Communications in Analysis and Geometry

Volume 19 (2011)

Number 1

On existence of the prescribing $k$-curvature problem on manifolds with boundary

Pages: 53 – 77



Yan He (Centre for Mathematical Sciences, Zhejiang University, Hangzhou, China)

Weimin Sheng (Department of Mathematics, Zhejiang University, Hangzhou, China)


In this paper, we study the problem of conformally deforming a metric to aprescribed $k$th order symmetric function of the eigenvalues of theSchouten tensor on compact Riemannian manifolds with totally geodesicboundary. We prove the solvability of the problem and the compactness of the solution setfor the case $k\geq n/2$, provided the conformal class admits a $k$-admissible metric.These results have been proved by Gursky and Viaclovsky,Trudinger and Wang for the manifolds without boundary, and by Jin et al. and S. Chen for the locally conformally flat manifolds with boundary.

Published 20 June 2011