---
abstract: 'Let $M$ be a $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp\ l\ \Bbb HP^k\sharp \ m\ \overline{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is a quaternionic projective space. When $k=4$, we also have $$Y(M\sharp\ l\ CaP^2\sharp \ m\ \overline{CaP^2})=Y(M),$$ where $CaP^2$ is a Cayley plane.'
author:
- |
    Chanyoung Sung[^1]\
    National Institute for Mathematical Sciences\
    385-16 Doryong-dong Yuseong-gu Daejeon Korea
title: |
    Connected sums with $\Bbb HP^{n}$ or $CaP^{2}$\
    and the Yamabe invariant
---

Introduction
============

The Yamabe invariant is an invariant of a smooth closed manifold defined using the scalar curvature. Let $M$ be a closed smooth $n$-manifold. By the well-known solution of the Yamabe problem, each conformal class of a smooth Riemannian metric on $M$ contains a so-called *Yamabe metric* which has constant scalar curvature. Moreover, letting $$[g]=\{\varphi g \mid \varphi:M
\rightarrow \Bbb R^+ \  \textrm{is smooth} \}$$ be the conformal class of a Riemannian metric $g$, a Yamabe metric of $[g]$ actually realizes $$Y(M,[g]):=\inf_{\tilde{g} \in
[g]} \frac{\int_M s_{\tilde{g}}\ dV_{\tilde{g}}}{(\int_M
dV_{\tilde{g}})^{\frac{n-2}{n}}},$$ where $s_{\tilde{g}}$ and $dV_{\tilde{g}}$ respectively denote the scalar curvature and the volume element of $\tilde{g}$. The value $Y(M,[g])$, which is the value of the scalar curvature of a Yamabe metric with the total volume $1$ is the *Yamabe constant* of the conformal class.

In a quest of a “best” Yamabe metric or more ambitiously a “canonical” metric on $M$, one naturally takes the supremum of the Yamabe constants over the set of all conformal classes on $M$. This is possible because by Aubin’s theorem [@aubin], the Yamabe constant of any conformal class on any $n$-manifold is always bounded by that of the unit $n$-sphere $S^n(1) \subset \Bbb
R^{n+1}$, which is $n(n-1)(\textrm{Vol}(S^n(1)))^{2/n}$. The *Yamabe invariant* of $M$, $Y(M)$, is then defined as the supremum of the Yamabe constants over the set of all conformal classes on $M$. This supremum is not always attained, but if it is attained by a metric which is the unique Yamabe metric with total volume $1$ in its conformal class, then the metric has to be an Einstein metric.([@ander]) In general, one can hope a singular or degenerate Einstein metric leading to a kind of a “geometrization” from a maximizing sequence of Yamabe metrics. It is also noteworthy that the Yamabe invariant is a topological invariant of a closed manifold depending only on the smooth structure of the manifold.

The Yamabe invariant of a compact orientable surfaces is just $4\pi \chi(M)$ where $\chi(M)$ denotes the Euler characteristic of $M$ by the Gauss-Bonnet theorem. In higher dimensions, it is not an easy task to compute the Yamabe invariant. Nevertheless recently there have been much progresses in dimension $3$ and $4$. In dimension $3$, the geometrization by the Ricci flow gives a lot of answers, and in dimension $4$, the Spin$^{c}$ structure and the Dirac operator are keys for computing the Yamabe invariant. In particular, LeBrun [@lb1; @lb3] showed that if $M$ is a compact Kähler surface whose Kodaira dimension is not equal to $- \infty$, then $$Y(M)=-4\sqrt{2}\pi\sqrt{(2\chi+3\sigma)(\tilde{M})},$$ where $\sigma$ denotes the signature and $\tilde{M}$ is the minimal model of $M$. Now based on this evidence, one can ask if the blowing-up does not change the Yamabe invariant of a closed orientable $4$-manifold with nonpostive Yamabe invariant, namely

\[ques1\] Let $M$ be a closed orientable $4$-manifold with $Y(M)\leq 0$. Is there an orientation of $M$ such that $Y(M\sharp\ l\
\overline{\Bbb CP^{2}})=Y(M)$ for any integer $l>0$? What about in higher dimensions?

