Richard Melvin Schoen (born October 23, 1950) 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.
Richard Melvin Schoen (born October 23, 1950) 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.
Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the University of Dayton in mathematics. He then received his PhD in 1977 from Stanford University. After faculty positions at the Courant Institute, NYU, University of California, Berkeley, and University of California, San Diego, he was Professor at Stanford University from 1987–2014, as Bass Professor of Humanities and Sciences since 1992. [16] He is currently Distinguished Professor and Excellence in Teaching Chair at the University of California, Irvine. [17] His surname is pronounced "Shane."
Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship in 1979. [2] Schoen is a 1983 MacArthur Fellow. [3] He has been invited to speak at the International Congress of Mathematicians (ICM) three times, including twice as a Plenary Speaker. [18] In 1983 he was an Invited Speaker at the ICM in Warsaw, in 1986 he was a Plenary Speaker at the ICM in Berkeley, and in 2010 he was a Plenary Speaker at the ICM in Hyderabad. For his work on the Yamabe problem, Schoen was awarded the Bôcher Memorial Prize in 1989. [5] He was elected to the American Academy of Arts and Sciences in 1988 and to the National Academy of Sciences in 1991, became Fellow of the American Association for the Advancement of Science in 1995, and won a Guggenheim Fellowship in 1996. [4] [6] [7] In 2012 he became a Fellow of the American Mathematical Society. [8] He received the 2014–15 Dean’s Award for Lifetime Achievements in Teaching from Stanford University. [9] In 2015, he was elected Vice President of the American Mathematical Society. [19] He was awarded an Honorary Doctor of Science from the University of Warwick in 2015. [20] He received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman. [21] In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by Kazan Federal University, and the Rolf Schock Prize. [22] [23] [24]
He has had over 44 doctoral students, including Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves. [25]
Schoen has investigated the use of analytic techniques in global differential geometry, with a number of fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.
In 1976, Schoen and Shing-Tung Yau used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by James Eells and Joseph Sampson to noncompact settings. [26] [27] By identifying a certain interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel conditions on the domain leading to the same conclusion. These rigidity theorems are complemented by their existence theorem for harmonic maps on noncompact domains, as a simple corollary of Richard Hamilton's resolution of the Dirichlet boundary-value problem. [28] As a consequence they found some striking geometric results, such as that certain noncompact manifolds do not admit any complete metrics of nonnegative Ricci curvature.
In two papers from the 1980s, Schoen and Karen Uhlenbeck made a foundational contribution to the regularity theory of energy-minimizing harmonic maps. The techniques they developed, making extensive use of monotonicity formulas, have been very influential in the field of geometric analysis and have been adapted to a number of other problems. Fundamental conclusions of theirs include compactness theorems for sets of harmonic maps and control over the size of corresponding singular sets. Leon Simon applied such results to obtain a clear picture of the small-scale geometry of energy-minimizing harmonic maps. [29]
Later, Mikhael Gromov had the insight that an extension of the theory of harmonic maps, to allow values in metric spaces rather than Riemannian manifolds, would have a number of significant applications, with analogues of the classical Eells−Sampson rigidity theorem giving novel rigidity theorems for lattices. The intense analytical details of such a theory were worked out by Schoen. Further foundations of this new context for harmonic maps were laid out by Schoen and Nicholas Korevaar.
In 1979, Schoen and his former doctoral supervisor, Shing-Tung Yau, made a number of highly influential contributions to the study of positive scalar curvature. By an elementary but novel combination of the Gauss equation, the formula for second variation of area, and the Gauss-Bonnet theorem, Schoen and Yau were able to rule out the existence of several types of stable minimal surfaces in three-dimensional manifolds of positive scalar curvature. By contrasting this result with an analytically deep theorem of theirs establishing the existence of such surfaces, they were able to achieve constraints on which manifolds can admit a metric of positive scalar curvature. Schoen and Doris Fischer-Colbrie later undertook a broader study of stable minimal surfaces in 3-dimensional manifolds, using instead an analysis of the stability operator and its spectral properties.
