544 research outputs found
Constant mean curvature surfaces in 3-dimensional Thurston geometries
This is a survey on the global theory of constant mean curvature surfaces in
Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight
canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2
\times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie
group Sol3. We will focus on the problems of classifying compact CMC surfaces
and entire CMC graphs in these spaces. A collection of important open problems
of the theory is also presented
Harmonic maps and constant mean curvature surfaces in \H^2 \times \R
We introduce a hyperbolic Gauss map into the Poincare disk for any surface in
H^2xR with regular vertical projection, and prove that if the surface has
constant mean curvature H=1/2, this hyperbolic Gauss map is harmonic.
Conversely, we show that every nowhere holomorphic harmonic map from an open
simply connected Riemann surface into the Poincare disk is the hyperbolic Gauss
map of a two-parameter family of such surfaces. As an application we obtain
that any holomorphic quadratic differential on the surface can be realized as
the Abresch-Rosenberg holomorphic differential of some, and generically
infinitely many, complete surfaces with H=1/2 in H^2xR. A similar result
applies to minimal surfaces in the Heisenberg group Nil_3.
Finally, we classify all complete minimal vertical graphs in H^2xR.Comment: 37 pages, 1 figur
A characterization of constant mean curvature surfaces in homogeneous 3-manifolds
It has been recently shown by Abresch and Rosenberg that a certain Hopf
differential is holomorphic on every constant mean curvature surface in a
Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this
paper we describe all the surfaces with holomorphic Hopf differential in the
homogeneous 3-manifolds isometric to H^2xR or having isometry group isomorphic
either to the one of the universal cover of PSL(2,R), or to the one of a
certain class of Berger spheres. It turns out that, except for the case of
these Berger spheres, there exist some exceptional surfaces with holomorphic
Hopf differential and non-constant mean curvature.Comment: corrected typo
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