Balanced metrics for extremal K\'ahler metrics and Fano manifolds
- Resource Type
- Working Paper
- Authors
- Hashimoto, Yoshinori
- Source
- Subject
- Mathematics - Differential Geometry
Mathematics - Algebraic Geometry
Mathematics - Complex Variables
53C55 (Primary), 32Q26 (Secondary)
- Language
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.
Comment: 14 pages, to appear in Proceedings of Hayama Symposium 2022 on Complex Analysis in Several Variables