An inductive argument based upon the existence of stable minimal hypersurfaces allowed them to extend their results to higher dimensions. Further analytic techniques facilitated the application of topological surgeries on manifolds which admit metrics of positive scalar curvature, showing that the class of such manifolds is topologically rich. Mikhael Gromov and H. Blaine Lawson obtained similar results by other methods, also undertaking a deeper analysis of topological consequences. [30] [31]
By an extension of their techniques to noncompact manifolds, Schoen and Yau were able to settle the important Riemannian case of the positive mass theorem in general relativity, which can be viewed as a statement about the geometric behavior near infinity of noncompact manifolds with positive scalar curvature. Like their original results, the argument is based upon contradiction. A more constructive argument, using the theory of harmonic spinors instead of minimal hypersurfaces, was later found by Edward Witten. [32] [33] [34]
Schoen, Yau, and Leon Simon identified a simple combination of the Simons formula with the formula for second variation of area which yields important curvature estimates for stable minimal hypersurfaces of low dimensions. In 1983, Schoen obtained similar estimates in the special case of two-dimensional surfaces, making use of the existence of isothermal coordinates. Slightly weaker estimates were obtained by Schoen and Simon, although without any dimensional restriction. Fundamental consequences of the Schoen−Simon estimates include compactness theorems for stable minimal hypersurfaces as well as control over the size of "singular sets." In particular, the Schoen−Simon estimates are an important tool in the Almgren–Pitts min-max theory, which has found a number of applications.
The possible presence of singular sets restricts the dimensions in which Schoen and Yau's inductive arguments can be easily carried out. Meanwhile Witten's essential use of spinors restricts his results to topologically special cases. Thus the general case of the positive mass theorem in higher dimensions was left as a major open problem in Schoen and Yau's 1979 work. In 1988, they settled the problem in arbitrary dimension in the special case that the Weyl tensor is zero; this has been significant in conformal geometry. In 2017, they released a preprint claiming the general case, in which they deal directly with the singular sets of minimal hypersurfaces.
In 1960, Hidehiko Yamabe introduced the "Yamabe functional" on a conformal class of Riemannian metrics and demonstrated that a critical point would have constant scalar curvature. [35] He made partial progress towards proving that critical points must exist, which was taken further by Neil Trudinger and Thierry Aubin. [36] [37] Aubin's work, in particular, settled the cases of high dimension or when there exists a point where the Weyl tensor is nonzero. In 1984, Schoen settled the cases left open by Aubin's work, the decisive point of which rescaled the metric by the Green's function of the Laplace-Beltrami operator. This allowed an application of Schoen and Yau's positive mass theorem to the resulting metric, giving important asymptotic information about the original metric. The works of Yamabe, Trudinger, Aubin, and Schoen together comprise the solution of the Yamabe problem, which asserts that there is a metric of constant scalar curvature in every conformal class.
In 1989, Schoen was also able to adapt Karen Uhlenbeck's bubbling analysis, developed for other geometric-analytic problems, to the setting of constant scalar curvature. [38] [39] The uniqueness of critical points of the Yamabe functional, and more generally the compactness of the set of all critical points, is a subtle question first investigated by Schoen in 1991. Fuller results were later obtained by Simon Brendle, Marcus Khuri, Fernando Codá Marques, and Schoen.
In the 1980s, Richard Hamilton introduced the Ricci flow and proved a number of convergence results, most notably for two- and three-dimensional spaces. [40] [41] Although he and others found partial results in high dimensions, progress was stymied by the difficulty of understanding the complicated Riemann curvature tensor. [42] Simon Brendle and Schoen were able to prove that the positivity of Mario Micallef and John Moore's "isotropic curvature" is preserved by the Ricci flow in any dimension, a fact independently proven by Huy Nguyen. [43] [44] Brendle and Schoen were further able to relate their positivity condition to the positivity of sectional curvature and of curvature operator, which allowed them to exploit then-recent algebraic ideas of Christoph Böhm and Burkhard Wilking, thereby obtaining a new convergence theorem for Ricci flow. [45] A special case of their convergence theorem has the differentiable sphere theorem as a simple corollary, which had been a well-known conjecture in the study of positive sectional curvature for the past fifty years.
Textbooks
In the mathematical field of Riemannian geometry, the scalar curvature is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow, sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
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 he was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua.
Eugenio Calabi was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications.
In the mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental interest, as they are the solutions of Einstein's field equations in a vacuum with vanishing cosmological constant.
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.
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric product. The best-known is the Cheeger–Gromoll splitting theorem for Riemannian manifolds, although there has also been research into splitting of Lorentzian manifolds.
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.
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.
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.
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 differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold. First introduced by Richard S. Hamilton, Yamabe flow is for noncompact manifolds, and is the negative L2-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.
The Geometry Festival is an annual mathematics conference held in the United States.
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.
In mathematics, the Almgren–Pitts min-max theory is an analogue of Morse theory for hypersurfaces.
David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow of the American Mathematical Society since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman.
Joseph Harold Sampson Jr. was an American mathematician known for his work in mathematical analysis, geometry and topology, especially his work about harmonic maps in collaboration with James Eells. He obtained his Ph.D. in mathematics from Princeton University in 1951 under the supervision of Salomon Bochner.
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