Mathematical Research Letters

Volume 27 (2020)

Number 3

Perelman’s $W$-functional on manifolds with conical singularities

Pages: 665 – 685



Xianzhe Dai (Department of Mathematics, University of California at Santa Barbara)

Changliang Wang (Max Planck Institute for Mathematics, Bonn, Germany)


In this paper, we develop the theory of Perelman’s $W$-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of $W$-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvature satisfies certain condition near the singularities. We also obtain an asymptotic order for the minimizer near the singularities.

C. Wang is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and support. X. Dai gratefully acknowledges the partial support from the Simons Foundation.

Received 13 November 2018

Accepted 18 February 2019

Published 20 August 2020