Shing-Tung Yau

Shing-Tung Yau (; Chinese: 丘成桐; pinyin: Qiū Chéngtóng; born April 4, 1949) is an American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University.[1]

Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation.[2] Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied mathematics, engineering, and numerical analysis.

Yau was born in Shantou, Guangdong, China in Jiaoling County in 1949. His mother, Yeuk Lam Leung, was born in Meizhou; his father, Chen Ying Chiu, was a Chinese scholar of philosophy, history, literature, and economics.[YN19] He was the fifth of eight children, with Hakka ancestry.[3]

During the Communist takeover of China, when he was only a few months old, his family moved to Hong Kong; he was not able to revisit until 1979, at the invitation of Hua Luogeng, when China entered the reform and opening era.[YN19]. They had financial troubles from having lost all of their possessions, and his father and second-oldest sister died when he was thirteen. Yau began to read and appreciate his father's books, and became more devoted to schoolwork. After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969, without receiving a degree due to graduating early. He left his textbooks with his younger brother, Stephen Shing-Toung Yau, who then decided to major in mathematics as well.

Yau left for the Ph.D. program in mathematics at University of California, Berkeley in the fall of 1969. Over the winter break, he read the first issues of the Journal of Differential Geometry, and was deeply inspired by John Milnor's papers on geometric group theory.[4][YN19] Subsequently he formulated a generalization of Preissman's theorem, and developed his ideas further with Blaine Lawson over the next semester.[5] Using this work, he received his Ph.D. the following year, in 1971, under the supervision of Shiing-Shen Chern.[6]

He spent a year as a member of the Institute for Advanced Study at Princeton before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor at Stanford University.[7] From 1984 to 1987 he worked at University of California, San Diego.[8] Since 1987, he has been at Harvard University.[9]

In 1978, Yau became "stateless" after the British Consulate revoked his Hong Kong residency due to his United States permanent residency status.[10][11] Regarding his status when receiving his Fields Medal in 1982, Yau stated "I am proud to say that when I was awarded the Fields Medal in mathematics, I held no passport of any country and should certainly be considered Chinese."[12] Yau remained "stateless" until 1990, when he obtained United States citizenship.[10][13]

With science journalist Steve Nadis, Yau has written a non-technical account of Calabi-Yau manifolds and string theory,[YN10] a history of Harvard's mathematics department,[NY13] and an autobiography.[YN19]

Yau has made major contributions to the development of modern differential geometry and geometric analysis. As said by William Thurston in 1981:[14]

We have rarely had the opportunity to witness the spectacle of the work of one mathematician affecting, in a short span of years, the direction of whole areas of research. In the field of geometry, one of the most remarkable instances of such an occurrence during the last decade is given by the contributions of Shing-Tung Yau.

His most widely celebrated results include the resolution (with Shiu-Yuen Cheng) of the boundary-value problem for the Monge-Ampère equation, the positive mass theorem in the mathematical analysis of general relativity (achieved with Richard Schoen), the resolution of the Calabi conjecture, the topological theory of minimal surfaces (with William Meeks), the Donaldson-Uhlenbeck-Yau theorem (done with Karen Uhlenbeck), and the Cheng−Yau and Li−Yau gradient estimates for partial differential equations (found with Shiu-Yuen Cheng and Peter Li). Many of Yau's results (in addition to those of others) were written into textbooks co-authored with Schoen.[SY94][SY97]

In addition to his research, Yau is the founder and director of several mathematical institutes, mostly in China. John Coates has commented that "no other mathematician of our times has come close" to Yau's success at fundraising for mathematical activities in China and Hong Kong.[5] During a sabbatical year at National Tsinghua University in Taiwan, Yau was asked by Charles Kao to start a mathematical institute at the Chinese University of Hong Kong. After a few years of fundraising efforts, Yau established the multi-disciplinary Institute of Mathematical Sciences in 1993, with his frequent co-author Shiu-Yuen Cheng as associate director. In 1995, Yau assisted Yongxiang Lu with raising money from Ronnie Chan and Gerald Chan's Morningside Group for the new Morningside Center of Mathematics at the Chinese Academy of Sciences. Yau has also been involved with the Center of Mathematical Sciences at Zhejiang University,[15] at Tsinghua University,[16] at National Taiwan University,[17] and in Sanya.[18] More recently, in 2014, Yau raised money to establish the Center of Mathematical Sciences and Applications (of which he is the director), the Center for Green Buildings and Cities, and the Center for Immunological Research, all at Harvard University.[19]

