Tag - Riemannian geometry

In this talk I will report on some of the new results in the theory of constant mean curvature (CMC) surfaces M in Riemannian 3-manifolds N. I will mention just a few of the topics touched on in this talk. I first begin with the recent classification of CMC spheres in a homogeneous 3-manifold N and a sketch of its proof. The main result states that any two spheres in N with the same absolute mean curvature differ by an ambient isometry of N. Furthermore, the range of values of the mean curvature spheres are described in terms of the geometry of the universal cover X of N. In the case that X is diffeomorphic to ℝ³ then there exists a sphere of constant mean curvature H in N iff H is greater than half the Cheeger constant of X and otherwise there exists a sphere of constant mean curvature in N for every real number. These results generalize previous work of Hopf, of Abresch-Rosenberg and more recently of Danieil-Mira and of Meeks in the case of the Sol geometry.

Jointly with Tinaglia, we obtain curvature for embedded disks of fixed constant mean curvature H>0 in any fixed homogeneous 3-manifolds. In the ℝ³ setting this result implies that any complete embedded finite topology surface in ℝ³ of constant mean curvature is proper; this generalizes the previous work of Colding -Minicozzi in the case of minimal surfaces. Previous classification results then imply that the only complete embedded simply connected constant mean curvature surfaces in ℝ³ are the plane, the helicoid and round spheres. Another application of this work by Meeks-Tinaglia is to prove that complete embedded CMC surfaces of finite topology in a complete hyperbolic 3-manifold are proper if the mean curvature H is at least 1. On the other hand, Coskunuzer-Meeks-Tinaglia recently constructed for any H in [0,1) a non-proper, complete, stable embedded plane in hyperbolic 3-space having constant mean curvature H.

In 1982 Choi and Wang proved that an embedded closed minimal surface F in the the round 3-sphere S³ has a bound on its area that only depends on its genus; actually their result generalizes from the ambient space S³ to any closed 3-manifold M with positive Ricci curvature. This result was then used by Choi and Schoen to prove the compactness of the moduli space of such examples of fixed genus g in M. Tinaglia and I have been able to give the following related result in the case of connected closed surfaces M embedded in any Riemannian homology 3-sphere manifold N:

Theorem: For any H>0 and non-negative integer g, there exists a constant A(N,g,H) such that any closed surface embedded in M of genus g and constant mean curvature H has area at most A(N,g,H). This area estimate lead to a natural compactification of the moduli space of all such embedded constant mean curvature H examples in N with genus at most g, when H lies in a fixed compact interval [a,b] of positive numbers, and under a compact deformation of the Riemannian metric on N.

The recent classification of properly embedded minimal surfaces of genus 0 in ℝ³ given by Meeks-Perez-Ros, Lopez-Ros, Colin and of Meeks-Rosenberg play a role in the above area estimates, as do the curvature estimates of Meeks-Tinaglia for certain complete embedded CMC surfaces in a Riemannian 3-manifold.

At the end of my talk I will present a brief survey of some recent results on the existence and classification of CMC foliations of closed and non-closed 3-manifolds.