On the Spectrum of weighted Laplacian operator and its application to uniqueness of Kähler Einstein metrics

Abstract

The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of Kähler 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.

we correct one mistake in the previous version: the eigenfunctions of weighted Laplacian operator is not real valued in general. But this condition has not been used essentially in our proof

References in corpus (3)

• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• $\cF$-functional and geodesic stability