Affine spheres were introduced by Ţiţeica in [72, 73], and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine R which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics...

On the interior of a regular convex cone K ⊂ R there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K with parameter of order O(n), the universal barrier. This barrier is invariant with respect to the unimodular automorphism su...

S. T. Yau has done extremely deep and powerful work in differential geometry and partial differential equations. His resolution of the Calabi conjecture on the existence of KählerEinstein metrics, by solving a complex Monge-Ampère equation on Kähler manifolds, is of fundamental importance in both mathematics and physics. We would like to recall in this article the contributions of S. Y. Cheng a...

We prove Li–Yau-type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier–Stokes equations.

We prove Berezin–Li–Yau-type lower bounds with additional term for the eigenvalues of the Stokes operator and improve the previously known estimates for the Laplace operator. Generalizations to higher-order operators are given. Dedicated to Professor R.Temam on the occasion of his 70th birthday

We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a...

We derive a canonical asymptotic expansion up to infinite order of the Kähler–Einstein metric on a quasi-projective manifold, which can be compactified by adding a divisor with simple normal crossings. Characterized by the log filtration of the Cheng–Yau Hölder ring, the asymptotics are obtained by constructing an initial Kähler metric, deriving certain iteration formula and applying the isomor...