Mathematical Research Letters

Volume 21 (2014)

Number 4

Sharp gradient estimate and spectral rigidity for $p$-Laplacian

Pages: 885 – 904



Chiung-Jue Anna Sung (Department of Mathematics, National Tsing Hua University, Hsin-Chu, Taiwan)

Jiaping Wang (School of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.)


We derive a sharp gradient estimate for positive eigenfunctions of the $p$-Laplacian on a complete manifold with Ricci curvature bounded below. As an application, we study the rigidity of manifolds achieving the maximum value of the principal eigenvalue of the $p$-Laplacian.

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