Modeled on an earlier physics conference organized by Tsung-Dao Lee and Chen-Ning Yang, Yau proposed the International Congress of Chinese Mathematicians, which is now held every three years. The first congress was held at the Morningside Center from December 12 to 18, 1998. He co-organizes the annual "Journal of Differential Geometry" and "Current Developments in Mathematics" conferences. Yau is an editor-in-chief of the Journal of Differential Geometry,[20] Asian Journal of Mathematics,[21] and Advances in Theoretical and Mathematical Physics.[22] As of 2021, he has advised over seventy Ph.D. students.[6]

In Hong Kong, with the support of Ronnie Chan, Yau set up the Hang Lung Award for high school students. He has also organized and participated in meetings for high school and college students, such as the panel discussions Why Math? Ask Masters! in Hangzhou, July 2004, and The Wonder of Mathematics in Hong Kong, December 2004. Yau also co-initiated a series of books on popular mathematics, "Mathematics and Mathematical People".

In 2002 and 2003, Grigori Perelman posted preprints to the arXiv claiming to prove the Thurston geometrization conjecture and, as a special case, the renowned Poincaré conjecture. Although his work contained many new ideas and results, his proofs lacked detail on a number of technical arguments. Over the next few years, several mathematicians devoted their time to fill in details and provide expositions of Perelman's work to the mathematical community.[23] A well-known August 2006 article in the New Yorker written by Sylvia Nasar and David Gruber about the situation brought some professional disputes involving Yau to public attention.[12][13]

Yau claimed that Nasar and Gruber's article was defamatory and contained several falsehoods, and that they did not give him the opportunity to represent his own side of the disputes. He considered filing a lawsuit against the magazine, claiming professional damage, but says he decided that it wasn't sufficiently clear what such an action would achieve.[YN19] He established a public relations website, with letters responding to the New Yorker article from several mathematicians, including himself and two others quoted in the article.[28]

In his autobiography, Yau said that his statements in 2006 such as that Cao and Zhu gave "the first complete and detailed account of the proof of the Poincaré conjecture" should have been phrased more carefully. Although he does believe Cao and Zhu's work to be the first and most rigorously detailed account of Perelman's work, he says he should have clarified that they had "not surpassed Perelman's work in any way."[YN19] He has also maintained the view that (as of 2019) the final parts of Perelman's proof should be better understood by the mathematical community, with the corresponding possibility that there remain some unnoticed errors.

Yau has made a number of major research contributions, centered on differential geometry and its appearance in other fields of mathematics and science. In addition to his research, Yau has compiled influential sets of open problems in differential geometry, including both well-known old conjectures with new proposals and problems. Two of Yau's most widely cited problem lists from the 1980s have been updated with notes on progress as of 2014.[29] Particularly well-known are a conjecture on existence of minimal hypersurfaces and on the spectral geometry of minimal hypersurfaces.

In 1978, by studying the complex Monge–Ampère equation, Yau resolved the Calabi conjecture, which had been posed by Eugenio Calabi in 1954.[Y78a][30] As a special case, this showed that Kähler-Einstein metrics exist on any closed Kähler manifold whose first Chern class is nonpositive. Yau's method relied upon adapting earlier work of Calabi, Jürgen Moser, and Aleksei Pogorelov, developed for quasilinear elliptic partial differential equations and the real Monge–Ampère equation, to the setting of the complex Monge-Ampère equation.[31][32][33]

The understanding of the Calabi conjecture in the noncompact setting is less definitive. Gang Tian and Yau extended Yau's analysis of the complex Monge−Ampère equation to the noncompact setting, where the use of cutoff functions and corresponding integral estimates necessitated the conditional assumption of certain controlled geometry near infinity.[TY90] This reduces the problem to the question of existence of Kähler metrics with such asymptotic properties; they obtained such metrics for certain smooth quasi-projective complex varieties. They later extended their work to allow orbifold singularities.[TY91] With Brian Greene, Alfred Shapere, and Cumrun Vafa, Yau introduced an ansatz for a Kähler metric on the set of regular points of certain surjective holomorphic maps, with Ricci curvature approximately zero.[G+90] They were able to apply the Tian−Yau existence theorem to construct a Kähler metric which is exactly Ricci-flat. The Greene−Shapere−Vafa−Yau ansatz and its natural generalization, now known as a "semi-flat metric," has become important in several analyses of problems in Kähler geometry.[38]

The positive energy theorem, obtained by Yau in collaboration with his former doctoral student Richard Schoen, is often described in physical terms:

In Einstein's theory of general relativity, the gravitational energy of an isolated physical system is nonnegative.

