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Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.

We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite codimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyperplane of a 2-valued function satisfying a local C1, estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3−δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure.

Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperk¨ahler manifold, up to finite group factor.

A well-known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mn in the unit sphere S^(n+1)(1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.