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Regularity of solutions to Monge--Ampère equations on compact Hermitian manifolds

Quang-Tuan Dang

Abstract

We study the stability and Hölder continuity of solutions to degenerate complex Monge--Ampère equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern--Ricci flow.

Regularity of solutions to Monge--Ampère equations on compact Hermitian manifolds

Abstract

We study the stability and Hölder continuity of solutions to degenerate complex Monge--Ampère equations associated with a (non-closed) big form on compact Hermitian manifolds. We also show that the solution is globally continuous when the reference form is the pullback of a Hermitian metric. As a consequence, we establish a uniform diameter bound for the twisted Chern--Ricci flow.

Paper Structure

This paper contains 13 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-psh functions and Monge-Ampère measures
  4. Demailly's regularization
  5. Local Hölder continuity of Monge-Ampère potentials
  6. Stability
  7. Hölder continuity
  8. Global Continuity of Monge-Ampère Potentials on Resolutions of Singularities
  9. Continuity of potentials of Ricci flat currents
  10. Continuity of potentials of pluripotential Chern–Ricci flows.
  11. Geometric applications
  12. Modulus of continuity and diameter bound
  13. Estimates along Chern-Ricci flows

Key Result

Theorem A

Let $\mu=fdV_X$ be a measure absolutely continuous with respect to Lebesgue measure with density $0\leq f\in L^p(X,dV_X)$, $p>1$. Let $(\varphi,c)\in \mathop{\mathrm{\rm PSH}}\nolimits(X,\theta)\times (0,+\infty)$ be such that $\sup_X\varphi=0$, with a uniform constant $C_0>0$. Then $\varphi$ is Hölder continuous in $\Omega$.

Theorems & Definitions (33)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Proposition 2.2: BoucksomGuedjLu2025-volume
  • Theorem 3.1
  • Lemma 3.2: BoucksomGuedjLu2025-volume
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 23 more