However, it is a precise theorem of differential geometry and geometric analysis, in which physical systems are modeled by Riemannian manifolds with nonnegativity of a certain generalized scalar curvature. As such, Schoen and Yau's approach originated in their study of Riemannian manifolds of positive scalar curvature, which is of interest in and of itself. The starting point of Schoen and Yau's analysis is their identification of a simple but novel way of inserting the Gauss-Codazzi equations into the second variation formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold, which by the Gauss-Bonnet theorem highly constrains the possible topology of such a surface when the 3-manifold has positive scalar curvature.[SY79a][39]

Schoen and Yau exploited this observation by finding novel constructions of stable minimal hypersurfaces with various controlled properties.[SY79a] Some of their existence results were developed simultaneously with similar results of Jonathan Sacks and Karen Uhlenbeck.[40] Schoen and Yau adapted their work to the setting of certain asymptotically flat initial data sets in general relativity, where they showed that negativity of the mass would allow one to invoke the Plateau problem to construct stable minimal surfaces; the topology of such surfaces would be contradicted by (an extension of) their original observation on the Gauss-Bonnet theorem. This contradiction proved a Riemannian formulation of the positive mass theorem in general relativity.[SY79c]

Schoen and Yau extended this to the standard Lorentzian formulation of the positive mass theorem by studying a partial differential equation proposed by Pong-Soo Jang. They proved that solutions to the Jang equation exist away from the apparent horizons of black holes, at which solutions can diverge to infinity.[SY81] By relating the geometry of a Lorentzian initial data set to the geometry of the graph of a solution to the Jang equation, interpreted as a Riemannian initial data set, Schoen and Yau reduced the general Lorentzian formulation of the positive mass theorem to their previously-proved Riemannian formulation. Furthermore, by reverse-engineering their analysis of the Jang equation, they were able to establish that any sufficient concentration of energy in general relativity must be accompanied by an apparent horizon.[SY83]

Due to the use of the Gauss-Bonnet theorem, these results were originally restricted to the case of three-dimensional Riemannian manifolds and four-dimensional Lorentzian manifolds. Schoen and Yau established an induction on dimension by constructing Riemannian metrics of positive scalar curvature on minimal hypersurfaces of Riemannian manifolds which have positive scalar curvature.[SY79b] Such minimal hypersurfaces, which were constructed by means of geometric measure theory by Frederick Almgren and Herbert Federer, are generally not smooth in large dimensions, so these methods only directly apply up for Riemannian manifolds of dimension less than eight. Without any dimensional restriction, Schoen and Yau proved the positive mass theorem in the class of locally conformally flat manifolds.[SY88][30] In 2017, Schoen and Yau published a preprint claiming to resolve these difficulties, thereby proving the induction without dimensional restriction and verifying the Riemannian positive mass theorem in arbitrary dimension.

Gerhard Huisken and Yau made a further study of the asymptotic region of Riemannian manifolds with strictly positive mass. Huisken had earlier initiated the study of volume-preserving mean curvature flow of hypersurfaces of Euclidean space.[41] Huisken and Yau adapted his work to the Riemannian setting, proving a long-time existence and convergence theorem for the flow. As a corollary, they established a new geometric feature of positive-mass manifolds, which is that their asymptotic regions are foliated by surfaces of constant mean curvature.[HY96]

Traditionally, the maximum principle technique is only applied directly on compact spaces, as maxima are then guaranteed to exist. In 1967, Hideki Omori found a novel maximum principle which applies on noncompact Riemannian manifolds whose sectional curvatures are bounded below.[42] It is trivial that approximate maxima exist; Omori additionally proved the existence of approximate maxima where the values of the gradient and second derivatives are suitably controlled. Yau partially extended Omori's result to require only a lower bound on Ricci curvature; the result is known as the Omori−Yau maximum principle.[Y75b] Such generality is useful due to the appearance of Ricci curvature in the Bochner formula, where a lower bound is also typically used in algebraic manipulations. In addition to giving a very simple proof of the principle itself, Shiu-Yuen Cheng and Yau were able to show that the Ricci curvature assumption in the Omori−Yau maximum principle can be replaced by the assumption of the existence of cutoff functions with certain controllable geometry.[CY75][30]

