Balanced metrics for extremal Kähler metrics and Fano manifolds

Abstract

The first three sections of this paper are a survey of the author's work on balanced metrics and stability notions in algebraic geometry. The last section is devoted to proving the well-known result that a geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive, where we present a proof which relies only on the Hopf--Rinow theorem and extends to locally compact complete length metric spaces.

arXiv comment14 pages, to appear in Proceedings of Hayama Symposium 2022 on Complex Analysis in Several Variables

References in corpus (7)

• An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II
• K-semistability of cscK manifolds with transcendental cohomology class
• Scalar curvature and asymptotic Chow stability of projective bundles and blowups
• Quantisation of extremal Kähler metrics
• Relative stability associated to quantised extremal Kähler metrics
• Geometric quantization of coupled Kähler-Einstein metrics
• Anticanonically balanced metrics on Fano manifolds