Further one can also ask whether the analogous statement holds true for the “quaternionic blow-up”, i.e. a connected sum with a quaternionic projective space $\Bbb HP^{n}$, or even a connected sum with a Cayley plane $CaP^{2}$. The purpose of this paper is to prove an affirmative answer to this:

\[th1\] Let $M$ be a closed $4k$-manifold with $Y(M)\leq 0$. Then $$Y(M\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=Y(M),$$ where $l,m$ are nonnegative integers. When $k=4$, we also have $$Y(M\sharp\ l\ CaP^{2}\sharp \ m\ \overline{CaP^{2}})=Y(M).$$

Preliminaries
=============

A computationally useful formula for the Yamabe constant is $$|Y(M,[g])|=\inf_{\tilde{g}\in [g]}(\int_{M} {|s_{\tilde{g}}|}^{\frac{n}{2}}
d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where the infimum is attained only by a Yamabe metric. (For a proof, see [@lb3; @sung3].) So when $Y(M,[g])\leq 0$, this implies that $$Y(M,[g]) =
-\inf_{\tilde{g}\in [g]}(\int_{M} |s^{-}_{\tilde{g}}|^{\frac{n}{2}}
d\mu_{\tilde{g}})^{\frac{2}{n}},$$ where $s_{g}^-$ is defined as $\min\{s_g,0\}$. Therefore when $Y(M)\leq 0$, $$\begin{aligned}
\label{form1}
Y(M)=-\inf_{g}(\int_{M} |s_{g}|^{\frac{n}{2}}
d\mu_{g})^{\frac{2}{n}}=-\inf_{g}(\int_{M} |s_{g}^-|^{\frac{n}{2}}
d\mu_{g})^{\frac{2}{n}}.\end{aligned}$$ Also essential is Kobayashi’s connected sum formula [@koba; @sung2]. $$Y(M_{1}\sharp M_{2}) \geq
\left\{
\begin{array}{ll}  -( |Y(M_1)|^{\frac{n}{2}}+  |Y(M_2)|^{\frac{n}{2}} )^{\frac{2}{n}}
   &\mbox{if } Y(M_i)\leq 0 \ \forall i\\
  \min(Y(M_1),Y(M_2)) &\mbox{otherwise. }
\end{array}\right.$$

We also need to know about the geometry and topology of $\Bbb
HP^{k}$ and $CaP^{2}$. Both has the homogeneous Einstein metric of positive scalar curvature unique up to constant and can be viewed as the mapping cones of the (generalized) Hopf fibrations $\pi_{1}: S^{4k-1}\rightarrow \Bbb HP^{k-1}$ with $S^{3}$ fibers and $\pi_{2}: S^{15}\rightarrow S^{8}$ with $S^{7}$ fibers respectively.

These fibrations have the associated geometries of Riemannian submersion with totally geodesic fibers. In case of $\pi_{1}$, $S^{4k-1}$ and $S^{3}$ are endowed with the round metric of constant curvature $1$, and $\Bbb HP^{k-1}$ is given the homogeneous Einstein metric with curvature ranging between $1$ and $4$. In case of $\pi_{2}$, the total space and the fibers have the round metric of curvature $1$, but the base has the round metric of curvature $4$.

We will denote the round $n$-sphere with the metric of constant curvature $\frac{1}{a^{2}}$ by $S^{n}(a)$, i.e. the sphere of radius $a$ in the Euclidean $\Bbb R^{n+1}$.

Proof of Theorem
================

It’s enough to prove for one connected sum. Let $M'$ be $M\sharp\
\Bbb HP^{k}$ or $M\sharp \  \overline{\Bbb HP^{k}}$, and set $n=4k$. First recall that $\Bbb HP^{k}$ admits a metric of positive scalar curvature meaning that $Y(\Bbb HP^k)>0$. Thus by the connected sum formula, $Y(M')\geq Y(M).$ The idea of the proof is to surger out an $\Bbb HP^{k-1}$ in $M'$ by performing the Gromov-Lawson surgery [@GL] to get back $M$ without decreasing the Yamabe constant much.