Yau was able to directly apply the Omori−Yau principle to generalize the classical Schwarz−Pick lemma of complex analysis. Lars Ahlfors, among others, had previously generalized the lemma to the setting of Riemann surfaces.[43] With his methods, Yau was able to consider the setting of a mapping from a complete Kähler manifold (with a lower bound on Ricci curvature) to a Hermitian manifold with holomorphic bisectional curvature bounded above by a negative number.[Y78b][34]

Cheng and Yau extensively used their variant of the Omori−Yau principle to find Kähler−Einstein metrics on noncompact Kähler manifolds, under an ansatz developed by Charles Fefferman.[44] The estimates involved in the method of continuity were not as difficult as in Yau's earlier work on the Calabi conjecture, due to the fact that Cheng and Yau only considered Kähler−Einstein metrics with negative scalar curvature. The more subtle question, where Fefferman's earlier work became important, is to do with completeness. In particular, Cheng and Yau were able to find complete Kähler-Einstein metrics of negative scalar curvature on any bounded, smooth, and strictly pseudoconvex subset of complex Euclidean space.[CY80][34] These can be thought of as complex geometric analogues of the Poincaré ball model of hyperbolic space.

Yau's original application of the Omori−Yau maximum principle was to establish gradient estimates for a number of second-order elliptic partial differential equations.[Y75b] Given a function on a complete and smooth Riemannian manifold which satisfies various conditions relating the Laplacian to the function and gradient values, Yau applied the maximum principle to various complicated composite expressions to control the size of the gradient. Although the algebraic manipulations involved are complex, the conceptual form of Yau's proof is strikingly simple.

Yau's novel gradient estimates have come to be called "differential Harnack inequalities" since they can be integrated along arbitrary paths in to recover inequalities which are of the form of the classical Harnack inequalities, directly comparing the values of a solution to a differential equation at two different input points. By making use of Calabi's study of the distance function on a Riemannian manifold,[45] Yau and Shiu-Yuen Cheng gave a powerful localization of Yau's gradient estimates, using the same methods to simplify the proof of the Omori−Yau maximum principle.[CY75] Such estimates are widely quoted in the particular case of harmonic functions on a Riemannian manifold, although Yau and Cheng−Yau's original results cover more general scenarios.

In 1986, Yau and Peter Li made use of the same methods to study parabolic partial differential equations on Riemannian manifolds.[LY86] Richard Hamilton generalized their results in certain geometric settings to matrix inequalities.[46] Analogues of the Li−Yau and Hamilton−Li−Yau inequalities are of great importance in the theory of Ricci flow, where Hamilton proved a matrix differential Harnack inequality for the curvature operator of certain Ricci flows, and Grigori Perelman proved a differential Harnack inequality for the solutions of a backwards heat equation coupled with a Ricci flow.[47][48]

Cheng and Yau were able to use their differential Harnack estimates to show that, under certain geometric conditions, closed submanifolds of complete Riemannian or pseudo-Riemannian spaces are themselves complete. For instance, they showed that if M is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then the induced Riemannian metric on M is complete.[CY76a] Analogously, they showed that if M is an affine hypersphere of affine space which is topologically closed, then the induced affine metric on M is complete.[CY86] Such results are achieved by deriving a differential Harnack inequality for the (squared) distance function to a given point and integrating along intrinsically defined paths.

In 1985, Simon Donaldson showed that if M is a nonsingular projective variety of complex dimension two, then a holomorphic vector bundle over M admits a hermitian Yang-Mills connection if and only if the bundle is stable.[49] A result of Yau and Karen Uhlenbeck generalized Donaldson's result to allow M to be a compact Kähler manifold of any dimension.[UY86] The Uhlenbeck-Yau method relied upon elliptic partial differential equations while Donaldson's used parabolic partial differential equations, roughly in parallel to Eells and Sampson's epochal work on harmonic maps.[50]

The results of Donaldson and Uhlenbeck-Yau have since been extended by other authors.[51] Uhlenbeck and Yau's article is important in giving a clear reason that stability of the holomorphic vector bundle can be related to the analytic methods used in constructing a hermitian Yang-Mills connection. The essential mechanism is that if an approximating sequence of hermitian connections fails to converge to the required Yang-Mills connection, then they can be rescaled to converge to a subsheaf which can be verified to be destabilizing by Chern-Weil theory.

