Prescribing integral curvature equation

TL;DR

This paper introduces new integral-based curvature functions on spheres, establishes existence results for symmetric prescribed curvatures, and explores their properties and applications to conformal geometry and Yamabe problems.

Contribution

It formulates novel integral curvature functions, proves existence of symmetric solutions, and extends the theory to general manifolds and conformal covariance.

Findings

Existence of antipodally symmetric prescribed curvature functions on spheres.

Equivalence of integral and differential curvature functions for certain even orders.

Proposal of a general Yamabe type problem for integral curvature functions.

Abstract

In this paper we formulate new curvature functions on $\mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $\mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $\mathbb{S}^3$ is proved. Curvature function on general compact manifold as well as the conformal covariance property for the corresponding integral operator are also addressed, and a general Yamabe type problem is proposed.