Over the winter break, he read the first issues of the Journal of Differential Geometry, and was deeply inspired by John Milnor's papers on geometric group theory.

[11][14] With science journalist Steve Nadis, Yau has written a non-technical account of Calabi-Yau manifolds and string theory,[YN10][15] a history of Harvard's mathematics department,[NY13] a case for the construction of the Circular Electron Positron Collider in China,[NY15][16][17] an autobiography,[YN19][18] and a book on the relation of geometry to physics.

As said by William 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 (with Shiu-Yuen Cheng) of the boundary-value problem for the Monge-Ampère equation, the positive mass theorem in the mathematical analysis of general relativity (achieved with Richard Schoen), the resolution of the Calabi conjecture, the topological theory of minimal surfaces (with William Meeks), the Donaldson-Uhlenbeck-Yau theorem (done with Karen Uhlenbeck), and the Cheng−Yau and Li−Yau gradient estimates for partial differential equations (found with Shiu-Yuen Cheng and Peter Li).

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.

After a few years of fundraising efforts, Yau established the multi-disciplinary Institute of Mathematical Sciences in 1993, with his frequent co-author Shiu-Yuen Cheng as associate director.

[24] Modeled on an earlier physics conference organized by Tsung-Dao Lee and Chen-Ning Yang, Yau proposed the International Congress of Chinese Mathematicians, which is now held every three years.

[29] A well-known August 2006 article in the New Yorker written by Sylvia Nasar and David Gruber about the situation brought some professional disputes involving Yau to public attention.

[TY91] With Brian Greene, Alfred Shapere, and Cumrun Vafa, Yau introduced an ansatz for a Kähler metric on the set of regular points of certain surjective holomorphic maps, with Ricci curvature approximately zero.

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.

[SY79c][50] Schoen and Yau extended this to the full Lorentzian formulation of the positive mass theorem by studying a partial differential equation proposed by Pong-Soo Jang.

[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.

[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 an ansatz developed by Charles Fefferman.

In particular, Cheng and Yau were able to find complete Kähler-Einstein metrics of negative scalar curvature on any bounded, smooth, and strictly pseudoconvex subset of complex Euclidean space.

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.

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.

[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.

William Meeks and Yau produced some foundational results on minimal surfaces in three-dimensional manifolds, revisiting points left open by older work of Jesse Douglas and Charles Morrey.

[MY82][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.

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.

[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, 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.

[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 physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes proposed a formula of enumerative geometry which encodes the number of rational curves of any fixed degree in a general quintic hypersurface of four-dimensional complex projective space.

[37][44][70][71] In one of Yau's earliest papers, written with Blaine Lawson, a number of fundamental results were found on the topology of closed Riemannian manifolds with nonpositive curvature.

They established the special case of Riemannian metrics for which geodesic spheres have constant mean curvature, which they proved to be characterized by radial symmetry of the heat kernel.

Under the assumption that the symmetric "model" space under-estimates the Ricci curvature of the manifold itself, they carried out a direct calculation showing that the resulting function is a subsolution of the heat equation.

[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.

[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.

[GY03][83] With Tony Chan, Paul Thompson, and Yalin Wang, Gu and Yau applied their work to the problem of matching two brain surfaces, which is an important issue in medical imaging.

[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.