Collapsing geometry of hyperk\"ahler 4-manifolds and applications
Differential Geometry
2023-01-02 v2 Mathematical Physics
Analysis of PDEs
math.MP
Abstract
We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler 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 K\"ahler metric on the 2-sphere, or the unit interval. (2) Any complete hyperk\"ahler 4-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.
Keywords
Cite
@article{arxiv.2108.12991,
title = {Collapsing geometry of hyperk\"ahler 4-manifolds and applications},
author = {Song Sun and Ruobing Zhang},
journal= {arXiv preprint arXiv:2108.12991},
year = {2023}
}
Comments
A conjecture on the L2-curvature energy was added in Section 7.3. To appear in Acta Mathematica