A class of Neumann problems arising in conformal geometry

Weimin Sheng and Li-Xia Yuan

Vol. 270 (2014), No. 1, 211–235

DOI: 10.2140/pjm.2014.270.211

Abstract

In this paper, we solve a class of Neumann problems on a manifold with totally geodesic smooth boundary. As a consequence, we also solve the prescribing $k$-curvature problem of the modified Schouten tensor on such manifolds; that is, if the initial $k$-curvature of the modified Schouten tensor is positive for $\tau >n-1$ or negative for $\tau <1$, then there exists a conformal metric such that its $k$-curvature defined by the modified Schouten tensor equals some prescribed function and the boundary remains totally geodesic.

Keywords

$k$-curvature, modified Schouten tensor, Neumann problem, umbilic boundary

Mathematical Subject Classification 2010

Primary: 53C21

Secondary: 35J65

Milestones

Received: 28 December 2012

Revised: 11 February 2014

Accepted: 25 February 2014

Published: 2 August 2014