No abstract uploaded!

In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. %either a totally geodesic disk or a spherical cap. Our result also provides a proof of a conjecture proposed by Sternberg-Zumbrun in {\it J Reine Angew Math 503 (1998), 63--85}. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

We give simple conditions on an ambient manifold that are necessary and sufficient for isoperimetric inequalities to hold.

The weak mean curvature is lower semicontinuous under weak convergence of varifolds, that is, if ¹k ! ¹ weakly as varifolds then k ~H¹ kLp(¹)· lim infk!1 k ~H¹k kLp(¹k). In contrast, if Tk ! T weakly as integral currents, then ¹T may not have a locally bounded ¯rst variation even if k ~H¹Tk kL1(¹k) is bounded. In 1999, Luigi Ambrosio asked the question whether lower semi- continuity of the weak mean curvature is true when T is assumed to be smooth. This was proved in [AmMa03] for p > n = dim T in Rn+1 using results from [Sch04]. Here we prove this in any dimension and codimension down to the desired exponent p = 2. For p = n = 2, this corresponds to the Willmore functional. In a forthcoming joint work [RoSch06], main steps of the pre- sent article are used to prove a modi¯ed conjecture of De Giorgi that the sum of the area and the Willmore functional is the ¡-limit of a di®use Landau-Ginzburg approximation.

We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold$N$of ℂ^{$n$}to complex transversal wedges with edges containing$N$. We show that given any$v$∈ℂ^{$n$}∖($T$_{$p$}$N$+$iT$_{$p$}$N$), there exists a wedge of direction$v$whose edge contains a neighborhood of$p$in$N$, such that any continuous CR function defined locally near$p$extends holomorphically to that wedge.