On the Spectrum of weighted Laplacian operator and its application to uniqueness of K\"ahler Einstein metrics

TL;DR

This paper introduces a new proof for the uniqueness of Kähler Einstein metrics on Fano manifolds by analyzing the spectrum of a weighted Laplacian operator, utilizing approximation techniques without requiring additional regularity.

Contribution

It provides a novel proof of the Bando-Mabuchi theorem using weak geodesics and spectral analysis, avoiding the need for higher regularity assumptions.

Findings

Eigenfunctions converge to the first eigenspace of the weighted Laplacian.

The proof does not rely on regularity of geodesics, broadening applicability.

Spectral methods are effectively used in the weak setting.

Abstract

The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of K\"ahler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the weighed Laplacian operator directly. However, we can use smooth geodesics to approximate the weak one, then prove that a sequence of eigenfunctions will converge into the first eigenspace of the weighted Laplacian operator.