In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in R with negative curvature, Eugenio Calabi and Shing-Tung Yau had a question, about whether there was an immersed complete bounded minimal surface in R, Shing-Tung Yau is from Hong Kong and what type of American mathematician?
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Nadirashvili surface In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in R with negative curvature. The first example of such a surface was constructed by in . This simultaneously answered a question of Hadamard about whether there was an immersed complete bounded surface in R with negative curvature, and a question of Eugenio Calabi and Shing-Tung Yau about whether there was an immersed complete bounded minimal surface in R.
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Shing-Tung Yau Shing-Tung Yau ( ; ; born April 4, 1949) is a Hong Kong and naturalised American mathematician. He was awarded the Fields Medal in 1982. He is currently the William Caspar Graustein Professor of Mathematics at Harvard.
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Chen–Gackstatter surface In differential geometry, the Chen–Gackstatter surface family (or the Chen–Gackstatter–Thayer surface family) is a family of minimal surfaces that generalize the Enneper surface by adding handles, giving it nonzero topological genus.
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Bryant surface In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1. These surfaces take their name from the geometer Robert Bryant, who proved that every simply-connected minimal surface in 3-dimensional Euclidean space is isometric to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization.
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Angenent torus In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow (for which every embedded closed curve converges to a circle as it shrinks to a point), the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.
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Flatness (mathematics) In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. (See "curvature".)
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Incompressible surface In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold.
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Arnold–Givental conjecture The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of L on the number of intersection points of L with a Hamiltonian isotopic Lagrangian submanifold which intersects L transversally.
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Yamabe problem The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature, and takes its name from the mathematician Hidehiko Yamabe. claimed to have a solution, but discovered a critical error in his proof. The combined work of Neil Trudinger, Thierry Aubin, and Richard Schoen later provided a complete solution to the problem in 1984.
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Prescribed scalar curvature problem In Riemannian geometry, a branch of mathematics, the prescribed scalar curvature problem is as follows: given a closed, smooth manifold "M" and a smooth, real-valued function "ƒ" on "M", construct a Riemannian metric on "M" whose scalar curvature equals "ƒ". Due primarily to the work of J. Kazdan and F. Warner in the 1970s, this problem is well understood.
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V. Lakshmibai Venkatraman Lakshmibai is an Indian mathematician who works as a professor of mathematics at Northeastern University in Boston. Her research concerns algebraic geometry, the theory of algebraic groups, and representation theory, including in particular the theory of flag varieties and Schubert varieties.
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Hilbert's theorem (differential geometry) In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface formula_1 of constant negative gaussian curvature formula_2 immersed in formula_3. This theorem answers the question for the negative case of which surfaces in formula_3 can be obtained by isometrically immersing complete manifolds with constant curvature.
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Helicoid The helicoid, after the plane and the catenoid, is the third minimal surface to be known.
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Yoshiharu Kohayakawa Yoshiharu Kohayakawa (Japanese: コハヤカワヨシハル; born 1963) is a Japanese-Brazilian mathematician working on discrete mathematics and probability theory. He is known for his work on Szemerédi's regularity lemma, which he extended to sparser graphs.
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Richmond surface In differential geometry, a Richmond surface is a minimal surface first described by Herbert William Richmond in 1904. It is a family of surfaces with one planar end and one Enneper surface-like self-intersecting end.
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Saddle tower In differential geometry, a saddle tower is a minimal surface family generalizing the singly periodic Scherk's second surface so that it has "N"-fold ("N" > 2) symmetry around one axis.
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Clebsch surface In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface is a non-singular cubic surface studied by and all of whose 27 exceptional lines
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Kenji Fukaya Kenji Fukaya (Japanese: 深谷賢治, "Fukaya Kenji", born March 12, 1959) is a Japanese mathematician known for his work in symplectic geometry and Riemannian geometry. His many fundamental contributions to mathematics include the discovery of the Fukaya category. He is a permanent faculty member at the Simons Center for Geometry and Physics and a professor of mathematics at Stony Brook University.
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Celso Costa Celso José da Costa (born April 7, 1949 in Congonhinhas) is a Brazilian mathematician working in differential geometry. His research activity has focused in the construction and classification of minimal surfaces embedded in three-dimensional Euclidean space. He is best known for his discovery of Costa's minimal surface, which was described in 1982.
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Surface Evolver Surface Evolver is an interactive program for the study of surfaces shaped by surface tension and other energies, and subject to various constraints. A surface is implemented as a simplicial complex. The user defines an initial surface in a datafile. The Evolver evolves the surface toward minimal energy by a gradient descent method. The aim can be to find a minimal energy surface, or to model the process of evolution by mean curvature. The energy in the Evolver can be a combination of surface tension, gravitational energy, squared mean curvature, user-defined surface integrals, or knot energies. The Evolver can handle arbitrary topology, volume constraints, boundary constraints, boundary contact angles, prescribed mean curvature, crystalline integrands, gravity, and constraints expressed as surface integrals. The surface can be in an ambient space of arbitrary dimension, which can have a Riemannian metric, and the ambient space can be a quotient space under a group action.
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Mean curvature flow In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
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Sasakian manifold In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold formula_1 equipped with a special kind of Riemannian metric formula_2, called a "Sasakian" metric.