In [W], Witten considered the indices of elliptic operators on the free loop space CM of a spin manifold M. In particular the index of the formal signature operator on the loop space is exactly the elliptic genus of Landweber-Stong-Ochanine [LS],[O]. Motivated by physics, Witten made the conjectures about the rigidity of these elliptic operators which say that their 51-equivariant indices on M are independent of g G Sl. See [L] for the early history of the subject.

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.

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In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke– Santal´o type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.