Regularity of the first eigenvalue of the p-Laplacian and Yamabe invariant along geometric flows

Er-Min Wang and Yu Zheng

Vol. 254 (2011), No. 1, 239–255

DOI: 10.2140/pjm.2011.254.239

Abstract

We first prove that the first eigenvalue of the p-Laplace operator and the Yamabe invariant are both locally Lipschitz along geometric flows under weak assumptions without assumptions on curvature. Secondly, the Yamabe invariant is found to be directionally differentiable along geometric flows. As an application, an open question about the Yamabe metric and Einstein metric is partially answered.

Keywords

Dini derivative, locally Lipschitz, first eigenvalue, p-Laplace operator, Yamabe invariant, geometric flow

Mathematical Subject Classification 2010

Primary: 58C40

Secondary: 53C44

Milestones

Received: 20 September 2010

Revised: 16 May 2011

Accepted: 14 September 2011

Published: 7 February 2012