which is sharp as indicated in the Euclidean case. However even if M contains a small compact region where the Ricci curvature is not nonnegative, estimate (1.1) becomes very much different from (1.2) when r is large, due to the presence of the √ k term. Whether estimate (1.2) is stable under perturbation has been an open question for some time, in light of the known stability results on weaker...

We construct new complete Einstein metrics on a smoothly bounded strictly pseudoconvex domain of a Stein manifold. The approach that we take here is to deform the KählerEinstein metric constructed by Cheng and Yau, which generalizes a work of Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recasting the problem into the question of vanishing of an L cohomology and...

In this note, we obtain the rigidity of sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative Gaussian curvature, Li-Yau solutions to heat equations and related estimates Dirichlet Green’s Riemannian manifolds Ricci curvature. Moreover, also corresponding stability results.

Twenty years ago Yau, 56], generalized the classical Liouville theorem of complex analysis to open manifolds with nonnegative Ricci curvature. Speciically, he proved that a positive harmonic function on such a manifold must be constant. This theorem of Yau was considerably generalized by Cheng-Yau (see 15]) by means of a gradient estimate which implies the Harnack inequality. As a consequence o...

Twenty years ago Yau generalized the classical Liouville theo rem of complex analysis to open manifolds with nonnegative Ricci curva ture Speci cally he proved that a positive harmonic function on such a manifold must be constant This theorem of Yau was considerably generalized by Cheng Yau see by means of a gradient estimate which implies the Harnack inequality As a consequence of this gradien...

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. establish compactness properties of the set normalized quasi-convex functions and show local global comparison principles for twisted operators. then use Perron method to solve whose RHS involves an arbitrary probability measure, generalizing works Cheng-Yau, Delanoe, Caffarelli-Viaclovsky Hultgren-Onnheim. The intrinsi...

According to the conjecture of Calabi, on a complex manifold X with ample canonical bundle KX , there should exist a Kähler-Einstein metric g. Namely, a metric satisfying Ricg = −ωg, where ωg is the Kähler form of the Kähler metric g. The existence of such metric when X is compact was proved by Aubin and Yau ([23]) using complex Monge-Ampère equation. This important result has many applications...

For each invariant polynomial ?, we construct a global CR via the renormalized characteristic form of Cheng–Yau metric on strictly pseudoconvex domain. When degree ? is 0, agrees with total Q?-curvature. equal to dimension, primed pseudo-hermitian I?? which integrates corresponding invariant. These are generalizations I?-curvature five-manifolds, introduced by Case–Gover.

Departments of Pediatrics and Pediatric Surgery, National Taiwan University Hospital, National Taiwan University Medical College, Taipei; and Department of Pediatrics, Cardinal Tien Hospital, Taipei Hsien. Received: 12 July 2000. Revised: 21 August 2000. Accepted: 21 December 2000. Reprint requests and correspondence to: Dr. Kuo-Inn Tsou Yau, Department of Pediatrics, Cardinal Tien Hospital, 36...