Vol. 254, No. 1, 2011

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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

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.

Dini derivative, locally Lipschitz, first eigenvalue, p-Laplace operator, Yamabe invariant, geometric flow
Mathematical Subject Classification 2010
Primary: 58C40
Secondary: 53C44
Received: 20 September 2010
Revised: 16 May 2011
Accepted: 14 September 2011
Published: 7 February 2012
Er-Min Wang
Foundation Teaching Department
Zhengzhou Huaxin College
Xinzheng High-Tech Development Zone
Zhengzhou, Henan, 451100
Yu Zheng
Department of Mathematics
East China Normal University
Dong Chuan Road 500
Shanghai, 200241