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Fully non-linear elliptic equations on complex manifolds

Rirong Yuan

TL;DR

This work studies fully nonlinear elliptic equations of Hessian type on Hermitian manifolds, formulating $F(\chi+\sqrt{-1}\partial\overline{\partial}u)=\psi$ with $F=f(\lambda[\omega^{-1}(\chi+\sqrt{-1}\partial\overline{\partial}u)])$ on a Gårding-type cone $\Gamma$ and under ellipticity, concavity, nondegeneracy, plus the $\mathcal{C}$-subsolution condition. The authors prove global $C^{2,\alpha}$-estimates for closed manifolds via a refined blow-up argument and Evans–Krylov theory, and develop a quantitative boundary framework enabling sharp $C^{2}$-type boundary estimates to solve the Dirichlet problem, including degenerate right-hand sides, on manifolds with boundary and on product spaces. They introduce a set of technical tools—a refined subsolution lemma, a monotonicity result, blow-up and Liouville-type analysis, and a new quantitative eigenvalue lemma (Yuan’s lemma)—to control interior and boundary behavior under optimal cone conditions. The results extend solvability and regularity for general Hessian-type equations to curved, non-Kähler Hermitian manifolds, provide construction of subsolutions on products, and offer a robust framework for degenerate and boundary-influenced problems with potential geometric applications.

Abstract

In this paper, we study a broad class of fully nonlinear elliptic equations on Hermitian manifolds. On one hand, under the optimal structural assumptions we derive $C^{2,α}$-estimate for solutions of the equations on closed Hermitian manifolds. On the other hand, we treat the Dirichlet problem. In both cases, we prove the existence theorems with unbounded condition.

Fully non-linear elliptic equations on complex manifolds

TL;DR

This work studies fully nonlinear elliptic equations of Hessian type on Hermitian manifolds, formulating with on a Gårding-type cone and under ellipticity, concavity, nondegeneracy, plus the -subsolution condition. The authors prove global -estimates for closed manifolds via a refined blow-up argument and Evans–Krylov theory, and develop a quantitative boundary framework enabling sharp -type boundary estimates to solve the Dirichlet problem, including degenerate right-hand sides, on manifolds with boundary and on product spaces. They introduce a set of technical tools—a refined subsolution lemma, a monotonicity result, blow-up and Liouville-type analysis, and a new quantitative eigenvalue lemma (Yuan’s lemma)—to control interior and boundary behavior under optimal cone conditions. The results extend solvability and regularity for general Hessian-type equations to curved, non-Kähler Hermitian manifolds, provide construction of subsolutions on products, and offer a robust framework for degenerate and boundary-influenced problems with potential geometric applications.

Abstract

In this paper, we study a broad class of fully nonlinear elliptic equations on Hermitian manifolds. On one hand, under the optimal structural assumptions we derive -estimate for solutions of the equations on closed Hermitian manifolds. On the other hand, we treat the Dirichlet problem. In both cases, we prove the existence theorems with unbounded condition.
Paper Structure (27 sections, 39 theorems, 303 equations)

This paper contains 27 sections, 39 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Equations on closed manifolds
  3. The Dirichlet problem
  4. Notation
  5. Key lemmas
  6. A lemma for subsolution
  7. Monotonicity
  8. Lemmas for blow-up argument
  9. A quantitative lemma
  10. Second estimate and proof of Theorem \ref{['thm2-second-order']}
  11. Blow-up argument and completion of proof of Theorem \ref{['thm1-main']}
  12. Fully nonlinear elliptic equations with type 2 cone
  13. Quantitative boundary estimate: Set-up
  14. $C^0$-estimate and boundary gradient estimate
  15. A key lemma
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $(M, \omega)$ be a closed Hermitian manifold. In addition to elliptic, concave and nondegenerate, we assume $\psi\in C^\infty(M)$ and that equ1-main admits a $\mathcal{C}$-subsolution $\underline{u}\in C^2(M)$. Let $u\in C^4(M)$ be an admissible solution to equ1-main with $\sup_M u=0$, then for where $C$ is a positive constant depending on the data $(M, \omega)$, $\alpha$, $f$, $\psi$, $\chi$

Theorems & Definitions (72)

  • Definition 1.1: $\mathcal{C}$-subsolution
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 3.1
  • proof
  • ...and 62 more