Shing-Tung Yau

Shing-Tung Yau (/jaʊ/; Chinese: 丘成桐; pinyin: Qiū Chéngtóng; born April 4, 1949) 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.^[1][2]

Yau was born in Swatow, Canton, Republic of China, moved to British Hong Kong at a young age, and then moved 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.^[3] Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work are also seen in the mathematical and physical fields of 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.

Biography

Yau was born in

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.^[8] In 1976 he took a visiting faculty position with UCLA and married physicist Yu-Yun Kuo, who he knew from his time as a graduate student at Berkeley. From 1984 to 1987 he worked at University of California, San Diego.^[9] Since 1987, he has been at Harvard University.^[10] In April 2022, Yau announced a forthcoming move from Harvard to Tsinghua University.^[2]

According to Yau's autobiography, he became "

Yau has made major contributions to the development of modern differential geometry and geometric analysis. As said by William Thurston in 1981:^[15]

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

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,

• mirror symmetry. Although it is undisputed that Lian−Liu−Yau's article appeared after Givental's, they claim that his work contained gaps that were only filled in following work in their own publication; Givental claims that his original work was complete. Nasar and Gruber quote an anonymous mathematician as agreeing with Givental.^[25]
• In the 1980s, Yau's colleague
  Science Magazine covered the broader phenomena of such positions in China, with Tian and Yau as central figures.^[27]
• Nasar and Gruber say that, having allegedly not done any notable work since the middle of the 1980s, Yau tried to regain prominence by claiming that
  Xi-Ping Zhu and Yau's former student Huai-Dong Cao had solved the Thurston and Poincaré conjectures, only partially based on some of Perelman's ideas. Nasar and Gruber quoted Yau as agreeing with the acting director of one of Yau's mathematical centers, who at a press conference assigned Cao and Zhu thirty percent of the credit for resolving the conjectures, with Perelman receiving only twenty-five (with the rest going to Richard Hamilton). A few months later, a segment of NPR's All Things Considered covering the situation reviewed an audio recording of the press conference and found no such statement made by either Yau or the acting director.^[28]

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.^[29]

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.

Technical contributions to mathematics

Yau has made a number of major research contributions, centered on

In 1978, by studying the

• In
  Peter Kronheimer, although no proposals for general existence results, analogous to Calabi's conjecture, have been successfully identified in the case of the other groups.^[32]
• In
  Chern numbers of surfaces, in addition to homotopical characterizations of the complex structures of the complex projective plane and of quotients of the two-dimensional complex unit ball.^[31][35]
• A special case of the Calabi conjecture asserts that a Kähler metric of zero Ricci curvature must exist on any Kähler manifold whose first Chern class is zero.^[31] In string theory, it was discovered in 1985 by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten that these Calabi–Yau manifolds, due to their special holonomy, are the appropriate configuration spaces for superstrings. For this reason, Yau's resolution of the Calabi conjecture is considered to be of fundamental importance in modern string theory.^[36][37][38]

The understanding of the Calabi conjecture in the noncompact setting is less definitive.

The positive energy theorem, obtained by Yau in collaboration with his former doctoral student Richard Schoen, can be 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. The Gauss–Bonnet theorem then highly constrains the possible topology of such a surface when the ambient manifold has positive scalar curvature.^{[SY79a][41][42]}

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, using different techniques. Their fundamental result is on the existence of minimal immersions with prescribed topological behavior. As a consequence of their calculation with the Gauss–Bonnet theorem, they were able to conclude that certain topologically distinguished three-dimensional manifolds cannot have any Riemannian metric of nonnegative scalar curvature.^[43][44]

Schoen and Yau then adapted their work to the setting of certain Riemannian

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. 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][35][49]}

Cheng and Yau extensively used their variant of the Omori−Yau principle to find Kähler−Einstein metrics on noncompact Kähler manifolds, under an

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, 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.^[51][47]

In 1986, Yau and Peter Li made use of the same methods to study parabolic partial differential equations on Riemannian manifolds.^[LY86][47] Richard Hamilton generalized their results in certain geometric settings to matrix inequalities. 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.^[52][51]

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.

Donaldson−Uhlenbeck−Yau theorem

In 1985,

Like the Calabi–Yau theorem, the Donaldson–Uhlenbeck–Yau theorem is of interest in theoretical physics.[37] 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.^[36] Ji-Xiang Fu and Yau introduced an ansatz for the solution of Strominger's system on certain three-dimensional complex manifolds, reducing 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.

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.^[55] 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.

• The archetypical example of such questions is
  James Simons, Enrico Bombieri, Ennio De Giorgi, and Enrico Giusti in the 1960s. Their work asserts that a minimal hypersurface which is a graph over Euclidean space must be a plane in low dimensions, with counterexamples in high dimensions.^[56] The key point of the proof of planarity is the non-existence of conical and non-planar stable minimal hypersurfaces of Euclidean spaces of low dimension; this was given a simple proof by Richard Schoen, Leon Simon, and Yau.^[SSY75] Their technique of combining the Simons inequality with the formula for second variation of area has subsequently been used many times in the literature.^[41][57]
• Given the "threshold" dimension phenomena in the standard Bernstein problem, it is a somewhat surprising fact, due to Shiu-Yuen Cheng and Yau, that there is no dimensional restriction in the Lorentzian analogue: any spacelike hypersurface of multidimensional Minkowski space which is a graph over Euclidean space and has zero mean curvature must be a plane.^[CY76a] Their proof makes use of the maximum principle techniques which they had previously used to prove differential Harnack estimates.^[CY75] Later they made use of similar techniques to give a new proof of the classification of complete parabolic or elliptic affine hyperspheres in affine geometry.^[CY86]
• In one of his earliest papers, Yau considered the extension of the
  constant mean curvature condition to higher codimension, where the condition can be interpreted either as the mean curvature being parallel as a section of the normal bundle, or as the constancy of the length of the mean curvature. Under the former interpretation, he fully characterized the case of two-dimensional surfaces in Riemannian space forms, and found partial results under the (weaker) second interpretation.^[Y74] Some of his results were independently found by Bang-Yen Chen.^[58]
• Extending
  space forms which have constant scalar curvature.^[59] The key tool in their analysis was an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem.^[CY77b]

Outside of the setting of submanifold rigidity problems, Yau was able to adapt Jürgen Moser's method of proving Caccioppoli inequalities, thereby proving new rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a subharmonic function cannot be both positive and L^p integrable unless it is constant.^{[Y76][47][60]} Similarly, on a complete Kähler manifold, a holomorphic function cannot be L^p integrable unless it is constant.^[Y76]

Minkowski problem and Monge–Ampère equation

The Minkowski problem of classical differential geometry can be viewed as the problem of prescribing Gaussian curvature. In the 1950s, Louis Nirenberg and Aleksei Pogorelov resolved the problem for two-dimensional surfaces, making use of recent progress on the Monge–Ampère equation for two-dimensional domains. 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][61]

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.

Cheng and Yau's papers followed some ideas presented in 1971 by Pogorelov, although his publicly available works (at the time of Cheng and Yau's work) had lacked some significant detail.[62] Pogorelov also published a more detailed version of his original ideas, and the resolutions of the problems are commonly attributed to both Cheng–Yau and Pogorelov.^[63][61] The approaches of Cheng−Yau and Pogorelov are no longer commonly seen in the literature on the Monge–Ampère equation, as other authors, notably Luis Caffarelli, Nirenberg, and Joel Spruck, have developed direct techniques which yield more powerful results, and which do not require the auxiliary use of the Minkowski problem.^[63]

Affine spheres are naturally described by solutions of certain Monge–Ampère equations, so that their full understanding is significantly more complicated than that of Euclidean spheres, the latter not being based on partial differential equations. In the parabolic case, affine spheres were completely classified as paraboloids by successive work of Konrad Jörgens, Eugenio Calabi, and Pogorelov. The elliptic affine spheres were identified as ellipsoids by Calabi. The hyperbolic affine spheres exhibit more complicated phenomena. Cheng and Yau proved that they are asymptotic to convex cones, and conversely that every (uniformly) convex cone corresponds in such a way to some hyperbolic affine sphere.^[CY86] They were also able to provide new proofs of the previous classifications of Calabi and Jörgens–Calabi–Pogorelov.^[61][64]

Mirror symmetry

A

The works of Givental and of Lian−Liu−Yau confirm a prediction made by the more fundamental mirror symmetry conjecture of how three-dimensional Calabi−Yau manifolds can be paired off. However, their works do not logically depend on the conjecture itself, and so have no immediate bearing on its validity. With Andrew Strominger and Eric Zaslow, Yau proposed a geometric picture of how mirror symmetry might be systematically understood and proved to be true.^[SYZ96] Their 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. Mirror manifolds would then be characterized, in terms of this conjectural structure, by having dual foliations. The Strominger−Yau−Zaslow (SYZ) proposal has been modified and developed in various ways since 1996. The conceptual picture that it provides has had a significant influence in the study of mirror symmetry, and research on its various aspects is currently an active field. It can be contrasted with the alternative homological mirror symmetry proposal by Maxim Kontsevich. The viewpoint of the SYZ conjecture is on geometric phenomena in Calabi–Yau spaces, while Kontsevich's conjecture abstracts the problem to deal with purely algebraic structures and category theory.^{[32][39][65][66]}

Comparison geometry

In one of Yau's earliest papers, written with

A well-known result of Yau's, obtained independently by Calabi, shows that any noncompact Riemannian manifold of nonnegative Ricci curvature must have volume growth of at least a linear rate.[Y76]^[47] A second proof, using the Bishop–Gromov inequality instead of function theory, was later found by Cheeger, Mikhael Gromov, and Michael Taylor.

Spectral geometry

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][41] 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][72]

In the 1910s,

In 1982, Yau identified fourteen problems of interest in spectral geometry, including the above Pólya conjecture.[Y82b] A particular conjecture of Yau's, 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.^[77]

Discrete and computational geometry

Xianfeng Gu and Yau considered the numerical computation of

In the interest of finding general graph-theoretic contexts for their results, Chung and Yau introduced a notion of Ricci-flatness of a graph.^[79] A more flexible notion of Ricci curvature, dealing with Markov chains on metric spaces, was later introduced by Yann Ollivier. Yong Lin, Linyuan Lu, and Yau developed some of the basic theory of Ollivier's definition in the special context of graph theory, considering for instance the Ricci curvature of Erdös–Rényi random graphs.^[LLY11] Lin and Yau also considered the curvature–dimension inequalities introduced earlier by Dominique Bakry and Michel Émery, relating it and Ollivier's curvature to Chung–Yau's notion of Ricci-flatness.^[LY10] They were further able to prove general lower bounds on Bakry–Émery and Ollivier's curvatures in the case of locally finite graphs.^[82]

Honors and awards

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.

His awards include:

• 1975–1976,
  Sloan Fellow
  .
• 1981, Oswald Veblen Prize in Geometry.
• 1981,
  United States National Academy of Sciences.^[83]
• 1982, Fields Medal, for "his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
• 1982, elected to the American Academy of Arts and Sciences
• 1982, Guggenheim Fellowship.
• 1984–1985,
  MacArthur Fellow
  .
• 1991, Humboldt Research Award, Alexander von Humboldt Foundation, Germany.
• 1993, elected to the United States National Academy of Sciences
• 1994, Crafoord Prize.^[84]
• 1997, United States National Medal of Science.
• 2003, China International Scientific and Technological Cooperation Award, for "his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers."
• 2010, Wolf Prize in Mathematics, for "his work in geometric analysis and mathematical physics".^[85]
• 2018, Marcel Grossmann Awards, "for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics."^[86]
• 2023, Shaw Prize in Mathematical Sciences.^[87]

Major publications

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

Survey articles and publications of collected works.

Textbooks and technical monographs.

Popular books.

References