Shing-Tung Yau

In 1978, by studying the complex Monge–Ampère equation, Yau resolved the Calabi conjecture, which had been posed by Eugenio Calabi in 1954.^[Y78a] This showed that Kähler-Einstein metrics exist on any closed Kähler manifold whose first Chern class is nonpositive. Yau's method relied upon finding appropriate adaptations of 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.[11][12][13]

In differential geometry, Yau's theorem is significant in proving the general existence of closed manifolds of special holonomy; any simply-connected closed Kähler manifold which is Ricci flat must have its holonomy group contained in the special unitary group, according to the Ambrose-Singer theorem. Examples of compact Riemannian manifolds with other special holonomy groups have been found by Dominic Joyce and 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.[14][15]

In algebraic geometry, the existence of canonical metrics as proposed by Calabi allows one to give equally canonical representatives of characteristic classes by differential forms. Due to Yau's initial efforts at disproving the Calabi conjecture by showing that it would lead to contradictions in such contexts, he was able to draw striking corollaries to his primary theorem.^[Y77] In particular, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern numbers of surfaces, as well as homotopical characterizations of the complex structures of the complex projective plane and of quotients of the two-dimensional complex unit ball.

In string theory, it was discovered in 1985 by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten that Calabi-Yau manifolds, due to their special holonomy, are the appropriate configuration spaces for superstrings.[16] For this reason, Yau's existence theorem for Calabi-Yau manifolds is considered to be of fundamental importance in modern string theory.

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. Schoen and Yau's approach is based in their study of Riemannian manifolds of positive scalar curvature, which is considered to be of interest in and of itself.

Schoen and Yau identified 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.

Schoen and Yau exploited this observation by finding novel constructions of stable minimal hypersurfaces with various controlled properties. Some of their existence results were developed simultaneously with renowned results of Jonathan Sacks and Karen Uhlenbeck.[17] Their best-known result is in the setting of certain asymptotically flat initial data sets in general relativity, where they showed that the negativity of the mass would allow one to invoke the Plateau problem to construct stable minimal surfaces whose topology is 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.

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. 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.

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. 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 then eight. 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.

In 1975, Yau partially extended a result of Hideki Omori's which allows the application of the maximum principle on noncompact spaces, where maxima are not guaranteed to exist.[18]^[Y75]

Let (M, g) be a complete and smooth Riemannian manifold whose Ricci curvature is bounded below, and let u be a C² function on M which is bounded above. Then there exists a sequence p_k in M such that

${\displaystyle \lim _{k\to \infty }u(p_{k})=\sup _{M}u,\qquad \lim _{k\to \infty }{\big |}du(p_{k}){\big |}_{g}\to 0,\qquad \limsup _{k\to \infty }\Delta ^{g}u(p_{k})\leq 0.}$

Omori's formulation required the more restrictive assumption that the sectional curvatures of g are bounded below by a constant, although it allowed for a stronger conclusion, in which the Laplacian of u can be replaced by its hessian.

A direct application of the Omori-Yau principle, published in 1978, gives Yau's generalization of the classical Schwarz lemma of complex analysis.^[Y78b]

Cheng and Yau showed that the Ricci curvature assumption in the Omori-Yau maximum principle can be replaced by the assumption of the existence of smooth cutoff functions of certain controllable geometry.^[CY75] Using this as the primary tool to extend some of Yau's work in proving the Calabi conjecture, they were able to construct complex-geometric analogues to the Poincaré ball model of hyperbolic space. In particular, they showed that complete Kähler-Einstein metrics of negative scalar curvature exist on any bounded, smooth, and strictly pseudoconvex subset of a finite-dimensional complex vector space.^[CY80]

In Yau's paper on the Omori-Yau maximum principle, his primary application was to establish gradient estimates for a number of second-order elliptic partial differential equations.^[Y75] Given a complete and smooth Riemannian manifold (M, g) and a function f on M which satisfies a condition relating Δf to f and df, Yau applied the maximum principle to such expressions as

${\displaystyle u={\frac {f+c_{1}}{\sqrt {|df|_{g}^{2}+c_{2}}}}}$

to show that u must be bounded below by a positive constant. Such a conclusion amounts to an upper bound on the size of the gradient of log(f + c₁).

These novel estimates have come to be called "differential Harnack inequalities" since they can be integrated along arbitrary paths in M 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,[19] 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.[20] 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.[21][22]

Interestingly, 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.[23] 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.[24]

The results of Donaldson and Uhlenbeck-Yau have since been extended by other authors.[25] 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.

The Donaldson-Uhlenbeck-Yau theorem, relating the existence of solutions of a geometric partial differential equation with algebro-geometric stability, can be seen as a precursor of the later Yau-Tian-Donaldson conjecture, discussed below.