To prove by contradiction, let’s assume $Y(M')>Y(M)+2c > Y(M)$ for a constant $c>0$ such that $c$ satisfies $c< \frac{|Y(M)|}{2}$ if $Y(M)<0$. Let $g$ be an unit-volume Yamabe metric on $M'$ such that $s_g\equiv Y(M',[g])= Y(M)+2c$. Let $W$ be an $\Bbb HP^{k-1}\subset \Bbb HP^{k}$ embedded in $M'$. Take a $\delta$-tubular neighborhood $N(\delta)=\{x\in M'|\ dist_{g}(x,W)< \delta\}$ of $W$ for $\delta>0$. We will take $\delta$ small enough so that $N(\delta)$ is diffeomorphic to $\Bbb
HP^{k}- \{\textrm{a point}\}$ and the boundary of $N(\delta)$ is diffeomorphic to $S^{4k-1}$.

We perform a Gromov-Lawson surgery described in [@sung1; @sung2] on $N(\delta)$ along $W$ keeping the scalar curvature bigger than $s_{g}-c$ to get a cylindrical end isometric to $(S^{4k-1}\times
[0,1],\hat{g}+dt^2)$, where $(S^{4k-1},\hat{g})$ is a Riemannian submersion onto $(W,g_W=g|_{W})$ with totally geodesic fibers isometric to $S^{3}(\varepsilon)$, the round $3$-sphere of radius $\varepsilon\ll 1$. Here, the horizontal distribution is given by the connections on the normal bundle. By arranging $\varepsilon$ sufficiently small, $\hat{g}$ has positive scalar curvature. Moreover the volume of the deformed metric can be made arbitrarily small, say $\nu\ll 1$. (For a proof, one may refer to [@sung2]. Also a different method bypassing this is given in the remark below.)

Now let’s take a homotopy $H_{b}(t)=\lambda(t)g_W+(1-\lambda(t))g_{std}$ of smooth metrics on $W$ from $g_W$ to the homogeneous Einstein metric $g_{std}$ of $\Bbb HP^{k-1}$ with curvature ranging from $1$ to $4$, where $\lambda:[0,1]\rightarrow [0,1]$ is a smooth decreasing function with the property that it is $1$ for $t$ near $0$ and $0$ near $1$. This induces a homotopy $H_{1}(t)$ of smooth metrics on $S^{4k-1}$ through a Riemannian submersion with totally geodesic fibers $S^3(\varepsilon)$. And then we homotope the horizontal distribution to that of the Hopf fibration through a Riemannian submersion with totally geodesic fibers $S^3(\varepsilon)$. Let’s denote this homotopy on $S^{4k-1}$ be $H_2(t)$ for $t\in [1,2]$. When $\varepsilon$ is sufficiently small, $H_{1}(t)+dt^2$ and $H_{2}(t)+dt^2$ will give a metric of positive scalar curvature on $S^{4k-1}\times [0,2]$, because it is a Riemannian submersion with totally geodesic fibers onto $\Bbb HP^{k-1}\times [0,2]$. We concatenate this part to the above one obtained from the Gromov-Lawson surgery to get a smooth metric with the boundary isometric to the squashed sphere $S^{4k-1}$ coming from the Hopf fibration. Let’s denote this metric on the boundary by $h_\varepsilon$ for a later purpose.

We want to close it up by a $4k$-ball $B^{4k}$ equipped with a metric of positive scalar curvature. To construct such a metric we resort to the Gromov-Lawson surgery again. Take a sphere $S^{4k}$ with any metric of positive scalar curvature and let $p$ be any point on it. As before, we perform a Gromov-Lawson surgery in a sufficiently small neighborhood of $p$ to get a $4k$-ball with the positive scalar curvature and the cylindrical end isometric to $S^{4k-1}(\varepsilon')\times [0,1]$ for a $\varepsilon'>0$. And then we take a homothety of the whole thing by $\frac{1}{\varepsilon'}$ so that the boundary is isometric to the round sphere $(S^{4k-1}(1),h_{1})$. In order to glue this to the above obtained part, we have to homotope the metric on the boundary. We take a homotopy $H_{3}(t)=\lambda(t)h_{1}+(1-\lambda(t))h_{\varepsilon}$ for $t\in
[0,1]$.

