480 research outputs found
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
Construction of harmonic diffeomorphisms and minimal graphs
We study complete minimal graphs in HxR, which take asymptotic boundary
values plus and minus infinity on alternating sides of an ideal inscribed
polygon Γ in H. We give necessary and sufficient conditions on the
"lenghts" of the sides of the polygon (and all inscribed polygons in Γ)
that ensure the existence of such a graph. We then apply this to construct
entire minimal graphs in HxR that are conformally the complex plane C. The
vertical projection of such a graph yields a harmonic diffeomorphism from C
onto H, disproving a conjecture of Rick Schoen
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
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