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Fully non-linear elliptic equations on noncompact complex manifolds

Hanzhang Yin

TL;DR

This work develops a comprehensive framework for fully nonlinear elliptic equations on noncompact complex manifolds, establishing C0–C2 a priori estimates and existence results under bounded geometry and a bounded global Kähler potential. The authors implement ε-regularization and a continuity-type approach to solve F(A)=h(x,u) on complete Kähler/Hermitian manifolds, and then upgrade regularity via Evans–Krylov and bootstrapping. They apply the theory to Monge-Ampère-type equations, obtaining solutions that yield complete Kähler metrics with prescribed volume forms and complete Chern-Einstein metrics on Hermitian manifolds. The methodology also encompasses (n−1) Monge-Ampère equations, unifying several complex Hessian-type problems in noncompact settings and enabling geometric constructions on strictly pseudoconvex domains.

Abstract

In this paper, we obtain the a priori estimates and the existence results for solutions of a general class of fully non-linear equations on noncompact Kähler and Hermitian manifolds. As geometric applications, we can construct Kähler metrics with some prescribed volume forms on strictly pseudoconvex domains; and we can construct complete Chern-Einstein metrics on Hermitian manifolds with bounded geometry.

Fully non-linear elliptic equations on noncompact complex manifolds

TL;DR

This work develops a comprehensive framework for fully nonlinear elliptic equations on noncompact complex manifolds, establishing C0–C2 a priori estimates and existence results under bounded geometry and a bounded global Kähler potential. The authors implement ε-regularization and a continuity-type approach to solve F(A)=h(x,u) on complete Kähler/Hermitian manifolds, and then upgrade regularity via Evans–Krylov and bootstrapping. They apply the theory to Monge-Ampère-type equations, obtaining solutions that yield complete Kähler metrics with prescribed volume forms and complete Chern-Einstein metrics on Hermitian manifolds. The methodology also encompasses (n−1) Monge-Ampère equations, unifying several complex Hessian-type problems in noncompact settings and enabling geometric constructions on strictly pseudoconvex domains.

Abstract

In this paper, we obtain the a priori estimates and the existence results for solutions of a general class of fully non-linear equations on noncompact Kähler and Hermitian manifolds. As geometric applications, we can construct Kähler metrics with some prescribed volume forms on strictly pseudoconvex domains; and we can construct complete Chern-Einstein metrics on Hermitian manifolds with bounded geometry.

Paper Structure

This paper contains 7 sections, 23 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical metrics
  4. $C^{0}$-estimates
  5. $C^{1}$ and $C^{2}$-estimates
  6. The solvability of $F(A)=h$
  7. Monge-Ampère equation

Key Result

Theorem 1.2

Let $\Omega$ be a complete Kähler manifold with a bounded global Kähler potential $\varphi<0$, let $\alpha$ be a Hermitian metric on $\Omega$. Fix a real $(1,1)$-form $\chi$ on $(\Omega,\alpha)$ such that $\lambda[\alpha^{i\bar{p}}\chi_{j\bar{p}}]\in \Gamma$, given $h\in C^1(\Omega \times\mathbb{R}) Suppose that the above equation satisfies conditions (a1)-(a5), and assume that then there is a co

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Proposition 3.4
  • ...and 29 more