The metric $H_{3}(t)$ on $S^{4k-1}$ has positive scalar curvature for every $t\in [0,1]$.

Note that $h_{1}$ and $h_{\varepsilon}$ differ only by the size of the Hopf fiber. So for each $t$, $H_3(t)$ also has the same Riemannian submersion structure with the fiber isometric to the round $3$-sphere of radius $r(t):=\lambda(t)+(1-\lambda(t))\varepsilon$. By the O’Neill’s formula [@besse], $$\begin{aligned}
s_{H_{3}(t)}&=&\frac{1}{r^{2}(t)}s_{f}+s_{b}\circ
\pi-r^{2}(t)|A|^{2},\end{aligned}$$ where $s_{f}$, $s_{b}$, and $A$ denote the scalar curvature of the fiber and the base, and the integrability tensor for $t=0$ respectively. Thus $s_{H_{3}(t)}$ is constant for each $t$ and increases as $t$ increases. From the fact that $s_{H_{3}(0)}\equiv(4k-1)(4k-2)>0$, the result follows.

Nevertheless the metric $H_{3}(t)+dt^2$ on $S^{4k-1}\times [0,1]$ may not have positive scalar curvature in general. But due to Gromov and Lawson’s lemma in [@GL], for a sufficiently large constant $L>0$, $H_{3}(\frac{t}{L})+dt^2$ on $S^{4k-1}\times
[0,L]$ has positive scalar curvature. Now we have a desired $4k$-ball to be glued to the part made previously out of $M'$.

After the gluing, what we get is just $M$ with a specially devised smooth metric which we denote by $\bar{g}$. Remember that the scalar curvature of $\bar{g}$ is bigger than $s_{g}-c$.

Now we will derive a contradiction. In case that $Y(M)=0$, $$s_{\bar{g}}>s_{g}
-c=Y(M)+c> Y(M)=0,$$ which is a contradiction. In case of $Y(M)<0$, we do the surgery so that $\nu^{\frac{2}{n}}< \frac{2c}{|Y(M)+c|}$. Then noting that $s_{g}<0$, $$\begin{aligned}
-(\int_M |s^-_{\bar{g}}|^{\frac{n}{2}} d\mu_{\bar{g}})^{\frac{2}{n}}
&>& -(\int_{M'-N(\delta)} |s_{g}|^{\frac{n}{2}} d\mu_{g}+ |s_{g}-c|^{\frac{n}{2}}\nu)^{\frac{2}{n}}\\
&>& -(\int_{M'} |s_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}} + (s_{g}-c)\nu^{\frac{2}{n}}\\
&=&  Y(M',[g])  + (Y(M)+c)\nu^{\frac{2}{n}} \\
&>& (Y(M)+2c)-2c\\
&=& Y(M).\end{aligned}$$ This gives a contradiction to the formula (\[form1\]), and completes a proof for the $\Bbb HP^{k}$ case.

The case of $CaP^{2}$ can be proved in the same way using the fact that $CaP^{2}$ also admits a metric of positive scalar curvature, and is the mapping cone of the (generalized) Hopf fibration $\pi: S^{15}\rightarrow S^{8}$ with $S^{7}$ fibers as explained in the previous section.

