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Second order estimates for equations with sums of Hessian operators on Hermitian manifolds

Weisong Dong, Ruijia Zhang

Abstract

In this paper, we establish an a priori second-order estimate for admissible solutions satisfying a dynamic plurisubharmonic condition to equations involving sums of Hessian operators on compact Hermitian manifolds. The estimate is derived using a concavity inequality for complex sum-of-Hessian operators.

Second order estimates for equations with sums of Hessian operators on Hermitian manifolds

Abstract

In this paper, we establish an a priori second-order estimate for admissible solutions satisfying a dynamic plurisubharmonic condition to equations involving sums of Hessian operators on compact Hermitian manifolds. The estimate is derived using a concavity inequality for complex sum-of-Hessian operators.
Paper Structure (4 sections, 7 theorems, 159 equations)

This paper contains 4 sections, 7 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Concavity Lemma
  4. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $(M,\omega)$ be a compact Hermitian manifold of complex dimension $n \geq 2$, and let $u \in C^{\infty}(M)$. Assume that eqn satisfies Hypothesis (RR) and that for some constant $\varepsilon > 0$. Assume that the smooth real $(1,1)$-form $g$, defined in g, satisfies Condition con1 (or Condition con2) and g-cone for some sufficiently small $\delta > 0$. Then we obtain the following uniform sec

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 3 more