Weak solutions to Hessian type equations on compact Riemannian manifolds

TL;DR

This work proves the existence of weak (and viscosity) solutions to a Hessian-type fully nonlinear elliptic equation on compact Riemannian manifolds without boundary, by formulating $\mathfrak{F}(\chi + \nabla^2 \varphi) = e^{f}$ with structural assumptions on the operator. The authors implement a strategy based on an $L^{\infty}$ bound and a stability estimate, together with Alexandrov–Bakelman–Pucci maximum principle, to construct a limit pair $(\varphi,b)$ solving $\mathfrak{F}(\chi + \nabla^2 \varphi) = e^{b} e^{f}$ and to define weak $C^0$ and viscosity solutions when $e^{f}$ is nonnegative and not identically zero. A key contribution is the quantitative stability control between two potential solutions, enabling a compactness framework for the limit process. Under additional assumptions, they derive interior gradient estimates and Lipschitz bounds, leading to locally Lipschitz viscosity solutions in regions where $e^{f}$ is Lipschitz. Overall, the paper extends weak solution theory for real Hessian-type equations on manifolds and provides a robust toolkit (L∞ bounds, stability, ABP principles, and gradient estimates) for further geometric-analytic applications.

Abstract

In this paper, we shall study the existence of weak solutions to Hessian type equations on compact Riemannian manifolds without boundary.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: Proposition 11 in Szekelyhidi
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3