Since the smallness of $\nu$ was used only in the case of $Y(M)<0$, we will show a way of proof without using it when $Y(M)<0$. As done in LeBrun [@lb4], instead of doing surgery on $(N(\delta),g)$, we first take a conformal change $\varphi g$ of $(M',g)$ such that $\varphi \equiv 1$ outside $N(\delta)$ and the scalar curvature of $\varphi g$ is positive on a much smaller neighborhood $N(\delta')$ of $W$. Moreover one can arrange that it satisfies $$-(\int_{M'} |s^{-}_{\varphi g}|^{\frac{n}{2}} d\mu_{\varphi g})^{\frac{2}{n}} > -(\int_{M'} |s^{-}_{g}|^{\frac{n}{2}} d\mu_{g})^{\frac{2}{n}}-\epsilon$$ for any $\epsilon>0$. (This is possible because the codimension of $W$ is $\geq 3$.) Let’s just say $\epsilon < c$. Then we perform a Gromov-Lawson surgery on $(N(\delta'),\varphi g)$ keeping the scalar curvature positive. The rest is the same and finally we get $$\begin{aligned}
-(\int_M |s^-_{\bar{g}}|^{\frac{n}{2}} d\mu_{\bar{g}})^{\frac{2}{n}}
&=&-(\int_{M'} |s^{-}_{\varphi g}|^{\frac{n}{2}} d\mu_{\varphi g})^{\frac{2}{n}} \\ &>& (Y(M)+2c)-c\\ &>& Y(M).\end{aligned}$$

Example and Final remark
========================

Obviously the theorem is vacuous for the case of $\Bbb HP^{1}$ which is diffeomorphic to $S^{4}$.

Let $H$ be a closed Hadarmard-Cartan manifold, i.e. one with a metric of nonpositive sectional curvature. By the well-known theorem of Gromov and Lawson [@GL2] on the enlargeable manifolds, $H$ cannot carry a metric with positive scalar curvature. Therefore $Y(H)\leq0.$ Applying our theorem to $H$, one has $$Y(H\sharp\ l\ \Bbb HP^{k}\sharp \ m\ \overline{\Bbb
HP^{k}})=Y(H).$$

For a specific example, take $M=T^{n} \times H$, where $T^{n}$ is an $n$-dimensional torus and $H$ is as above, e.g. a product of closed real hyperbolic manifolds. Now since $M$ has an obvious $F$-structure, its Yamabe invariant is actually $0$ by collapsing the $T^{n}$-part. (Refer to Paternain and Petean [@pp].) Thus $$Y(M\sharp\ l\ \Bbb
HP^{k}\sharp \ m\ \overline{\Bbb HP^{k}})=0.$$

Similar examples can also be constructed for $CaP^{2}$.

Going back to the question \[ques1\] addressed in the introduction, our argument does not apply to the case of complex projective space $\Bbb CP^{k}$. We still have the fact that $\Bbb CP^{k}$ is the mapping cone of the Hopf fibration $\pi: S^{2k-1 }\rightarrow \Bbb CP^{k-1}$ with $S^{1}$ fibers. So the $\Bbb CP^{k-1}$ is embedded as a submanifold of codimension $2$ which is one less for the Gromov-Lawson surgery to work. Moreover the statement corresponding to the theorem \[th1\] can not be true at least in dimension $4$. This is because of Wall’s stabilization theorem [@wall]. Let $M$ be a simply-connected closed smooth $4$-manifold. Then there exists integers $l,m$ such that $$M \sharp\ l\ \Bbb CP^{2} \sharp\ m\ \overline{\Bbb CP^{2}}=
a\ \Bbb CP^{2} \sharp\ b\ \overline{\Bbb CP^{2}},$$ where $a=l+\frac{1}{2}(b_{2}(M)+ \sigma(M))$ and $b=m+\frac{1}{2}(b_{2}(M)- \sigma(M)).$ But we know that $Y(a\ \Bbb CP^{2} \sharp\ b\ \overline{\Bbb CP^{2}})>0.$ Thus the Yamabe invariant changes drastically by taking connected sums with both $\Bbb CP^{2}$ and $\overline{\Bbb CP^{2}}$. We do not know whether the stabilization phenomenon of the Yamabe invariant is prevalent also in higher dimensions. But at least the question \[ques1\] is worth investigating in dimension both $4$ and higher.

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[^1]: email address: cysung@nims.re.kr Key Words: Yamabe invariant, quaternionic projective space, Cayley plane, surgery  MS Classification(2000): 53C20,58E11,57R65  This work was supported by the Korea Research Council of Fundamental Science and Technology (KRCF), Grant No. C-RESEARCH-2006-11-NIMS.
