction. However, just smoothly, and slowly converge to this minimal surface in here. So this is Mn infinity. And this is Mn zero. Nothing happens. No singularity, nothing, no surgery. It just completely smoothly goes to the minimal surface in there, and that's stable. In fact, it could be weakly stable. If I had attached a cylinder here along here, then it would just be a weakly stable minimal surface. If I draw the picture slightly different, if I draw it like this, and there's no such sink in there, then in this case, I guess, if there's no serious things happening in between, the solution will just keep going, keep going, and end up here, the point, just like the convex solutions in Euclidean space. Let's make the three manifold more complicated. Suppose there's some topology here. Well, then if the surface starts moving, something has to go wrong, right? And well, what goes wrong is that at some stage here, you have sort of, yeah, I should use, yeah, maybe I use white or orange. You have a surgery time here because you have developed somewhere in neck. Here, you do the surgery here. And then the surfaces continue, you have two pieces, one piece here and one piece there. And now, depending on the topology, these could converge to two separate minimal surfaces. I wanted to use red for the limit. So this could be the M2 infinity. Yeah, it's number two. So this can happen, breaks up, goes into two. And of course, I could have put some tentacle here, then you could have one piece being stuck on a minimal surface. This keeps going, and then the other piece could contract to a point, but also be possible. And one other piece is important to notice that, yeah, let's do a application to general relativity. So in general relativity, you have these three many folds which have stars. Here's a star. And this is some N3g bar, where here near infinity, it looks like R3, so asymptotically flat. And you should think of this sitting in some Lorentzian four many fold. And it's a space-like slice. And here you have a black hole. So this is a horizon, sigma 2h. And in the simplest case, the mean curvature of a horizon is 0 in this three many fold. And here you have some star, which means a region of positive scalar curvature. And then you can do the following with this theorem. You could start with some huge sphere near infinity. So m2 0 is the boundary of some huge sphere around the origin in this asymptotically flat region. And then you let it flow. And then at some stage, there's my orange here, at some stage, a surgery will be necessary because it reaches this point. And then after the surgery, these guys here will contract in finite time to this point. And these other guys will take infinite time until they reach this horizon. So you can use mean curvature flow to sweep out asymptotically flat three many folds, the exterior region. So mean curvature flow can sweep out the asymptotically flat region and asymptotically of a space-like slice, a Lorentzian manifold, which models in isolated gravitating system. I should make one remark, I should give credit that I didn't do that yet, examples remarks. And one remark that I didn't make for this long-term existence, that for some time, for some finite time, after that everything is smooth and no more singularity can happen, for this part, we use ideas from Brian White. You see, Brian White has pushed another concept of weak solutions, the level set approach, to its limit. And he has shown that in this situation, you can also have level set solutions. He doesn't have quite as much control on the singularities. He just has an estimate on the house of dimension of the singular set. But he can also prove that his weak solution is going to be smooth near the final minimal surface. And we prove our result by proving that our solution, after a long time, is very close to Brian White's solution. And that's how we get our smoothness for a long time. So we should give a lot of credit here to Brian White. So the theorem here doesn't assume any assumption on the curvature of the three manifold. So for example here, it is known in general relativity that you would like to consider three many faults with non-negative scalar curvature. But for this result, that you can start near infinity and you move until you find the outermost minimal surface, you don't need the non-negative scalar curvature. If you have the non-negative scalar curvature, you can conclude that this limiting surface must be a two-sphere and not a torus or a higher genus surface. So only at that last step, where you show that the horizon actually is a two-sphere in this low-dimensional case, you would use an energy assumption on the Lorentz in many faults or so. Here, this is completely general. But what we can say is, since we're always cutting necks, in other words, we're reducing genus, what we can say is that the genus of this limiting minimal surface, so the genus of FM2 infinity, is certainly less than the genus of the initial surface. It's not so easy. What you cannot do is argue that each time you do a surgery, you reduce the genus. Because sometimes you might cut off a trivial sphere. If you have a neck here, and you just cut off this sphere, it doesn't change the topology.  