On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

  title={On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*},
  author={Shing-Tung Yau},
  journal={Communications on Pure and Applied Mathematics},
  • S. Yau
  • Published 1 May 1978
  • Mathematics
  • Communications on Pure and Applied Mathematics
Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation. 

A Liouville Theorem for the Complex Monge‐Ampère Equation on Product Manifolds

Let Y be a closed Calabi‐Yau manifold. Let ω be the Kähler form of a Ricci‐flat Kähler metric on ℂm×Y . We prove that if ω is uniformly bounded above and below by constant multiples of ωℂm+ωY , where

A Degenerate Monge--Ampère equation and the Boundary Classes of Kähler Cones

on an elliptic curve C, the line bundle L dual to the tautological line bundle of the projectivized bundle M = P(E) does not admit any smooth Hermitian metric with nonnegative curvature. In fact,

The calabi-yau equation on almost-kähler four-manifolds

Let (M, \omega) be a compact symplectic 4-manifold with a compatible almost complex structure J. The problem of finding a J-compatible symplectic form with prescribed volume form is an

Complex Hessian Equations on Some Compact Kähler Manifolds

On a compact connected -dimensional Kahler manifold with Kahler form, given a smooth function and an integer , the corresponding complex elliptic th Hessian equation is solved by the continuity method under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative.

Mixed volume forms and a complex equation of Monge-Ampère type on Kähler manifolds of positive curvature

We consider a generalization of the Calabi problem. In the analytic set-up on a Kähler manifold, it leads to a complex Monge-Ampère equation containing the mixed discriminant of the given and unknown


Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive

The Kahler-Ricci flow on Kahler manifolds with 2 traceless bisectional curvature operator

It was proved by H. Chen earlier that the property of the sum of any two eigenvalues of the curvature operator is positive is preserved under the ricci flow in all dimensional. By a recent result of

The Obstruction Theory Of Kähler-Einstein Metrics

Note that the Ricci tensor Ricg corresponding to a Kähler metric g yields a representative of the first Chern class via c1(M) = i 2π Ricg(J ·, ·) ∈ H(M,Z) (with J the complex structure on M). Thus,

A class of the non-degenerate complex quotient equations on compact Kähler manifolds

  • Jundong Zhou
  • Mathematics
    Communications on Pure & Applied Analysis
  • 2021
In this paper, we are concerned with the equations that are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix on compact K \begin{document}$ \ddot{a}



Calabi's conjecture and some new results in algebraic geometry.

  • S. Yau
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1977
A proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold is announced and some new results in algebraic geometry and differential geometry are proved, including that the only Köhler structure on a complex projective space is the standard one.

Characteristic Classes of Hermitian Manifolds

In recent years the works of Stiefel,1 Whitney,2 Pontrjagin,3 Steenrod,4 Feldbau,5 Ehresmann,6 etc. have added considerably to our knowledge of the topology of manifolds with a differentiable

On the regularity of the solution of the n‐dimensional Minkowski problem

where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find

Multiple Integrals in the Calculus of Variations

Semi-classical results.- The spaces Hmp and Hmp0.- Existence theorems.- Differentiability of weak solutions.- Regularity theorems for the solutions of general elliptic systems and boundary value

Geometric Measure Theory

Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple

Métriques riemanniennes et courbure

Nous allons etudier certains changements de metrique sur les varietes Riemanniennes, et examiner dans quelle mesure, on peut modifier les proprietes de la courbure. Un probleme fondamental de la

The space of Kiihler metrics

  • Proc. Internat. Congress Math. Amsterdam,
  • 1954