In the interest of an appropriately general formulation of supersymmetry, Andrew Strominger included the hermitian Yang-Mills condition as part of his Strominger system, a proposal for the extension of the Calabi−Yau condition to non-Kähler manifolds.[52] Ji-Xiang Fu and Yau introduced an ansatz for the solution of Strominger's system on certain three-dimensional complex manifolds, which reduced the problem to a complex Monge−Ampère equation, which they solved.[FY08]

Yau's solution of the Calabi conjecture had given a reasonably complete answer to the question of how Kähler metrics on compact complex manifolds of nonpositive first Chern class can be deformed into Kähler-Einstein metrics.[Y78a] Akito Futaki showed that the existence of holomorphic vector fields can act as an obstruction to the direct extension of these results to the case when the complex manifold has positive first Chern class.[53] A proposal of Calabi's suggested that Kähler-Einstein metrics exist on any compact Kähler manifolds with positive first Chern class which admit no holomorphic vector fields.[Y82b] During the 1980s, Yau and others came to understand that this criterion could not be sufficient. Inspired by the Donaldson−Uhlenbeck−Yau theorem, Yau proposed that the existence of Kähler-Einstein metrics must be linked to stability of the complex manifold in the sense of geometric invariant theory, with the idea of studying holomorphic vector fields along projective embeddings, rather than holomorphic vector fields on the manifold itself.[Y93][Y14] Subsequent research of Gang Tian and Simon Donaldson refined this conjecture, which became known as the "Yau-Tian-Donaldson conjecture" relating Kähler-Einstein metrics and K-stability. The problem was resolved in 2015 by Xiuxiong Chen, Donaldson, and Song Sun, who were awarded the Oswald Veblen prize for their work.[54][55][56]

In 1982, Li and Yau resolved the Willmore conjecture in the non-embedded case.[LY82] More precisely, they established that, given any smooth immersion of a closed surface in the 3-sphere which fails to be an embedding, the Willmore energy is bounded below by 8π. This is complemented by a 2012 result of Fernando Marques and André Neves, which says that in the alternative case of a smooth embedding of the 2-dimensional torus S1 × S1, the Willmore energy is bounded below by 2π2.[57] Together, these results comprise the full Willmore conjecture, as originally formulated by Thomas Willmore in 1965. Although their assumptions and conclusions are quite similar, the methods of Li−Yau and Marques−Neves are distinct. Nonetheless, they both rely on structurally similar minimax schemes. Marques and Neves made novel use of the Almgren–Pitts min-max theory of the area functional from geometric measure theory; Li and Yau's approach depended on their new "conformal invariant", which is a min-max quantity based on the Dirichlet energy. The main work of their article is devoted to relating their conformal invariant to other geometric quantities.

William Meeks and Yau produced some foundational results on minimal surfaces in three-dimensional manifolds, revisiting points left open by older work of Jesse Douglas and Charles Morrey.[MY82][39] Following these foundations, Meeks, Leon Simon, and Yau gave a number of fundamental results on surfaces in three-dimensional Riemannian manifolds which minimize area within their homology class.[MSY82] They were able to give a number of striking applications. For example, they showed that if M is an orientable 3-manifold such that every smooth embedding of a 2-sphere can be extended to a smooth embedding of the unit ball, then the same is true of any covering space of M. Interestingly, Meeks-Simon-Yau's paper and Hamilton's foundational paper on Ricci flow, published in the same year, have a result in common, obtained by very distinct methods: any simply-connected compact 3-dimensional Riemannian manifold with positive Ricci curvature is diffeomorphic to the 3-sphere.

In the geometry of submanifolds, both the extrinsic and intrinsic geometries are significant. These are reflected by the intrinsic Riemannian metric and the second fundamental form. Many geometers have considered the phenomena which arise from restricting these data to some form of constancy. This includes as special cases the problems of minimal surfaces, constant mean curvature, and submanifolds whose metric has constant scalar curvature.

