A compactness theorem for hyperkaehler 4-manifolds with boundary

TL;DR

This paper establishes a compactness theorem for hyperkaehler 4-manifolds with boundary, showing smooth convergence under specific topological and boundary curvature conditions, and extends the result to torsion-free hypersymplectic triples.

Contribution

It provides a new compactness criterion for hyperkaehler 4-manifolds with boundary, linking boundary convergence to interior convergence, and generalizes to hypersymplectic triples.

Findings

Hyperkaehler triples converge smoothly if boundary restrictions do.

Positive mean curvature on boundary is crucial.

Results extend to torsion-free hypersymplectic triples.

Abstract

In this paper, we study the compactness of a boundary value problem for hyperkaehler 4-manifolds. We show that under certain topological conditions and the positive mean curvature condition on the boundary, a sequence of hyperkaehler triples converges smoothly up to diffeomorphisms if and only if their restrictions to the boundary converge smoothly up to diffeomorphisms. We also generalize this result to torsion-free hypersymplectic triples.