Prescribing the symmetric function of the eigenvalues of the Schouten tensor

TL;DR

This paper investigates the problem of conformally deforming metrics on compact Riemannian manifolds with boundary to achieve a specified symmetric function of the Schouten tensor's eigenvalues, proving solvability and compactness under certain conditions.

Contribution

It establishes the existence and compactness of solutions for conformal deformation problems involving symmetric functions of the Schouten tensor's eigenvalues on manifolds with boundary.

Findings

Proves solvability of the prescribed eigenvalue problem.

Shows compactness of the solution set under non-negative Ricci tensor.

Provides conditions for conformal deformation on manifolds with boundary.

Abstract

In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.