Perelman's functionals on manifolds with non-isolated conical singularities

TL;DR

This paper extends Perelman's functionals to manifolds with non-isolated conical and more general singularities, analyzing spectral properties, defining functionals, and studying eigenfunction asymptotics near singularities.

Contribution

It introduces a spectral approach to define Perelman's functionals on singular manifolds and analyzes their spectral and asymptotic properties.

Findings

Spectrum of Schrödinger operator is discrete with finite multiplicities.

The $ au$-functional's infimum is finite on these singular manifolds.

Eigenfunctions exhibit specific asymptotic behaviors near singularities.

Abstract

In this article, we define Perelman's functionals on manifolds with non-isolated conical singularities by starting from a spectral point of view for the Perelman's $\lambda$-functional. (Our definition of non-isolated conical singularities includes isolated conical singularities.) We prove that the spectrum of Schr\"odinger operator $-4\Delta + R$ on manifolds with non-isolated conical singularities consists of discrete eigenvalues with finite multiplicities, provided that scalar curvatures of cross sections of cones have a certain lower bound. This enables us to define the $\lambda$-functional on these singular manifolds, and further, to prove that the infimum of $W$-functional is finite, with the help of some weighted Sobolev inequalities. Furthermore, we obtain some asymptotic behavior of eigenfunctions and the minimizer of the $W$-functional near the singularity, and a more refined optimal partial asymptotic expansion for eigenfunctions near isolated conical singularities. We also study the spectrum of $-4\Delta + R$ and Perelman's functionals on manifolds with more general singularities, i.e. the $r^{\alpha}$-horn singularities which serve as prototypes of algebraic singularities.