2,613 research outputs found
A general halfspace theorem for constant mean curvature surfaces
In this paper, we prove a general halfspace theorem for constant mean
curvature surfaces. Under certain hypotheses, we prove that, in an ambient
space M^3, any constant mean curvature H_0 surface on one side of a constant
mean curvature H_0 surface \Sigma_0 is an equidistant surface to \Sigma_0. The
main hypotheses of the theorem are that \Sigma_0 is parabolic and the mean
curvature of the equidistant surfaces to \Sigma_0 evolves in a certain way.Comment: 3 figures, sign mistakes at the beginning of Section 6 are correcte
The Plateau problem at infinity for horizontal ends and genus 1
In this paper, we study Alexandrov-embedded r-noids with genus 1 and
horizontal ends. Such minimal surfaces are of two types and we build several
examples of the first one. We prove that if a polygon bounds an immersed
polygonal disk, it is the flux polygon of an r-noid with genus 1 of the first
type. We also study the case of polygons which are invariant under a rotation.
The construction of these surfaces is based on the resolution of the Dirichlet
problem for the minimal surface equation on an unbounded domain.Comment: 63 page
On minimal spheres of area and rigidity
Let be a complete Riemannian -manifold with sectional curvatures
between and . A minimal -sphere immersed in has area at least
. If an embedded minimal sphere has area , then is isometric to
the unit -sphere or to a quotient of the product of the unit -sphere with
, with the product metric. We also obtain a rigidity theorem for
the existence of hyperbolic cusps. Let be a complete Riemannian
-manifold with sectional curvatures bounded above by . Suppose there is
a -torus embedded in with mean curvature one. Then the mean convex
component of bounded by is a hyperbolic cusp;,i.e., it is isometric to
with the constant curvature metric:
with a flat metric on .Comment: 8 page
Minimal surfaces near short geodesics in hyperbolic -manifolds
If is a finite volume complete hyperbolic -manifold, the quantity
is defined as the infimum of the areas of closed minimal
surfaces in . In this paper we study the continuity property of the
functional with respect to the geometric convergence of
hyperbolic manifolds. We prove that it is lower semi-continuous and even
continuous if is realized by a minimal surface satisfying
some hypotheses. Understanding the interaction between minimal surfaces and
short geodesics in is the main theme of this paperComment: 35 pages, 4 figure
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