Outside of the setting of submanifold rigidity problems, Yau was able to adapt Jürgen Moser's method of proving Caccioppoli inequalities,[32] thereby proving new rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a smooth subharmonic function cannot be both positive and Lp integrable unless it is constant.[Y76] Similarly, on a complete Kähler manifold, a holomorphic function cannot be Lp integrable unless it is constant.[Y76]

The Minkowski problem of classical differential geometry can be viewed as the problem of prescribing Gaussian curvature. In 1953, Louis Nirenberg resolved the problem for two-dimensional surfaces, making use of recent progress on the Monge-Ampère equation for two-dimensional domains.[62][63] By the 1970s, higher-dimensional understanding of the Monge-Ampère equation was still lacking. In 1976, Shiu-Yuen Cheng and Yau resolved the Minkowski problem in general dimensions via the method of continuity, making use of fully geometric estimates instead of the theory of the Monge-Ampère equation.[CY76b]

As a consequence of their resolution of the Minkowski problem, Cheng and Yau were able to make progress on the understanding of the Monge-Ampère equation.[CY77a] The key observation is that the Legendre transform of a solution of the Monge–Ampère equation has its graph's Gaussian curvature prescribed by a simple formula depending on the "right-hand side" of the Monge–Ampère equation. As a consequence, they were able to prove the general solvability of the Dirichlet problem for the Monge-Ampère equation, which at the time had been an open question except for two-dimensional domains.

Cheng and Yau's papers followed the schematic form of earlier work of Aleksei Pogorelov, although his published works (at the time of Cheng and Yau's work) had been published without significant detail. The approach of Cheng−Yau and Pogorelov is no longer commonly seen in the literature on the Monge–Ampère equation, as other authors, such as Luis Caffarelli, Nirenberg, and Joel Spruck, have made direct progress on fundamental questions, without using the Minkowski problem.

A "Calabi−Yau manifold" refers to a compact Kähler manifold which is Ricci-flat; according to Yau's verification of the Calabi conjecture, such manifolds are known to exist.[Y78a] Mirror symmetry, which is a proposal of physicists dating to the late 80s, postulates that Calabi−Yau manifolds of complex dimension three can be grouped into pairs which share characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula of enumerative geometry which encodes the number of rational curves of any fixed degree in a general quintic hypersurface of four-dimensional complex projective space.[64] Bong Lian, Kefeng Liu, and Yau gave a rigorous proof that this formula holds.[LLY97] Alexander Givental had earlier given a proof of the mirror formulas; according to Lian, Liu, and Yau, the details of his proof were only successfully filled in following their own publication.[65][24]

The approaches of Givental and of Lian−Liu−Yau are formally independent of the conjectural picture of whether three-dimensional Calabi−Yau manifolds can in fact be grouped as physicists claim. With Andrew Strominger and Eric Zaslow, Yau proposed a geometric picture of how this grouping might be systematically understood.[SYZ96] The essential idea is that a Calabi−Yau manifold with complex dimension three should be foliated by "special Lagrangian" tori, which are certain types of three-dimensional minimal submanifolds of the six-dimensional Riemannian manifold underlying the Calabi−Yau structure. Given one three-dimensional Calabi−Yau manifold, one constructs its "mirror" by looking its torus foliation, dualizing each torus, and reconstructing the three-dimensional Calabi−Yau manifold, which will now have a new structure.

The Strominger−Yau−Zaslow (SYZ) proposal, although not stated very precisely, is now understood to be overly optimistic. One must allow for various degenerations and singularities; even so, there is still no single precise form of the SYZ conjecture. Nonetheless, its conceptual picture has been enormously influential in the study of mirror symmetry, and research on its various facets is currently an active field. It can be contrasted with an alternative (and equally influential) proposal by Maxim Kontsevich known as homological mirror symmetry, which deals with purely algebraic structures.[66]

In one of Yau's earliest papers, written with Blaine Lawson, a number of fundamental results were found on the topology of closed Riemannian manifolds with nonpositive curvature.[LY72] Their "flat torus theorem" characterizes the existence of a flat and totally geodesic immersed torus in terms of the algebra of the fundamental group.[67] The "splitting theorem" says that the splitting of the fundamental group as a maximally noncommutative direct product implies the isometric splitting of the manifold itself.[67] Similar results were obtained at the same time by Detlef Gromoll and Joseph Wolf. Their results have been extended to the broader context of isometric group actions on metric spaces of nonpositive curvature.[68]

