Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.
In mathematics, the uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
Shing-Tung Yau is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and Professor Emeritus at Harvard University. Until 2022, Yau was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua.
Richard Streit Hamilton is an American mathematician who serves as the Davies Professor of Mathematics at Columbia University. He is known for contributions to geometric analysis and partial differential equations. Hamilton is best known for foundational contributions to the theory of the Ricci flow and the development of a corresponding program of techniques and ideas for resolving the Poincaré conjecture and geometrization conjecture in the field of geometric topology. Grigori Perelman built upon Hamilton's results to prove the conjectures, and was awarded a Millennium Prize for his work. However, Perelman declined the award, regarding Hamilton's contribution as being equal to his own.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.
Richard Melvin Schoen is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from University of California at Berkeley. Cheng then spent some years as a post-doctoral fellow and assistant professor at Princeton University and the State University of New York at Stony Brook. Then he became a full professor at University of California at Los Angeles. Cheng chaired the Mathematics departments of both the Chinese University of Hong Kong and the Hong Kong University of Science and Technology in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the American Mathematical Society.
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.
Dennis M. DeTurck is an American mathematician known for his work in partial differential equations and Riemannian geometry, in particular contributions to the theory of the Ricci flow and the prescribed Ricci curvature problem. He first used the DeTurck trick to give an alternative proof of the short time existence of the Ricci flow, which has found other uses since then.
In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties.
Thierry Aubin was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry and non-linear partial differential equations. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold can be conformally rescaled to produce a manifold of constant scalar curvature. Along with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of compact Einstein manifolds. Aubin was the first mathematician to propose the Cartan–Hadamard conjecture.
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:
Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature.
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.
Leon Melvyn Simon, born in 1945, is a Leroy P. Steele Prize and Bôcher Prize-winning mathematician, known for deep contributions to the fields of geometric analysis, geometric measure theory, and partial differential equations. He is currently Professor Emeritus in the Mathematics Department at Stanford University.
Gerhard Huisken is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.
Paul C. Yang is a Taiwanese-American mathematician specializing in differential geometry, partial differential equations and CR manifolds. He is best known for his work in Conformal geometry for his study of extremal metrics and his research on scalar curvature and Q-curvature. In CR Geometry he is known for his work on the CR embedding problem, the CR Paneitz operator and for introducing the Q' curvature in CR Geometry.
Shi Yuguang is a Chinese mathematician at Peking University. His areas of research are geometric analysis and differential geometry.
Frank Wilson Warner III is an American mathematician, specializing in differential geometry.
