Vol. 6, No. 5, 2013

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Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

Ben Andrews and Julie Clutterbuck

Vol. 6 (2013), No. 5, 1013–1024

We derive sharp estimates on the modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.

eigenvalue lower bound, heat equation, modulus of continuity
Mathematical Subject Classification 2010
Primary: 35K05, 35K55, 35P15
Received: 1 April 2012
Accepted: 21 May 2013
Published: 3 November 2013
Ben Andrews
Mathematical Sciences Institute
Australian National University
Building 27
Canberra ACT 0200
Mathematical Sciences Center
Tsinghua University
Julie Clutterbuck
Mathematical Sciences Institute
Australian National University
Canberra ACT 0200