Collapsing geometry of hyperkähler 4-manifolds and applications

2021/08/30 by Sun, Song, Zhang, Ruobing · 3 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2108.12991

Abstract

We investigate the collapsing geometry of hyperkähler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperkähler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3-torus by an involution, a singular special Kähler metric on the 2-sphere, or the unit interval. (2) Any complete hyperkähler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.

Cited by

• Compactification of metric moduli space of K3 surfaces
• Ricci Flow on ALF manifolds
• On multiscale aspects of Kähler-Einstein metrics and algebraic geometry

Related

• ALF gravitational instantons and collapsing Ricci-flat metrics on the K3\n surface
• Hyper-Kähler instantons, symmetries, and flat spaces
• Hyperkähler manifolds
• H-projectively Equivalent Kahler Manifolds and Gravitational Instantons
• Quotients of gravitational instantons