Structure theorems for embedded disks with mean curvature bounded in $L^p$

Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve

3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genus-one immersed constant mean curvature surfaces with boundary C that are approximated b...

Embedded Minimal Disks with Prescribed Curvature Blowup

We construct a sequence of compact embedded minimal disks in a ball in R, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x3 − axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial a...

Properly embedded surfaces with constant mean curvature

In this paper we prove a maximum principle at infinity for properly embedded surfaces with constant mean curvature H > 0 in the 3-dimensional Euclidean space. We show that no one of these surfaces can lie in the mean convex side of another properly embedded H surface. We also prove that, under natural assumptions, if the surface lies in the slab |x3| < 1/2H and is symmetric with respect to the ...

Eigenvalue estimates for submanifolds with locally bounded mean curvature

We give lower bounds estimates for the first Dirichilet eigenvalues for domains Ω in submanifolds M ⊂ N with locally bounded mean curvature. These lower bounds depend on the local injectivity radius, local upper bound for sectional curvature of N and local bound for the mean cuvature of M . For sumanifolds with bounded mean curvature of Hadamard manifolds these lower bounds depends only on the ...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form