Remarks on Hessian quotient equations on Riemannian manifolds

TL;DR

This paper establishes second order a priori estimates for Hessian quotient equations on two-dimensional Riemannian manifolds, highlighting differences from complex geometry cases.

Contribution

It introduces new test functions and leverages concavity properties to obtain estimates, advancing understanding of Hessian quotient equations in Riemannian geometry.

Findings

Unobstructed second order estimates in 2D Riemannian case

Introduction of new test functions for analysis

Identification of differences with complex case obstructions

Abstract

We consider Hessian quotient equations in Riemannian setting related to a problem posed by Delano\"e and Urbas. We prove unobstructed second order a priori estimate for the real Hessian quotient equation via the maximum principle argument on Riemannian manifolds in dimension two. This is achieved by introducing new test function and exploiting some fine concavity properties of quotient operator. This result demonstrates that there is intriguing difference between the real case and the complex case, as there are known obstructions for $J$-equation in complex geometry.