62 research outputs found
Structure theorems for embedded disks with mean curvature bounded in L^P
After appropriate normalizations an embedded disk whose second fundamental
form has large norm contains a multi-valued graph, provided the L^P norm of the
mean curvature is sufficiently small. This generalizes to non-minimal surfaces
a well known result of Colding and Minicozzi
Curvature estimates for minimal surfaces with total boundary curvature less than 4\pi
We establish a curvature estimate for classical minimal surfaces with total
boundary curvature less than 4\pi. The main application is a bound on the genus
of these surfaces depending solely on the geometry of the boundary curve. We
also prove that the set of simple closed curves with total curvature less than
4\pi and which do not bound an orientable compact embedded minimal surface of
genus greater than g, for any given g, is open in the C^{2,\alpha} topology.Comment: 7 page
Topological Type of Limit Laminations of Embedded Minimal Disks
We consider two natural classes of minimal laminations in three-manifolds.
Both classes may be thought of as limits - in different senses - of embedded
minimal disks. In both cases, we prove that, under a natural geometric
assumption on the three-manifold, the leaves of these laminations are
topologically either disks, annuli or Moebius bands. This answers a question
posed by Hoffman and White.Comment: 21 pages. Published versio
Curvature estimates for surfaces with bounded mean curvature
Estimates for the norm of the second fundamental form, , play a crucial
role in studying the geometry of surfaces. In fact, when is bounded the
surface cannot bend too sharply. In this paper we prove that for an embedded
geodesic disk with bounded norm of , is bounded at interior
points, provided that the norm of its mean curvature is sufficiently
small, . In doing this we generalize some renowned estimates on for
minimal surfaces.Comment: 17 pages, no figures, submitte
The rigidity of embedded constant mean curvature surfaces
We study the rigidity of complete, embedded constant mean curvature surfaces
in R^3. Among other things, we prove that when such a surface has finite genus,
then intrinsic isometries of the surface extend to isometries of R^3 or its
isometry group contains an index two subgroup of isometries that extend.Comment: 10 page
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