In 1982, Li and Yau proved the following statement:

Let f : M → S³ be a smooth immersion which is not an embedding. If S³ is given its standard Riemannian metric and M is a closed smooth two-dimensional surface, then

${\displaystyle \int _{M}H^{2}\,d\mu \geq 8\pi }$

where H is the mean curvature of f and dμ is the induced Riemannian volume form on M.

This is complemented by a 2012 result of Fernando Marques and André Neves, which says that in the alternative case that f is a smooth embedding of S¹ × S¹, then the conclusion holds with 8π replaced by 2π².[26] Together, these results comprise the Willmore conjecture, as originally formulated by Thomas Willmore in 1965.

Although their assumptions and conclusions are similar, the methods of Li-Yau and Marques-Neves are distinct. Marques and Neves made novel use of the Almgren–Pitts min-max theory of geometric measure theory. Li and Yau introduced a new "conformal invariant": given a Riemannian manifold (M,g) and a positive integer n, they define

${\displaystyle V_{c}{\big (}(M,g),n{\big )}=\inf _{f:M\to S^{n}}\left\{\sup _{F:S^{n}\to S^{n}}{\Big \{}\int _{M}{\big |}d(F\circ \varphi ){\big |}^{2}\,d\mu _{g}:F{\text{ a conformal diffeomorphism}}{\Big \}}:f{\text{ conformal}}\right\}.}$

The main work of their article is in relating their conformal invariant to other geometric quantities. It is interesting that despite the logical independence of Li-Yau and Marques-Neves' proofs, they both rely on conceptually similar minimax schemes.

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. Following these foundations, Meeks, Simon, and Yau gave a number of fundamental results on surfaces in three-dimensional Riemannian manifolds which minimize area within their homology class. They were able to give a number of striking applications. For example:

If M is an orientable 3-manifold such that every smooth embedded 2-sphere is the boundary of a region diffeomorphic to an open ball in ℝ³, 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: any simply-connected compact 3-dimensional Riemannian manifold with positive Ricci curvature is diffeomorphic to the 3-sphere.

The following is a well-known result:[27][28]

Let u be a real-valued function on ℝⁿ. Suppose that the graph of u has vanishing mean curvature as a hypersurface of ℝⁿ⁺¹. If n is less than nine, then this implies that u is of the form u(x) = a⋅x + b, while this implication does not hold if n is greater than or equal to nine.

The key point of the proof is the non-existence of conical and non-planar stable hypersurfaces of Euclidean spaces of low dimension; this was given a simple proof by Schoen, Leon Simon, and Yau. Given the "threshold" dimension of nine in the above result, it is a somewhat surprising fact, due to Cheng and Yau, that there is no dimensional restriction in the Lorentzian version:

Let u be a real-valued function on ℝⁿ. Suppose that the graph of u is a spacelike hypersurface of the Minkowski space ℝ^n,1 which has vanishing mean curvature. Then u is of the form u(x) = a⋅x + b.

Their proof makes use of the maximum principle techniques which they had previously used to prove differential Harnack estimates. In an article published in 1986, they made use of similar techniques to give a new proof of the classification of complete parabolic or elliptic affine hyperspheres.

By adapting Jürgen Moser's method of proving Caccioppoli inequalities,[29] Yau proved new rigidity results for functions on complete Riemannian manifolds, for instance showing that if u is a smooth and positive function on a complete Riemannian manifold, then ∆u ≥ 0 together with the L^p integrability of u implies that u must be constant. Similarly, on a complete Kähler manifold, every holomorphic complex-valued function which is L^p-integrable must be constant.

Via an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem, Cheng and Yau produced novel rigidity theorems characterizing hypersurfaces of space forms by their intrinsic geometry.

Yau's paper of 1974, according to Robert Osserman's review, contains an "astonishing variety" of results on submanifolds of space forms which have parallel or constant-length mean curvature vector. The principal results are on reduction of codimension.

In 1953, Louis Nirenberg gave the solution to the two-dimensional Minkowski problem of classical differential geometry. In 1976 and 1977, Cheng and Yau gave solutions of the multidimensional Minkowski problem and the boundary-value problem for the Monge–Ampère equation. Their solution of the Monge–Ampère equation made use of the Minkowski problem, via the Legendre transform, the observation being 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. This approach is no longer commonly seen in the literature on the Monge–Ampère equation, which tends to rely on more direct, purely analytical methods. Nonetheless Cheng and Yau's papers were the first published results which gave a complete solution to these results; in schematic form they followed earlier work of Aleksei Pogorelov, although his published works had failed to address some significant technical details.

