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Modulus of continuity for solutions to complex Monge-Ampère equations on Hermitian manifolds

Junbang Liu

TL;DR

This work extends the uniform log‑continuity of solutions to complex Monge–Ampère equations from the Kähler setting to compact Hermitian manifolds. It proves that for any $p>n$ and $0<α<\frac{p}{n}-1$, solutions satisfy $|u(x)-u(y)|≤C/|\log dist_{ω_M}(x,y)|^α$, with constants depending on geometric data and integral bounds on $F$. The approach adapts the Guo–Phong–Tong PDE framework, constructing a local comparison metric, establishing a robust stability estimate, and employing holomorphic mollification to transfer pointwise control into a global modulus of continuity; it also extends to bounded solutions via an Orlicz‑norm stability theory. The results yield a diameter bound and provide a unified method for extending log‑continuity to broader classes of complex Hessian equations on Hermitian manifolds, with potential implications for regularity theory in non‑Kähler settings.

Abstract

In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Ampère equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the Kähler case.

Modulus of continuity for solutions to complex Monge-Ampère equations on Hermitian manifolds

TL;DR

This work extends the uniform log‑continuity of solutions to complex Monge–Ampère equations from the Kähler setting to compact Hermitian manifolds. It proves that for any and , solutions satisfy , with constants depending on geometric data and integral bounds on . The approach adapts the Guo–Phong–Tong PDE framework, constructing a local comparison metric, establishing a robust stability estimate, and employing holomorphic mollification to transfer pointwise control into a global modulus of continuity; it also extends to bounded solutions via an Orlicz‑norm stability theory. The results yield a diameter bound and provide a unified method for extending log‑continuity to broader classes of complex Hessian equations on Hermitian manifolds, with potential implications for regularity theory in non‑Kähler settings.

Abstract

In this note, we give a proof of the uniform log-continuity of the solution to complex Monge-Ampère equations on compact Hermitian manifolds, which is a generalization of the result of Guo-Phong-Tong-Wang in the Kähler case.
Paper Structure (4 sections, 14 theorems, 66 equations)

This paper contains 4 sections, 14 theorems, 66 equations.

Table of Contents

  1. introduction
  2. stability estimate
  3. log-continuity
  4. Extending to bounded solutions

Key Result

Theorem 1.1

Given $p>n$, for any $0<\alpha<\frac{p}{n}-1$, there exists a constant $C$ depending on $M,\omega_M,p,\alpha$, and upper bound of $\int_M(|F|^p+1)e^F\omega_M^n$, positive lower bound of $\int_Me^{\frac{1}{n}F}\omega_M^n$, such that solutions to maequation satisfies

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 16 more