On the existence of extremal metrics

Xingwang Xu

Vol. 174 (1996), No. 2, 555–568

DOI: 10.2140/pjm.1996.174.555

Abstract

We study the well known variational problem proposed by Calabi: Minimize the functional ∫ _M s_g² dv_g among all metrics in a given Kahler class. We are able to establish the existence of the extremal when the closed Riemann surface has genus different from zero. We have also given a different proof of the result originally proved by Calabi that: On a closed Riemann surface, the extremal metric has constant scalar curvature on a closed Riemann surface, the extremal metric has constant scalar curvature, which originally is proved by Calabi.

Mathematical Subject Classification 2000

Primary: 58E11

Secondary: 53C55

Milestones

Received: 29 November 1993

Revised: 22 February 1994

Published: 1 June 1996