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. Mirror symmetry, which is a proposal of physicists beginning in the late 80s, postulates that Calabi-Yau manifolds of complex dimension 3 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, given any positive integer d, encodes the number of rational curves of degree d in a general quintic hypersurface of four-dimensional complex projective space.[30] Bong Lian, Kefeng Liu, and Yau gave a rigorous proof that this formula holds. 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.[31][32]

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. 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.[33]

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 two-dimensional manifold without boundary, the first eigenvalue is bounded above by an explicit formula depending only on the genus and volume of the manifold.

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. In 1960, George Pólya conjectured that the Weyl behavior gives control of each individual eigenvalue, and not only of their asymptotic distribution. Li and Yau, in 1983, proved a weakened version controlling the average of the first k eigenvalues for arbitrary k. To date, the non-averaged Polya conjecture remains open.

Li and Yau's 1980 article gave a number of inequalities for eigenvalues (for both standard types of 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.

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 recent progress as of 2014.[34]

In 1982, William Thurston published his renowned geometrization conjecture, asserting that in an arbitrary closed 3-manifold, one could find embedded two-dimensional spheres and tori which disconnect the 3-manifold into pieces which admit uniform "geometric" structures. In the same year, Richard Hamilton published his epochal work on the Ricci flow, using a convergence theorem for a parabolic partial differential equation to prove that certain non-uniform geometric structures on 3-manifolds could be deformed into uniform geometric structures.

Although it is often attributed to Hamilton, he has observed that Yau is responsible for the insight that a precise understanding of the failure of convergence for Hamilton's differential equation could suffice to prove the existence of the relevant spheres and tori in Thurston's conjecture. This insight stimulated Hamilton's further research in the 1990s on singularities of the Ricci flow, and culminated with Grigori Perelman's preprints on the problem in 2002 and 2003. The geometrization conjecture is now commonly recognized as having been resolved through the work of Hamilton and Perelman.

In 1981, the Almgren–Pitts min-max theory in geometric measure theory was used to prove the existence of at least one minimal hypersurface of any closed smooth three-dimensional Riemannian manifold. Yau, in 1982, conjectured that infinitely many such immersed hypersurfaces must always exist. Kei Irie, Fernando Codá Marques, and André Neves solved this problem, for manifolds of dimension three through seven, in the generic case.[35] Antoine Song later released a preprint (not yet published) claiming that Yau's conjecture holds without the genericity assumption in the same dimension range.[36]

Yau's solution of the Calabi conjecture gave an essentially complete answer to the question of how Kähler metrics on complex manifolds of nonpositive first Chern class can be deformed into Kähler-Einstein metrics. Akito Futaki showed that the existence of holomorphic vector fields can act as an obstruction to the extension of these results to the case when the complex manifold has positive first Chern class. A proposal of Calabi's, appearing in Yau's "Problem section", was that Kähler-Einstein metrics exist on any compact Kähler manifolds with positive first Chern class which admit no holomorphic vector fields. During the 1980s, Yau came to believe that this criterion would not be sufficient, and that the existence of Kähler-Einstein metrics in this setting must be linked to stability of the complex manifold in the sense of geometric invariant theory. Yau's understanding of this question was updated in the 1990s publication "Open problems in geometry". Subsequent research of Gang Tian and Simon Donaldson refined this conjecture, which became known as the "Yau-Tian-Donaldson conjecture." The problem was resolved in 2015 due to Xiuxiong Chen, Donaldson, and Song Sun, who were awarded the Oswald Veblen prize for their work.[37][38][39]

In 1980, Yau conjectured that on a smooth closed Riemannian manifold, the size of the zero set of eigenfunctions of the Laplacian would grow at a price rate in accordance with the size of the eigenvalue. Following a number of partial results, the conjecture was resolved in 2018 by Alexander Logunov and Eugenia Malinnikova, who were awarded the Clay research award in part for their work.[40][41][42][43][44]

Yau's other major contributions include the resolution of the Frankel conjecture with Yum-Tong Siu (a more general solution being due to Shigefumi Mori and an extension due to Ngaiming Mok), work with William Meeks on the embeddedness and equivariance of solutions of the Plateau problem (which became a key part of the solution of the Smith conjecture in geometric topology), partial extensions of the Calabi conjecture to noncompact settings with Gang Tian, and a study of the existence of large spheres of constant mean curvature in asymptotically flat Riemannian manifolds with Gerhard Huisken.

Some of Yau's more recent notable contributions include work with Ji-Xiang Fu and Jun Li on the Strominger system, work with Yong Lin on the Ricci curvature of graphs, work with Kefeng Liu and Xiaofeng Sun on the geometry of the moduli space of Riemann surfaces, work with Dario Martelli and James Sparks on Sasaki–Einstein metrics, and work with Mu-Tao Wang on conserved quantities in general relativity.