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Complex Monge-Ampère equation in Orlicz space and Diameter Bound

Lei Zhang, Zhenlei Zhang

Abstract

In this paper, we establish diameter bounds for compact Kähler manifolds equipped with Kähler metrics $ω$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Ampère equation in Orlicz spaces, encompassing $L^{\infty}$ and stability estimates. This is achieved by employing Kołodziej's approach \cite{Ko98} and the argument of Guo-Phong-Tong-Wang \cite{GuPhToWa21}, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm \cite{GuPhSoSt24-1}, we derive the uniform (local/global) estimates of the Green's function and its gradient for the associated Kähler metric $ω$.

Complex Monge-Ampère equation in Orlicz space and Diameter Bound

Abstract

In this paper, we establish diameter bounds for compact Kähler manifolds equipped with Kähler metrics , assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Ampère equation in Orlicz spaces, encompassing and stability estimates. This is achieved by employing Kołodziej's approach \cite{Ko98} and the argument of Guo-Phong-Tong-Wang \cite{GuPhToWa21}, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm \cite{GuPhSoSt24-1}, we derive the uniform (local/global) estimates of the Green's function and its gradient for the associated Kähler metric .
Paper Structure (22 sections, 17 theorems, 286 equations)

This paper contains 22 sections, 17 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. A priori estimates for CMA in Orlicz space
  3. Geometric estimates of Kähler metrics
  4. Preliminary results
  5. Orlicz space
  6. Integrability conditions for $N$-function
  7. Growth conditions for $N$-function
  8. Complex Monge-Ampère equation in Orlicz space
  9. Measure domination by capacity
  10. $L^\infty$-estimate for CMA
  11. Stability estimate of psh functions around CMA
  12. Green's function and geometric bound
  13. Refined integral estimate of Green function and diameter bound
  14. Local integral bound of Green function and volume of geodesic balls
  15. Slow growth $N$-functions
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\Phi$ be an $N$-function satisfying (integrability 0). If $F\in L^\Phi$ and its Luxemburg norm then, the solution $\varphi_0$ of CMA (MA: smooth case) admits an a priori $L^\infty$-estimate where

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1: Orlicz space
  • Definition 2.2
  • Definition 3.1: Relative Capacity BEGZ10FGS20
  • Lemma 3.2
  • Lemma 3.3
  • ...and 27 more