Collapsing geometry of hyperkähler 4-manifolds and applications
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.
arXiv commentA conjecture on the L2-curvature energy was added in Section 7.3. To appear in Acta Mathematica
References in corpus (9)
- Metric measure spaces with Riemannian Ricci curvature bounded from below
- Gravitational Instantons of Type D_k
- Hyperkahler Metrics from Periodic Monopoles
- Gravitational instantons with faster than quadratic curvature decay (II)
- Gravitational instantons with faster than quadratic curvature decay (III)
- Complex structure degenerations and collapsing of Calabi-Yau metrics
- PL density invariant for type II degenerating K3 surfaces, Moduli compactification and hyperKahler metrics
- Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing
- Gravitational instantons with quadratic volume growth