Collapsing geometry of hyperk\"ahler 4-manifolds and applications
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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.
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Cited by 2 Pith papers
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On toric self-dual Einstein gravitational instantons
Toric self-dual Einstein instantons with negative cosmological constant satisfying a global conformal Kähler extension condition are precisely the Calderbank-Pedersen-Singer multipole solutions.
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Special Lagrangian submanifolds and circle collapse on K3
Constructs degenerating special Lagrangian two-spheres and tori in collapsing K3 surfaces that lift from affine lines on a three-dimensional base, including connections between Taub-NUT bubbles.
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