Some Dirichlet problems arising from conformal geometry

Qi-Rui Li and Weimin Sheng

Vol. 251 (2011), No. 2, 337–359

DOI: 10.2140/pjm.2011.251.337

Abstract

We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ric_g and scalar curvature R_g satisfy

This result generalizes Aviles and McOwen’s in the scalar curvature case.

Keywords

modified Schouten tensor, Dirichlet problem, complete metric, prescribed curvature

Mathematical Subject Classification 2000

Primary: 53C21

Secondary: 53C23

Milestones

Received: 14 May 2010

Revised: 23 February 2011

Accepted: 15 March 2011

Published: 3 June 2011