Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

Ruijia Zhang

TL;DR

This paper develops a semi-convexity–based concavity framework for the $k$-Hessian operator, enabling domain-free interior $C^2$ estimates, a Liouville-type rigidity for the equation $\sigma_k(D^2u)=1$, and streamlined proofs of Guan–Ren–Wang curvature bounds. The approach hinges on a concavity inequality for $q_k=\sigma_k/\sigma_{k-1}$ under the semi-convexity condition and leverages these estimates to address interior, global, and Liouville problems for $k$-Hessian equations. Key contributions include a generalization of Pogorelov-type interior estimates with independence from the domain, a rigidity result for entire semi-convex solutions with quadratic growth, and a concise, self-contained derivation of global curvature estimates for convex hypersurfaces. These results advance rigidity and curvature theory for fully nonlinear Hessian equations with applications in geometric analysis.

Abstract

We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations.

$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

TL;DR

This paper develops a semi-convexity–based concavity framework for the -Hessian operator, enabling domain-free interior estimates, a Liouville-type rigidity for the equation , and streamlined proofs of Guan–Ren–Wang curvature bounds. The approach hinges on a concavity inequality for under the semi-convexity condition and leverages these estimates to address interior, global, and Liouville problems for -Hessian equations. Key contributions include a generalization of Pogorelov-type interior estimates with independence from the domain, a rigidity result for entire semi-convex solutions with quadratic growth, and a concise, self-contained derivation of global curvature estimates for convex hypersurfaces. These results advance rigidity and curvature theory for fully nonlinear Hessian equations with applications in geometric analysis.

Abstract

We derive a concavity inequality for -Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the -Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for -curvature equations.
Paper Structure (7 sections, 18 theorems, 195 equations)

This paper contains 7 sections, 18 theorems, 195 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Symmetric function
  4. A concavity inequality
  5. Interior estimates of k-Hessian equations with Dirichlet boundary conditions
  6. A Liouville problem for general k-Hessian equation
  7. Global curvature estimate for convex hypersurfaces

Key Result

Theorem 1.1

Assume that $f\in C^2(\Omega\times R\times R^n)$ satisfies $f>f_0>0$ for a constant $f_0>0$ in $\overline{\Omega}$. Then, for any semi-convex $k$-admissible solution of the Dirichlet problem of the equation eq1 provided $2<k\leqslant n-1$, we have where $\alpha>0$ and $C>0$ depend on $n$, $k$, $f_0$, $\|f\|_{C^{1,1}}$, $\|u\|_{C^1}$ but are independent of $\Omega$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.1
  • Lemma 1.2: Lu2
  • Theorem 1.4: Guan-Ren-Wang, GRW
  • Theorem 1.5: Guan-Ren-Wang, GRW
  • Lemma 2.1
  • proof
  • ...and 22 more