In this article our main goal is to establish an L2 version of the twisted invariance of plurigenera. We equally simplify the original “two towers” approach of Yum-Tong Siu.

We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the L2 metric. The primary motivation for studying this problem comes from Teichm¨uller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichm¨uller space with respect to a class of metrics that generalize the Weil-Petersson metric.

We derive large time upper bounds for heat kernels on vector bundles of differential forms on a class of non-compact Riemannian manifolds under certain curvature conditions.

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In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the radius of the smallest extrinsic ball which contains the domain.