But you know that the genus will never go up. And I want to give you one more example that can happen. Now, this three-mini-fold could have one of these things that you need in spaceship enterprise. There is a three-mini-fold that looks like this. And it has this bridge here. It allows you to go from one part of the universe to the other part of the universe. And then, of course, your initial surface has to do some more surgery here. And then the surfaces would move in here, and there, and there. And it could happen that there is just one minimal surface in this whole bridge. These minimal surfaces, you would have two pieces, two separate pieces of your solution, which approach the same minimal surface inside the three-mini-fold from two different sides. This can also happen. But this is the worst that can happen. Because there's, again, a theorem of Brian White that you cannot have this would contradict, essentially, the monotonicity formula, density formula, that you cannot have three solutions of mean curvature flow coming arbitrarily close if they are embedded. Sorry, more than three. You can have at most three, one from the left, one from the right, and the minimal surface in the middle. You cannot have more than three. That's the result of Brian White, very general result for minimal surfaces and solutions of mean curvature flow if they are embedded. And you see, it can happen. So for the physicists, of course, they would separate the thing here and treat it as two different ends as seen from infinity. Now, last few minutes, what does this have to do with isoparametric? What can you do about isoparametric inequalities now? Well, it turns out you can now try to use this technique, for example, in such asymptotically flat stream manifolds and say something about the isoparametric behavior of these asymptotically flat stream manifolds. And those of you who know these things in general relativity, there's a concept of mass for these asymptotically flat stream manifolds. Sort of measures how big the stuff inside is. So there's this famous positive mass theorem. If this three-manifold N3g bar has a scalar curvature of g bar greater or equal to 0, and the thing is asymptotically flat, the mass, due to Anouwitz-Deser-Miesner of this three-manifold, is defined by the physicists as some constant by the two-sphere near infinity. So you take huge spheres near infinity of some derivatives of the metric, dGij dxj minus dGii dxj times the normal d sigma. Think of this like a Newtonian, a flux integral, like the metric is the Newtonian potential, and this is a normal integral over the normal derivative of the potential. You know in Newtonian mechanics this would give you the mass, the manifold turns out this flux integral near infinity is under suitable decay estimates of this metric through the Euclidean metric is a geometric invariant that replaces the Newton flux integral. And the positive mass theorem that this thing says that if the scalar curvature is greater or equal to 0, this thing is non-negative with equality if, not only if, N3g bar is equal to r3 with the standard metric, and this positive mass theorem is due to Schoenen-Yau, around 79. And you see this flux integral, at first you don't see that it is independent of the coordinate system. And there's a lot of work to be done. And it uses first derivatives of the metric. So it's not a nice quantity to work with. And positive mass theorem is very, very hard to prove actually using minimal surface theorem and so on. It turns out you can prove this theorem using inverse mean curvature flow. And you can do more using mean curvature flow. If you combine inverse mean curvature flow, which Toti talked about, and mean curvature flow, then you get an isoparametric control on very large regions. Remember, Toti said something about certain integrals being monotone and so on. You can use the inverse mean curvature flow to get a lower bound on the integral h squared, integral of the mean curvature squared, on any outward minimizing surface in the Schoenen-Yau fault. And that was the key step. Remember, at the beginning of this lecture, when I proved the isoparametric inequality in r3 for convex surfaces, the key point was to have a good lower bound on integral h squared. Now, it turns out you can use inverse mean curvature flow on each of these slices that you get from mean curvature flow. So we take mean curvature flow starting at the big yellow surface near infinity, and we flow inverse. On each of these surfaces coming from mean curvature flow, I can use inverse mean curvature flow to get a lower bound on integral of the mean curvature squared. And then I use that lower bound to prove an isoparametric inequality. And that isoparametric inequality in the end turns out to characterize the mass. So the theorem I want to finish with is that you can characterize the mass of such a 3-manifold as the isoparametric deficit. So you take the limb-soup domains of the boundaries of domains tending to infinity. So take larger and larger domains. And then you take the extra volume that you can put in compared to the Euclidean thing. And the Euclidean thing, we have the isoparametric inequality would say that of the boundary to the 3 over 2 is always bigger than 6 square root of pi times the volume. So in Euclidean space, this would almost be negative. But if you are in this manifold and the mass is positive, if the mass is positive, you can do better. And you just have to rescale the things. You have to put a 2 here and divide by the area because mass scales like area. So you have to put in the right scaling. But then you get this thing. You can characterize the mass by how much the manifold differs isoparametrically from Euclidean space. And this is the one result that I could explain what I'm doing to my teenage children. I said, if you have a trampoline, then if it's flat, nobody on the trampoline, then the isoparametric inequality tells you that with a rope of length 2 pi, you can bound an area equal to pi. But if you, my child, stand on the trampoline, you have this tent. And then there's a mass on the trampoline. And then because of this tent, with the same rope, you can bound more area as just pi. And if I'm standing on the trampoline, it may get the black hole. But anyway, you can get more area. And it is exactly the same thing in general relativity. If you have a heavy star bending the three manifold that slices through spacetime, then you can put more volume into the same area than you can do in Euclidean space. And you can, in fact, use the isoparametric defect. How much more volume you can put is an exact measure for the mass in general relativity. And I think that's a good point to stop.