First eigenvalue of the $p$-Laplace operator along the Ricci flow

Details First eigenvalue of the $p$-Laplace operator along the Ricci flow First eigenvalue of the $p$-Laplace operator along the Ricci flow

2009-12-24

arxiv.org/abs/0912.4775v2

Mathematics

Differential Geometry

28 pages, added statements and references, deleted verbose statements and corrected typos

Scientific paper

In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the $p$-Laplace operator along the Ricci flow on closed manifolds. We show that the first $p$-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature assumptions. In particular, for an orientable closed surface, we construct various monotonic quantities and prove that the first $p$-eigenvalue is differentiable almost everywhere along the Ricci flow without any curvature assumption, and therefore derive a $p$-eigenvalue comparison-type theorem when its Euler characteristic is negative.

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