$C^{1,1}$ regularity of geodesics in the space of volume forms

TL;DR

This paper establishes a $C^{1,1}$ regularity estimate for solutions of certain nonlinear equations and applies it to prove the same regularity for geodesics in the space of volume forms, advancing understanding of geometric analysis.

Contribution

The paper introduces a $C^{1,1}$ estimate for fully nonlinear equations and applies it to demonstrate regularity of geodesics in the space of volume forms, a novel result in geometric analysis.

Findings

Proved $C^{1,1}$ regularity for solutions of Chen-He equations.

Established $C^{1,1}$ regularity of geodesics in the space of volume forms.

Enhanced understanding of geometric structures in volume form spaces.

Abstract

We prove a $C^{1,1}$ estimate for solutions of a class of fully nonlinear equations introduced by Chen-He. As an application, we prove the $C^{1,1}$ regularity of geodesics in the space of volume forms.