C^{1,alpha}theory for the prescribed mean curvature equation with Dirichlet data

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a $C^{1,\alpha}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C^{1,\alpha}

CMC hypersurfaces with bounded Morse index

We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids. We also provide bounds on the area of separating CMC surfaces of bounded (Morse) index and use this, together with the previous results, to bound their genus.Comment: Fixed a tex-compiling issu

C1,αC^{1,\alpha } C 1 , α -regularity for surfaces with H∈LpH\in L^p H ∈ L p

In this paper, we prove several results on the geometry of surfaces immersed in with small or bounded norm of . For instance, we prove that if the norm of and the norm of , , are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded norm of , not necessarily small, then such a disk is graphical away from its boundary, provided that the norm of is sufficiently small, . These results are related to previous work of Schoen-Simon (Surfaces with quasiconformal Gauss map. Princeton University Press, Princeton, vol 103, pp 127-146, 1983) and Colding-Minicozzi (Ann Math 160:69-92, 2004).</p