Jeff Cheeger and Yau studied the heat kernel on a Riemannian manifold. They established the special case of Riemannian metrics for which geodesic spheres have constant mean curvature, which they proved to be characterized by radial symmetry of the heat kernel.[CY81] Specializing to rotationally symmetric metrics, they used the exponential map to transplant the heat kernel to a geodesic ball on a general Riemannian manifold. Under the assumption that the symmetric "model" space under-estimates the Ricci curvature of the manifold itself, they carried out a direct calculation showing that the resulting function is a subsolution of the heat equation. As a consequence, they obtained a lower estimate of the heat kernel on a general Riemannian manifold in terms of lower bounds on its Ricci curvature.[69] In the special case of nonnegative Ricci curvature, Peter Li and Yau were able to use their gradient estimates to amplify and improve the Cheeger−Yau estimate.[LY86]

Given a smooth compact Riemannian manifold, with or without boundary, spectral geometry studies the eigenvalues of the Laplace-Beltrami operator, which in the case that the manifold has a boundary is coupled with a choice of boundary condition, usually Dirichlet or Neumann conditions. Paul Yang and Yau showed that in the case of a closed two-dimensional manifold, the first eigenvalue is bounded above by an explicit formula depending only on the genus and volume of the manifold.[YY80][39] Earlier, Yau had modified Jeff Cheeger's analysis of the Cheeger constant so as to be able to estimate the first eigenvalue from below in terms of geometric data.[Y75a][70]

Hermann Weyl, in the 1910s, showed that in the case of Dirichlet boundary conditions on a smooth and bounded open subset of the plane, the eigenvalues have an asymptotic behavior which is dictated entirely by the area contained in the region. His result is known as Weyl's law. In 1960, George Pólya conjectured that the Weyl law actually gives control of each individual eigenvalue, and not only of their asymptotic distribution.[71] Li and Yau, in 1983, proved a weakened version controlling the average of the first k eigenvalues for arbitrary k.[LY83][71] To date, the non-averaged Polya conjecture remains open.

In 1980, Li and Yau identified a number of new inequalities for eigenvalues (for Dirichlet and Neumann boundary conditions in addition to the boundaryless case), all based on the maximum principle and the pointwise differential Harnack estimates as pioneered five years earlier by Yau and by Cheng−Yau.[LY80][72] The same techniques were utilized in collaboration with Isadore Singer, Bun Wong, and Shing-Toung Yau to establish a gradient estimate for the quotient of the first two eigenfunctions.[S+85] Analogously to Yau's integration of gradient estimates to find Harnack inequalities, they were able to integrate their gradient estimate to obtain control of the "fundamental gap," which is the difference between the first two eigenvalues.

In 1982, Yau identified fourteen problems of interest in spectral geometry, including the above Pólya conjecture.[Y82b] A particular conjecture of long-standing interest, on the control of the size of level sets of eigenfunctions by the value of the corresponding eigenvalue, was resolved by Alexander Logunov and Eugenia Malinnikova, who were awarded the 2017 Clay research award in part for their work.[73]

Xianfeng Gu and Yau considered the numerical computation of conformal maps between two-dimensional manifolds (presented as discretized meshes), and in particular the computation of uniformizing maps as predicted by the uniformization theorem. In the case of genus-zero surfaces, a map is conformal if and only if it is harmonic, and so Gu and Yau are able to compute conformal maps by direct minimization of a discretized Dirichlet energy.[GY02] In the case of higher genus, the uniformizing maps are computed from their gradients, as determined from the Hodge theory of closed and harmonic 1-forms.[GY02] The main work is thus to identify numerically effective discretizations of the classical theory. Their approach is sufficiently flexible to deal with general surfaces with boundary.[GY03]

With Tony Chan, Paul Thompson, and Yalin Wang, Gu and Yau applied their work to the problem of matching two brain surfaces, which is an important issue in medical imaging. In the most-relevant genus-zero case, conformal maps are only well-defined up to the action of the Möbius group. By further optimizing a Dirichlet-type energy which measures the mismatch of brain landmarks such as the central sulcus, one can obtain mappings which are uniquely defined by such neurological features.[G+04]

Yau has received honorary professorships from many Chinese universities, including Hunan Normal University, Peking University, Nankai University, and Tsinghua University. He has honorary degrees from many international universities, including Harvard University, Chinese University of Hong Kong, and University of Waterloo. He is a foreign member of the National Academies of Sciences of China, India, and Russia.

Research articles. Yau is the author of over five hundred articles. The following, among the most cited, are surveyed above: