Shing-Tung Yau (/jaʊ/;Chinese:丘成桐;pinyin:Qiū Chéngtóng; born April 4, 1949) is a Chinese-Americanmathematician. He is the director of the Yau Mathematical Sciences Center atTsinghua University and professor emeritus atHarvard University. Until 2022, Yau was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua.[1][2]
Yau was born inShantou in 1949, moved toBritish Hong Kong at a young age, and then moved to the United States in 1969. He was awarded theFields Medal in 1982, in recognition of his contributions topartial differential equations, theCalabi conjecture, thepositive energy theorem, and theMonge–Ampère equation.[3] Yau is considered one of the major contributors to the development of moderndifferential geometry andgeometric analysis.The impact of Yau's work are also seen in the mathematical and physical fields ofconvex geometry,algebraic geometry,enumerative geometry,mirror symmetry,general relativity, andstring theory, while his work has also touched uponapplied mathematics,engineering, andnumerical analysis.
Yau was born inShantou,Guangdong, Republic of China in 1949 toHakka parents.[YN19] His ancestral hometown isJiaoling County, China.[YN19] His mother, Yeuk Lam Leung, was fromMeixian District, China; his father, Chen Ying Chiu (丘鎭英), was a Republic of China Kuomintang scholar of philosophy, history, literature, and economics.[YN19] He was the fifth of eight children.[4]
During the Communist takeover of mainland China when he was only a few months old, his family moved toBritish Hong Kong where his schooling was (except for English classes) entirely in theCantonese language instead of his parents' nativeHakka Chinese language.[YN19] He was not able to revisit until 1979, at the invitation ofHua Luogeng, when mainland China entered thereform and opening era.[YN19] They lived inYuen Long at first, and then moved toShatin in 1954.[YN19] They had financial troubles from having lost all of their possessions, and his father and second-oldest sister died when he was thirteen.[YN19] Yau began to read and appreciate his father's books, and became more devoted to schoolwork. After graduating fromPui Ching Middle School, he studied mathematics at theChinese University of Hong Kong from 1966 to 1969, without receiving a degree due to graduating early.[YN19] 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 atUniversity of California, Berkeley in the fall of 1969. Over the winter break, he read the first issues of theJournal of Differential Geometry, and was deeply inspired byJohn Milnor's papers ongeometric group theory.[5][YN19] Subsequently, he formulated a generalization ofPreissman's theorem, and developed his ideas further withBlaine Lawson over the next semester.[6] Using this work, he received his Ph.D. the following year, in 1971, under the supervision ofShiing-Shen Chern.[7]
He spent a year as a member of theInstitute for Advanced Study atPrinceton before joiningStony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor atStanford University.[8] In 1976, he took a visiting faculty position atUCLA and married physicist Yu-Yun Kuo, whom he knew from his time as a graduate student at Berkeley.[8] In 1979, he moved back to the Institute for Advanced Study and he became a professor there in 1980.[8] In 1984, he took a chair professorship atUniversity of California, San Diego.[9] In 1987, he moved toHarvard University.[8][10] In April 2022, Yau retired from Harvard, where he was William Caspar Graustein Professor of Mathematics Emeritus.[8] In the same year, he moved toTsinghua University as a professor of mathematics.[8][2]
According to Yau's autobiography, he became "stateless" in 1978 after the British Consulate revoked his Hong Kong residency due to hisUnited States permanent residency status.[11][12] Regarding his status when receiving hisFields 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."[13] Yau remained "stateless" until 1990, when he obtained United States citizenship.[11][14]
Withscience journalist Steve Nadis, Yau has written a non-technical account ofCalabi-Yau manifolds andstring theory,[YN10][15] a history of Harvard's mathematics department,[NY13] a case for the construction of theCircular Electron Positron Collider in China,[NY15][16][17] an autobiography,[YN19][18] and a book on the relation of geometry to physics.[NY24]
Yau has made major contributions to the development of moderndifferential geometry andgeometric analysis. As said byWilliam Thurston in 1981:[19]
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 (withShiu-Yuen Cheng) of theboundary-value problem for theMonge-Ampère equation, thepositive mass theorem in the mathematical analysis ofgeneral relativity (achieved withRichard Schoen), the resolution of theCalabi conjecture, thetopological theory ofminimal surfaces (withWilliam Meeks), the Donaldson-Uhlenbeck-Yau theorem (done withKaren Uhlenbeck), and the Cheng−Yau and Li−Yau gradient estimates forpartial differential equations (found with Shiu-Yuen Cheng andPeter 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 mainland China and Hong Kong.[6] During a sabbatical year atNational Tsinghua University inTaiwan, Yau was asked byCharles Kao to start a mathematical institute at theChinese 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-authorShiu-Yuen Cheng as associate director. In 1995, Yau assistedYongxiang Lu with raising money fromRonnie Chan andGerald Chan'sMorningside Group for the new Morningside Center of Mathematics at theChinese Academy of Sciences. Yau has also been involved with theCenter of Mathematical Sciences atZhejiang University,[20] atTsinghua University,[21] atNational Taiwan University,[22] and inSanya.[23] 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.[24]
Modeled on an earlier physics conference organized byTsung-Dao Lee andChen-Ning Yang, Yau proposed theInternational 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 theJournal of Differential Geometry,[25]Asian Journal of Mathematics,[26] andAdvances in Theoretical and Mathematical Physics.[27] As of 2021, he has advised over seventy Ph.D. students.[7]
In Hong Kong, with the support ofRonnie 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 discussionsWhy Math? Ask Masters! inHangzhou, July 2004, andThe 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 thearXiv claiming to prove theThurston geometrization conjecture and, as a special case, the renownedPoincaré conjecture. Although his work contained many new ideas and results, his proofs lacked detail on a number of technical arguments.[28] 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.[29] A well-knownAugust 2006 article in theNew Yorker written bySylvia Nasar andDavid Gruber about the situation brought some professional disputes involving Yau to public attention.[13][14]
Yau claimed that Nasar and Gruber's article wasdefamatory 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 theNew Yorker article from several mathematicians, including himself and two others quoted in the article.[34]
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 ondifferential geometry and its appearance in other fields of mathematics and science. In addition to his research, Yau has compiled influential sets ofopen 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.[35] Particularly well-known are aconjecture on existence of minimal hypersurfaces and on thespectral geometry of minimal hypersurfaces.
In 1978, by studying thecomplex Monge–Ampère equation, Yau resolved theCalabi conjecture, which had been posed byEugenio Calabi in 1954.[Y78a] As a special case, this showed thatKähler-Einstein metrics exist on anyclosedKähler manifold whosefirst Chern class is nonpositive. Yau's method adapted earlier work of Calabi,Jürgen Moser, andAleksei Pogorelov, developed for quasilinear ellipticpartial differential equations and the realMonge–Ampère equation, to the setting of the complex Monge–Ampère equation.[36][37][38][39]
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 certainsmooth quasi-projective complex varieties. They later extended their work to alloworbifold singularities.[TY91] WithBrian Greene, Alfred Shapere, andCumrun Vafa, Yau introduced anansatz 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 asemi-flat metric, has become important in several analyses of problems in Kähler geometry.[44][45]
The positive energy theorem, obtained by Yau in collaboration with his former doctoral studentRichard Schoen, can be described in physical terms:
In Einstein's theory ofgeneral relativity, the gravitational energy of an isolated physical system is nonnegative.
However, it is a precise theorem ofdifferential geometry andgeometric analysis, in which physical systems are modeled by Riemannian manifolds with nonnegativity of a certain generalizedscalar curvature. As such, Schoen and Yau's approach originated in their study ofRiemannian 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 theGauss–Codazzi equations into the second variation formula for the area of a stableminimal hypersurface of a three-dimensional Riemannian manifold. TheGauss–Bonnet theorem then highly constrains the possible topology of such a surface when the ambient manifold has positive scalar curvature.[SY79a][46][47]
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 andKaren 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.[48][49]
Schoen and Yau then adapted their work to the setting of certain Riemannianasymptotically flat initial data sets ingeneral relativity. They proved that negativity of the mass would allow one to invoke thePlateau problem to construct stable minimal surfaces which aregeodesically complete. A noncompact analogue of their calculation with the Gauss–Bonnet theorem then provides a logical contradiction to the negativity of mass. As such, they were able to prove the positive mass theorem in the special case of their Riemannian initial data sets.[SY79c][50]
Schoen and Yau extended this to the fullLorentzian formulation of the positive mass theorem by studying apartial differential equation proposed by Pong-Soo Jang. They proved that solutions to the Jang equation exist away from theapparent 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 such a solution to the Jang equation, interpreting the latter as a Riemannian initial data set, Schoen and Yau proved the full positive energy theorem.[50] 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 onminimal hypersurfaces of Riemannian manifolds which have positive scalar curvature.[SY79b] Such minimal hypersurfaces, which were constructed by means ofgeometric measure theory byFrederick Almgren andHerbert 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 oflocally conformally flat manifolds.[SY88][36] 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 ofvolume-preserving mean curvature flow of hypersurfaces ofEuclidean space.[51] 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 ofconstant mean curvature.[HY96]
Traditionally, themaximum principle technique is only applied directly oncompact spaces, as maxima are then guaranteed to exist. In 1967,Hideki Omori found a novel maximum principle which applies on noncompactRiemannian manifolds whosesectional curvatures are bounded below. It is trivial thatapproximate 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 onRicci curvature; the result is known as the Omori−Yau maximum principle.[Y75b] Such generality is useful due to the appearance of Ricci curvature in theBochner 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 ofcutoff functions with certain controllable geometry.[CY75][36][52][53][54]
Yau was able to directly apply the Omori−Yau principle to generalize the classicalSchwarz−Pick lemma ofcomplex analysis.Lars Ahlfors, among others, had previously generalized the lemma to the setting ofRiemann surfaces. With his methods, Yau was able to consider the setting of a mapping from a completeKähler manifold (with a lower bound on Ricci curvature) to aHermitian manifold with holomorphic bisectional curvature bounded above by a negative number.[Y78b][40][54]
Cheng and Yau extensively used their variant of the Omori−Yau principle to find Kähler−Einstein metrics on noncompact Kähler manifolds, under anansatz developed byCharles Fefferman. 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 withgeodesic completeness. In particular, Cheng and Yau were able to find complete Kähler-Einstein metrics of negative scalar curvature on any bounded, smooth, andstrictly pseudoconvex subset ofcomplex Euclidean space.[CY80] These can be thought of as complex geometric analogues of the Poincaré ball model ofhyperbolic space.[40][55]
Yau's original application of the Omori−Yau maximum principle was to establish gradient estimates for a number of second-orderelliptic partial differential equations.[Y75b] Given a function on a complete and smooth Riemannian manifold which satisfies various conditions relating theLaplacian 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.[56][52]
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 classicalHarnack 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 andShiu-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.[56][52]
In 1986, Yau andPeter Li made use of the same methods to studyparabolic partial differential equations on Riemannian manifolds.[LY86][52]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 ofRicci flow, where Hamilton proved a matrix differential Harnack inequality for the curvature operator of certain Ricci flows, andGrigori Perelman proved a differential Harnack inequality for the solutions of a backwards heat equation coupled with a Ricci flow.[57][56]
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 ifM is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then the induced Riemannian metric onM is complete.[CY76a] Analogously, they showed that ifM is an affine hypersphere of affine space which is topologically closed, then the induced affine metric onM 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, over anonsingular projective variety of complex dimension two, aholomorphic vector bundle admits ahermitian Yang–Mills connection if and only if the bundle is stable. A result of Yau andKaren Uhlenbeck generalized Donaldson's result to allow a compactKähler 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 onharmonic maps. The results of Donaldson and Uhlenbeck–Yau have since been extended by other authors. 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 byChern–Weil theory.[38][58]
Like the Calabi–Yau theorem, the Donaldson–Uhlenbeck–Yau theorem is of interest in theoretical physics.[42] In the interest of an appropriately general formulation ofsupersymmetry,Andrew Strominger included the hermitian Yang–Mills condition as part of hisStrominger system, a proposal for the extension of the Calabi−Yau condition to non-Kähler manifolds.[41] Ji-Xiang Fu and Yau introduced anansatz for the solution of Strominger's system on certain three-dimensionalcomplex 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.[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.[40] 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 ofgeometric invariant theory, with the idea of studying holomorphic vector fields along projective embeddings, rather than holomorphic vector fields on the manifold itself.[Y93][Y14a] Subsequent research ofGang Tian andSimon Donaldson refined this conjecture, which became known as theYau–Tian–Donaldson conjecture relating Kähler–Einstein metrics andK-stability. In 2019,Xiuxiong Chen, Donaldson, andSong Sun were awarded theOswald Veblen Prize for resolution of the conjecture.[59]
In 1982, Li and Yau resolved theWillmore conjecture in the non-embedded case.[LY82] More precisely, they established that, given any smooth immersion of aclosed surface in the 3-sphere which fails to be an embedding, theWillmore energy is bounded below by 8π. This is complemented by a 2012 result ofFernando Marques andAndré Neves, which says that in the alternative case of a smooth embedding of the 2-dimensional torusS1 ×S1, the Willmore energy is bounded below by 2π2.[60] Together, these results comprise the full Willmore conjecture, as originally formulated byThomas 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 theAlmgren–Pitts min-max theory of the area functional fromgeometric measure theory; Li and Yau's approach depended on their new "conformal invariant", which is a min-max quantity based on theDirichlet 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 ofJesse Douglas andCharles Morrey.[MY82][46] 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 ifM 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 ofM. Interestingly, Meeks-Simon-Yau's paper and Hamilton's foundational paper onRicci 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 ofsubmanifolds, both the extrinsic and intrinsic geometries are significant. These are reflected by the intrinsicRiemannian metric and thesecond 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 ofminimal surfaces,constant mean curvature, and submanifolds whose metric has constantscalar curvature.
Outside of the setting of submanifold rigidity problems, Yau was able to adaptJürgen 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 asubharmonic function cannot be both positive andLp integrable unless it is constant.[Y76][52][65] Similarly, on a completeKähler manifold, aholomorphic function cannot be Lp integrable unless it is constant.[Y76]
TheMinkowski problem of classical differential geometry can be viewed as the problem of prescribingGaussian curvature. In the 1950s,Louis Nirenberg andAleksei Pogorelov resolved the problem for two-dimensional surfaces, making use of recent progress on theMonge–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 themethod of continuity, making use of fully geometric estimates instead of the theory of the Monge–Ampère equation.[CY76b][66]
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 theLegendre 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 theDirichlet problem for the Monge–Ampère equation, which at the time had been a major open question except for two-dimensional domains.[66]
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.[67] 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.[68][66] The approaches of Cheng−Yau and Pogorelov are no longer commonly seen in the literature on the Monge–Ampère equation, as other authors, notablyLuis Caffarelli, Nirenberg, andJoel Spruck, have developed direct techniques which yield more powerful results, and which do not require the auxiliary use of the Minkowski problem.[68]
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 onpartial differential equations. In theparabolic case, affine spheres were completely classified asparaboloids by successive work ofKonrad Jörgens,Eugenio Calabi, and Pogorelov. Theelliptic affine spheres were identified asellipsoids by Calabi. Thehyperbolic 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.[66][69]
ACalabi–Yau manifold is a compact Kähler manifold which is Ricci-flat; as a special case of Yau's verification of the Calabi conjecture, such manifolds are known to exist.[Y78a] Mirror symmetry, which is a proposal developed by theoretical physicists dating from the late 1980s, postulates that Calabi−Yau manifolds of complex dimension three can be grouped into pairs which share certain characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicistsPhilip Candelas,Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula ofenumerative geometry which encodes the number ofrational curves of any fixed degree in a general quintic hypersurface of four-dimensionalcomplex projective space. Bong Lian,Kefeng Liu, and Yau gave a rigorous proof that this formula holds.[LLY97] A year earlier,Alexander Givental had published 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.[30] The proofs of Givental and Lian–Liu–Yau have some overlap but are distinct approaches to the problem, and each have since been given textbook expositions.[70][71]
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. WithAndrew Strominger andEric 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 byspecial Lagrangian tori, which are certain types of three-dimensionalminimal 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 havingdual 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 alternativehomological mirror symmetry proposal byMaxim 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 andcategory theory.[37][44][70][71]
In one of Yau's earliest papers, written withBlaine Lawson, a number of fundamental results were found on the topology ofclosed Riemannian manifolds with nonpositive curvature.[LY72] Theirflat torus theorem characterizes the existence of a flat andtotally geodesic immersed torus in terms of the algebra of thefundamental group. Thesplitting theorem says that the splitting of the fundamental group as a maximally noncommutativedirect product implies the isometric splitting of the manifold itself. Similar results were obtained at the same time byDetlef Gromoll andJoseph Wolf.[72][73] Their results have been extended to the broader context of isometric group actions onmetric spaces of nonpositive curvature.[74]
Jeff Cheeger and Yau studied theheat kernel on a Riemannian manifold. They established the special case of Riemannian metrics for which geodesic spheres haveconstant mean curvature, which they proved to be characterized by radial symmetry of the heat kernel.[CY81] Specializing to rotationally symmetric metrics, they used theexponential 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 theRicci curvature of the manifold itself, they carried out a direct calculation showing that the resulting function is a subsolution of theheat 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.[75][76] 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][52]
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][52] A second proof, using theBishop–Gromov inequality instead of function theory, was later found by Cheeger,Mikhael Gromov, andMichael Taylor.
Given a smooth compact Riemannian manifold, with or without boundary, spectral geometry studies the eigenvalues of theLaplace–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 aclosed two-dimensional manifold, the first eigenvalue is bounded above by an explicit formula depending only on the genus and volume of the manifold.[YY80][46] Earlier, Yau had modifiedJeff Cheeger's analysis of theCheeger constant so as to be able to estimate the first eigenvalue from below in terms of geometric data.[Y75a][77]
In the 1910s,Hermann Weyl 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 asWeyl'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. Li and Yau proved a weakened version of Pólya's conjecture, obtaining control of theaverages of the eigenvalues by the expression in the Weyl law.[LY83][78]
In 1980, Li and Yau identified a number of new inequalities for Laplace–Beltrami eigenvalues, all based on the maximum principle and the differential Harnack estimates as pioneered five years earlier by Yau and Cheng−Yau.[LY80] Their result on lower bounds based on geometric data is particularly well-known,[79][56][52] and was the first of its kind to not require any conditional assumptions.[80] Around the same time, a similar inequality was obtained byisoperimetric methods byMikhael Gromov, although his result is weaker than Li and Yau's.[75] In collaboration withIsadore Singer, Bun Wong, andShing-Toung Yau, Yau used the Li–Yau methodology 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 thefundamental gap, which is the difference between the first two eigenvalues. The work of Singer–Wong–Yau–Yau initiated a series of works by various authors in which new estimates on the fundamental gap were found and improved.[81]
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 ofeigenfunctions by the value of the corresponding eigenvalue, was resolved byAlexander Logunov andEugenia Malinnikova, who were awarded the 2017Clay Research Award in part for their work.[82]
Xianfeng Gu and Yau considered the numerical computation ofconformal maps between two-dimensional manifolds (presented as discretized meshes), and in particular the computation of uniformizing maps as predicted by theuniformization 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 discretizedDirichlet 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][83] WithTony 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 inmedical imaging. In the most-relevant genus-zero case, conformal maps are only well-defined up to the action of theMöbius group. By further optimizing a Dirichlet-type energy which measures the mismatch of brain landmarks such as thecentral sulcus, they obtained mappings which are well-defined by such neurological features.[G+04]
In the field ofgraph theory,Fan Chung and Yau extensively developed analogues of notions and results from Riemannian geometry. These results on differential Harnack inequalities,Sobolev inequalities, andheat kernel analysis, found partly in collaboration withRonald Graham and Alexander Grigor'yan, were later written into textbook form as the last few chapters of her well-known book "Spectral Graph Theory".[84] Later, they introduced aGreen's function as defined for graphs, amounting to apseudo-inverse of thegraph Laplacian.[CY00] Their work is naturally applicable to the study ofhitting times forrandom walks and related topics.[85][86]
In the interest of finding general graph-theoretic contexts for their results, Chung and Yau introduced a notion ofRicci-flatness of a graph.[84] A more flexible notion of Ricci curvature, dealing withMarkov chains onmetric 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 ofErdős–Rényi random graphs.[LLY11] Lin and Yau also considered thecurvature–dimension inequalities introduced earlier byDominique 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.[87]
Yau has received honorary professorships from many Chinese universities, includingHunan Normal University,Peking University,Nankai University, andTsinghua University. He has honorary degrees from many international universities, includingHarvard University,Chinese University of Hong Kong, andUniversity of Waterloo. He is a foreign member of the National Academies of Sciences of China, India, and Russia.
His awards include:
Research articles.Yau is the author of over five hundred articles. The following, among the most cited, are surveyed above:
| LY72. | Lawson, H. Blaine Jr.; Yau, Shing Tung (1972)."Compact manifolds of nonpositive curvature".Journal of Differential Geometry.7 (1–2):211–228.doi:10.4310/jdg/1214430828.MR 0334083.Zbl 0266.53035. |
| Y74. | Yau, Shing Tung (1974). "Submanifolds with constant mean curvature. I".American Journal of Mathematics.96 (2):346–366.doi:10.2307/2373638.JSTOR 2373638.MR 0370443.Zbl 0304.53041. |
| CY75. | Cheng, S. Y.; Yau, S. T. (1975). "Differential equations on Riemannian manifolds and their geometric applications".Communications on Pure and Applied Mathematics.28 (3):333–354.doi:10.1002/cpa.3160280303.MR 0385749.Zbl 0312.53031. |
| SSY75. | Schoen, R.;Simon, L.; Yau, S. T. (1975)."Curvature estimates for minimal hypersurfaces".Acta Mathematica.134 (3–4):275–288.doi:10.1007/BF02392104.MR 0423263.Zbl 0323.53039. |
| Y75a. | Yau, Shing Tung (1975)."Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold".Annales Scientifiques de l'École Normale Supérieure. Quatrième Série.8 (4):487–507.doi:10.24033/asens.1299.MR 0397619.Zbl 0325.53039. |
| Y75b. | Yau, Shing Tung (1975). "Harmonic functions on complete Riemannian manifolds".Communications on Pure and Applied Mathematics.28 (2):201–228.doi:10.1002/cpa.3160280203.MR 0431040.Zbl 0291.31002. |
| CY76a. | Cheng, Shiu Yuen; Yau, Shing Tung (1976). "Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces".Annals of Mathematics. Second Series.104 (3):407–419.doi:10.2307/1970963.JSTOR 1970963.MR 0431061.Zbl 0352.53021. |
| CY76b. | Cheng, Shiu Yuen; Yau, Shing Tung (1976). "On the regularity of the solution of then-dimensional Minkowski problem".Communications on Pure and Applied Mathematics.29 (5):495–516.doi:10.1002/cpa.3160290504.MR 0423267.Zbl 0363.53030. |
| SY76. | Schoen, Richard; Yau, Shing Tung (1976). "Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature".Commentarii Mathematici Helvetici.51 (3):333–341.doi:10.1007/BF02568161.MR 0438388.S2CID 120845708.Zbl 0361.53040. |
| Y76. | Yau, Shing Tung (1976)."Some function-theoretic properties of complete Riemannian manifold and their applications to geometry".Indiana University Mathematics Journal.25 (7):659–670.doi:10.1512/iumj.1976.25.25051.MR 0417452.Zbl 0335.53041. (Erratum: doi:10.1512/iumj.1982.31.31044) |
| CY77a. | Cheng, Shiu Yuen; Yau, Shing Tung (1977). "On the regularity of the Monge–Ampère equationdet(∂2u/∂xi∂xj) = F(x,u)".Communications on Pure and Applied Mathematics.30 (1):41–68.doi:10.1002/cpa.3160300104.MR 0437805.Zbl 0347.35019. |
| CY77b. | Cheng, Shiu Yuen; Yau, Shing Tung (1977)."Hypersurfaces with constant scalar curvature".Mathematische Annalen.225 (3):195–204.doi:10.1007/BF01425237.MR 0431043.S2CID 33626481.Zbl 0349.53041. |
| Y77. |
| Y78a. | Yau, Shing Tung (1978). "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I".Communications on Pure and Applied Mathematics.31 (3):339–411.doi:10.1002/cpa.3160310304.MR 0480350.Zbl 0369.53059. |
| Y78b. | Yau, Shing Tung (1978). "A general Schwarz lemma for Kähler manifolds".American Journal of Mathematics.100 (1):197–203.doi:10.2307/2373880.JSTOR 2373880.MR 0486659.Zbl 0424.53040. |
| SY79a. | Schoen, R.; Yau, Shing Tung (1979). "Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature".Annals of Mathematics. Second Series.110 (1):127–142.doi:10.2307/1971247.JSTOR 1971247.MR 0541332.Zbl 0431.53051. |
| SY79b. | Schoen, R.; Yau, S. T. (1979)."On the structure of manifolds with positive scalar curvature".Manuscripta Mathematica.28 (1–3):159–183.doi:10.1007/BF01647970.MR 0535700.S2CID 121008386.Zbl 0423.53032. |
| SY79c. |
| CY80. | Cheng, Shiu Yuen; Yau, Shing Tung (1980). "On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation".Communications on Pure and Applied Mathematics.33 (4):507–544.doi:10.1002/cpa.3160330404.MR 0575736.Zbl 0506.53031. |
| LY80. | Li, Peter; Yau, Shing Tung (1980). "Estimates of eigenvalues of a compact Riemannian manifold". InOsserman, Robert;Weinstein, Alan (eds.).Geometry of the Laplace Operator. University of Hawaii, Honolulu (March 27–30, 1979). Proceedings of Symposia in Pure Mathematics. Vol. 36. Providence, RI:American Mathematical Society. pp. 205–239.doi:10.1090/pspum/036.ISBN 978-0-8218-1439-0.MR 0573435.Zbl 0441.58014. |
| SY80. | Siu, Yum Tong; Yau, Shing Tung (1980)."Compact Kähler manifolds of positive bisectional curvature".Inventiones Mathematicae.59 (2):189–204.Bibcode:1980InMat..59..189S.doi:10.1007/BF01390043.MR 0577360.S2CID 120664058.Zbl 0442.53056. |
| YY80. | Yang, Paul C.; Yau, Shing Tung (1980)."Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds".Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV.7 (1):55–63.MR 0577325.Zbl 0446.58017. |
| CY81. | Cheeger, Jeff; Yau, Shing-Tung (1981). "A lower bound for the heat kernel".Communications on Pure and Applied Mathematics.34 (4):465–480.doi:10.1002/cpa.3160340404.MR 0615626.Zbl 0481.35003. |
| CLY81. | Cheng, Siu Yuen;Li, Peter; Yau, Shing-Tung (1981). "On the upper estimate of the heat kernel of a complete Riemannian manifold".American Journal of Mathematics.103 (5):1021–1063.doi:10.2307/2374257.JSTOR 2374257.MR 0630777.Zbl 0484.53035. |
| SY81. | Schoen, Richard; Yau, Shing Tung (1981)."Proof of the positive mass theorem. II".Communications in Mathematical Physics.79 (2):231–260.Bibcode:1981CMaPh..79..231S.doi:10.1007/BF01942062.MR 0612249.S2CID 59473203.Zbl 0494.53028. |
| LY82. |
| MSY82. | Meeks, William III;Simon, Leon; Yau, Shing Tung (1982). "Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature".Annals of Mathematics. Second Series.116 (3):621–659.doi:10.2307/2007026.JSTOR 2007026.MR 0678484.Zbl 0521.53007. |
| LY83. | Li, Peter; Yau, Shing Tung (1983)."On the Schrödinger equation and the eigenvalue problem".Communications in Mathematical Physics.88 (3):309–318.Bibcode:1983CMaPh..88..309L.doi:10.1007/BF01213210.MR 0701919.S2CID 120055958.Zbl 0554.35029. |
| SY83. |
| S+85. | Singer, I. M.; Wong, Bun; Yau, Shing-Tung;Yau, Stephen S.-T. (1985)."An estimate of the gap of the first two eigenvalues in the Schrödinger operator".Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV.12 (2):319–333.MR 0829055.Zbl 0603.35070. |
| CY86. | Cheng, Shiu Yuen; Yau, Shing-Tung (1986). "Complete affine hypersurfaces. I. The completeness of affine metrics".Communications on Pure and Applied Mathematics.39 (6):839–866.doi:10.1002/cpa.3160390606.MR 0859275.Zbl 0623.53002. |
| LY86. | Li, Peter; Yau, Shing-Tung (1986)."On the parabolic kernel of the Schrödinger operator".Acta Mathematica.156 (3–4):153–201.doi:10.1007/bf02399203.MR 0834612.Zbl 0611.58045. |
| UY86. | Uhlenbeck, K.; Yau, S.-T. (1986). "On the existence of Hermitian–Yang–Mills connections in stable vector bundles".Communications on Pure and Applied Mathematics.39 (S):257–293.doi:10.1002/cpa.3160390714.MR 0861491.Zbl 0615.58045. (Erratum: doi:10.1002/cpa.3160420505) |
| SY88. |
| G+90. | Greene, Brian R.; Shapere, Alfred;Vafa, Cumrun; Yau, Shing-Tung (1990). "Stringy cosmic strings and noncompact Calabi–Yau manifolds".Nuclear Physics B.337 (1):1–36.Bibcode:1990NuPhB.337....1G.doi:10.1016/0550-3213(90)90248-C.MR 1059826.Zbl 0744.53045. |
| TY90. | Tian, G.; Yau, Shing-Tung (1990)."Complete Kähler manifolds with zero Ricci curvature. I".Journal of the American Mathematical Society.3 (3):579–609.doi:10.1090/S0894-0347-1990-1040196-6.MR 1040196.Zbl 0719.53041. |
| TY91. | Tian, Gang; Yau, Shing-Tung (1991)."Complete Kähler manifolds with zero Ricci curvature. II".Inventiones Mathematicae.106 (1):27–60.Bibcode:1991InMat.106...27T.doi:10.1007/BF01243902.MR 1123371.S2CID 122638262.Zbl 0766.53053. |
| HY96. | Huisken, Gerhard; Yau, Shing-Tung (1996). "Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature".Inventiones Mathematicae.124 (1–3):281–311.Bibcode:1996InMat.124..281H.doi:10.1007/s002220050054.hdl:11858/00-001M-0000-0013-5B63-3.MR 1369419.S2CID 122669931.Zbl 0858.53071. |
| SYZ96. | Strominger, Andrew; Yau, Shing-Tung;Zaslow, Eric (1996). "Mirror symmetry is T-duality".Nuclear Physics B.479 (1–2):243–259.arXiv:hep-th/9606040.Bibcode:1996NuPhB.479..243S.doi:10.1016/0550-3213(96)00434-8.MR 1429831.S2CID 14586676.Zbl 0896.14024. |
| LLY97. | Lian, Bong H.;Liu, Kefeng; Yau, Shing-Tung (1997)."Mirror principle. I".Asian Journal of Mathematics.1 (4):729–763.arXiv:alg-geom/9712011.Bibcode:1997alg.geom.12011L.doi:10.4310/AJM.1997.v1.n4.a5.MR 1621573.Zbl 0953.14026. |
| CY00. | Chung, Fan; Yau, S.-T. (2000)."Discrete Green's functions".Journal of Combinatorial Theory. Series A.91 (1–2):191–214.doi:10.1006/jcta.2000.3094.MR 1779780.Zbl 0963.65120. |
| GY02. | Gu, Xianfeng; Yau, Shing-Tung (2002)."Computing conformal structures of surfaces".Communications in Information and Systems.2 (2):121–145.arXiv:cs/0212043.Bibcode:2002cs.......12043G.doi:10.4310/CIS.2002.v2.n2.a2.MR 1958012.Zbl 1092.14514. |
| GY03. | Gu, Xianfeng; Yau, Shing Tung (2003). "Global conformal surface parameterization". InKobbelt, Leif;Schroeder, Peter; Hoppe, Hugues (eds.).Eurographics Symposium on Geometry Processing (Aachen, Germany, June 23–25, 2003). Goslar, Germany:Eurographics Association. pp. 127–137.doi:10.2312/SGP/SGP03/127-137. |
| G+04. | Gu, Xianfeng; Wang, Yalin;Chan, Tony F.;Thompson, Paul M.; Yau, Shing-Tung (2004). "Genus zero surface conformal mapping and its application to brain surface mapping".IEEE Transactions on Medical Imaging.28 (8):949–958.doi:10.1109/TMI.2004.831226.PMID 15338729. |
| FY08. | Fu, Ji-Xiang; Yau, Shing-Tung (2008)."The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampère equation".Journal of Differential Geometry.78 (3):369–428.arXiv:hep-th/0604063.doi:10.4310/jdg/1207834550.MR 2396248.Zbl 1141.53036. |
| LY10. | Lin, Yong; Yau, Shing-Tung (2010)."Ricci curvature and eigenvalue estimate on locally finite graphs".Mathematical Research Letters.17 (2):343–356.doi:10.4310/MRL.2010.v17.n2.a13.MR 2644381.Zbl 1232.31003. |
| LLY11. | Lin, Yong; Lu, Linyuan; Yau, Shing-Tung (2011)."Ricci curvature of graphs".Tohoku Mathematical Journal. Second Series.63 (4):605–627.doi:10.2748/tmj/1325886283.MR 2872958.Zbl 1237.05204. |
Survey articles and publications of collected works.
| Y82a. | Yau, Shing Tung (1982). "Survey on partial differential equations in differential geometry". In Yau, Shing-Tung (ed.).Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ:Princeton University Press. pp. 3–71.doi:10.1515/9781400881918-002.ISBN 978-1-4008-8191-8.MR 0645729.Zbl 0478.53001. |
| Y82b. | Yau, Shing Tung (1982). "Problem section". In Yau, Shing-Tung (ed.).Seminar on Differential Geometry. Annals of Mathematics Studies. Vol. 102. Princeton, NJ:Princeton University Press. pp. 669–706.doi:10.1515/9781400881918-035.ISBN 978-1-4008-8191-8.MR 0645762.Zbl 0479.53001. |
| Y87. | Yau, Shing-Tung (1987)."Nonlinear analysis in geometry".L'Enseignement Mathématique. Revue Internationale. 2e Série.33 (1–2):109–158.doi:10.5169/seals-87888.MR 0896385.Zbl 0631.53002. |
| Y93. | Yau, Shing-Tung (1993). "Open problems in geometry". InGreene, Robert; Yau, S. T. (eds.).Differential Geometry: Partial Differential Equations on Manifolds. American Mathematical Society Summer Institute on Differential Geometry (University of California, Los Angeles, July 9–27, 1990). Proceedings of Symposia in Pure Mathematics. Vol. 54. Providence, RI:American Mathematical Society. pp. 1–28.doi:10.1090/pspum/054.1.ISBN 978-0-8218-1494-9.MR 1216573.Zbl 0801.53001. |
| Y00. | Yau, S.-T. (2000)."Review of geometry and analysis".Asian Journal of Mathematics.4 (1):235–278.doi:10.4310/AJM.2000.v4.n1.a16.MR 1803723.Zbl 1031.53004. |
| Y06. | Yau, Shing-Tung (2006). "Perspectives on geometric analysis". In Yau, Shing-Tung (ed.).Essays in geometry in memory of S.S. Chern. Surveys in Differential Geometry. Vol. 10. Somerville, MA: International Press. pp. 275–379.arXiv:math/0602363.doi:10.4310/SDG.2005.v10.n1.a8.MR 2408227.Zbl 1138.53004. |
| Y14a. | Ji, Lizhen;Li, Peter;Liu, Kefeng;Schoen, Richard, eds. (2014a).Selected expository works of Shing-Tung Yau with commentary. Vol. I. Advanced Lectures in Mathematics. Vol. 28. Somerville, MA: International Press.ISBN 978-1-57146-293-0.MR 3307244.Zbl 1401.01045. |
| Y14b. | Ji, Lizhen;Li, Peter;Liu, Kefeng;Schoen, Richard, eds. (2014b).Selected expository works of Shing-Tung Yau with commentary. Vol. II. Advanced Lectures in Mathematics. Vol. 29. Somerville, MA: International Press.ISBN 978-1-57146-294-7.MR 3307245.Zbl 1401.01046. |
| Y19a. | Cao, Huai-Dong;Li, Jun;Schoen, Richard, eds. (2019a).Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 1: Metric geometry and minimal submanifolds. Somerville, MA: International Press.ISBN 978-1-57146-376-0.Zbl 1412.01037. |
| Y19b. | Cao, Huai-Dong;Li, Jun;Schoen, Richard, eds. (2019b).Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 2: Metric geometry and harmonic functions. Somerville, MA: International Press.ISBN 978-1-57146-377-7.Zbl 1412.01038. |
| Y19c. | Cao, Huai-Dong;Li, Jun;Schoen, Richard, eds. (2019c).Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 3: Eigenvalues and general relativity. Somerville, MA: International Press.ISBN 978-1-57146-378-4.Zbl 1412.01039. |
| Y19d. | Cao, Huai-Dong;Li, Jun;Schoen, Richard, eds. (2019d).Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 4: Kähler geometry I. Somerville, MA: International Press.ISBN 978-1-57146-379-1.Zbl 1412.01040. |
| Y19e. | Cao, Huai-Dong;Li, Jun;Schoen, Richard, eds. (2019e).Selected works of Shing-Tung Yau. Part I: 1971–1991. Volume 5: Kähler geometry II. Somerville, MA: International Press.ISBN 978-1-57146-380-7.Zbl 1412.01041. |
Textbooks and technical monographs.
| SY94. | Schoen, R.; Yau, S.-T. (1994).Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 1. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang, Jia Qing Zhong and Yi Chao Xu. Translated from the Chinese by Ding and S. Y. Cheng. Preface translated from the Chinese by Kaising Tso. Cambridge, MA: International Press.ISBN 1-57146-012-8.MR 1333601.Zbl 0830.53001. |
| SY97. | Schoen, R.; Yau, S. T. (1997).Lectures on harmonic maps. Conference Proceedings and Lecture Notes in Geometry and Topology. Vol. 2. Cambridge, MA: International Press.ISBN 1-57146-002-0.MR 1474501.Zbl 0886.53004. |
| SY98. | Salaff, Stephen; Yau, Shing-Tung (1998).Ordinary differential equations (Second ed.). Cambridge, MA: International Press.ISBN 1-57146-065-9.MR 1691427.Zbl 1089.34500. |
| GY08. | Gu, Xianfeng David; Yau, Shing-Tung (2008).Computational conformal geometry. Advanced Lectures in Mathematics. Vol. 3. Somerville, MA: International Press.ISBN 978-1-57146-171-1.MR 2439718. |
Popular books.
| YN10. | Yau, Shing-Tung; Nadis, Steve (2010).The shape of inner space. String theory and the geometry of the universe's hidden dimensions. New York:Basic Books.ISBN 978-0-465-02023-2.MR 2722198.Zbl 1235.00025. |
| NY13. | Nadis, Steve; Yau, Shing-Tung (2013).A history in sum. 150 years of mathematics at Harvard (1825–1975). Cambridge, MA:Harvard University Press.ISBN 978-0-674-72500-3.MR 3100544.Zbl 1290.01005. |
| NY15. | Nadis, Steve; Yau, Shing-Tung (2015).From the Great Wall to the great collider: China and the quest to uncover the inner workings of the universe. Somerville, MA: International Press.ISBN 978-1571463104. |
| YN19. | Yau, Shing-Tung; Nadis, Steve (2019).The shape of a life. One mathematician's search for the universe's hidden geometry. New Haven, CT:Yale University Press.ISBN 978-0-300-23590-6.MR 3930611.Zbl 1435.32001. |
| NY24. | Nadis, Steve; Yau, Shing-Tung (2024).The gravity of math: how geometry rules the universe. New York:Basic Books.ISBN 978-1541604292. |
He became a United States citizen in 1990.
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