The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions

TL;DR

This work proves Pogorelov-type interior $C^2$ estimates for the sum Hessian equation $S_k(D^2u)=\sigma_k(D^2u)+\alpha\sigma_{k-1}(D^2u)$ under $(k-1)$-convexity and semi-convexity, and extends the results to parabolic versions with $(k-1)$-convex-monotone solutions. The authors derive two parabolic Pogorelov bounds, one with $(-u)^{1+\delta}\Delta u\le C$ (when gradient dependence is absent) and another with $(-u)^{\gamma_0}\Delta u\le C$ (with semi-convexity and gradient terms), enabling parabolic rigidity results. Utilizing these estimates, they establish rigidity theorems: in the elliptic setting, entire $(k-1)$-convex, semi-convex solutions with quadratic growth must be quadratic polynomials; in the parabolic setting, entire $(k-1)$-convex-monotone solutions with quadratic growth take the form $u(x,t)=-mt+p(x)$ with $p$ quadratic. A Warren-type non-polynomial example demonstrates the necessity of the stated assumptions for rigidity.

Abstract

In this paper, we primarily study the Pogorelov-type $C^2$ estimates for $(k-1)$-convex solutions of the sum Hessian equation under the assumption of semi-convexity, and apply these estimates to obtain a rigidity theorem for global solutions satisfying the corresponding conditions. Furthermore, we investigate the Pogorelov estimates and rigidity theorems for solutions to the parabolic versions under similar conditions. These results extend the works of \cite{He-Sheng-Xiang-Zhang-2022-CCM} and \cite{Liu-Ren-2023-JFA}.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 1.8
• Lemma 2.2: See Ball
• Lemma 2.3: See LRW1Liu-Ren-2023-JFA