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Viscosity solution to complex Hessian equations on compact Hermitian manifolds

Jingrui Cheng, Yulun Xu

TL;DR

This work proves the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold under a determinant domination condition, and establishes uniqueness when the right-hand side is strictly monotone in the unknown. It combines continuity-path methods, a priori $C^{2,\alpha}$-type estimates via a Székelyhidi framework, and GPT-style $L^\infty$ stability to obtain viscosity solutions. A key contribution is the introduction of sup/inf convolution techniques on manifolds to handle uniqueness, along with a detailed treatment of the case when the RHS is independent of the unknown, which reduces to monotonicity properties of the solvability constant $c(G)$. The results extend the viscosity-solutions theory for complex Hessian equations beyond domains to compact Hermitian manifolds and provide groundwork for further analysis of degenerate RHS and subsolution frameworks.

Abstract

We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.

Viscosity solution to complex Hessian equations on compact Hermitian manifolds

TL;DR

This work proves the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold under a determinant domination condition, and establishes uniqueness when the right-hand side is strictly monotone in the unknown. It combines continuity-path methods, a priori -type estimates via a Székelyhidi framework, and GPT-style stability to obtain viscosity solutions. A key contribution is the introduction of sup/inf convolution techniques on manifolds to handle uniqueness, along with a detailed treatment of the case when the RHS is independent of the unknown, which reduces to monotonicity properties of the solvability constant . The results extend the viscosity-solutions theory for complex Hessian equations beyond domains to compact Hermitian manifolds and provide groundwork for further analysis of degenerate RHS and subsolution frameworks.

Abstract

We prove the existence of viscosity solutions to complex Hessian equations on a compact Hermitian manifold that satisfy a determinant domination condition. This viscosity solution is shown to be unique when the right hand is strictly monotone increasing in terms of the solution. When the right hand side does not depend on the solution, we reduces it to the strict monotonicity of the solvability constant.
Paper Structure (11 sections, 50 theorems, 225 equations)

This paper contains 11 sections, 50 theorems, 225 equations.

Table of Contents

  1. introduction
  2. Notations and preliminaries
  3. existence of solution with smooth right hand side
  4. Openness of the coutinuity path
  5. Closedness of the continuity path
  6. stability estimate and existence of viscosity solutions
  7. uniqueness issues
  8. sup/inf convolution of the viscosity solution
  9. when the right hand side has strict monotonicity
  10. when the right hand side does not depend on $\varphi$
  11. Acknowlegement

Key Result

Proposition 1.2

Let $p(x)$ be a homogeneous polynomial of degree $N$ on $\mathbb{R}^n$. Denote $e=(1,\cdots,1)\in \mathbb{R}^n$. Assume that: Then $p(x)^{\frac{1}{N}}\ge c(x_1\cdots x_n)^{\frac{1}{n}}$ for some $c>0$ on $\{x_1>0,\cdots,x_n>0\}$.

Theorems & Definitions (94)

  • Proposition 1.2
  • Theorem 1.1
  • Remark 1.3
  • Theorem 1.2
  • Proposition 1.4
  • Proposition 1.6
  • Proposition 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 84 more