We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in $\mathbb{C}^{n}, n \geq3$ , is not subelliptic.

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NRODUCTION LET y be a rectifiable Jordan curve in three-dimensional euclidean space. Answering an old question, whether y can bound a surface with minimal area, Douglas [l I] and Rad6 [45](independently) found a minimal surface spanning y which is parametrized by the disk. This minimal surface has minimal area among all Lipschitz maps from the disk into R3 which span y. The question whether this solution has branch points or not was finally settled by Osserman [42], who proved that there are no interior true branch points, and by Gulliver [lS], who proved that there are no interior false branch points.

The purpose of this paper is to classify the Möbius homogeneous hypersurfaces with two distinct principal curvatures in$S$^{$n$+1}under the Möbius transformation group. Additionally, we give a classification of the Möbius homogeneous hypersurfaces in$S$^{4}.

We study the heat kernel of a regular symmetric Dirichlet form on a metric space with doubling measure, in particular, a connection between the properties of the jump measure and the long time behaviour of the heat kernel. Under appropriate optimal hypotheses, we obtain the Hölder regularity and lower estimates of the heat kernel.