Corpus ID: 118444347

# Constant mean curvature spheres in homogeneous three-spheres

@article{Meeks2013ConstantMC,
title={Constant mean curvature spheres in homogeneous three-spheres},
author={William H. Iii Meeks and Pablo Mira and Joaqu{\'i}n P{\'e}rez and Antonio Ros},
journal={arXiv: Differential Geometry},
year={2013}
}
• W. Meeks, +1 author A. Ros
• Published 2013
• Mathematics
• arXiv: Differential Geometry
We give a complete classification of the immersed constant mean curvature spheres in a three-sphere with an arbitrary homogenous metric, by proving that for each $H\in\mathbb{R}$, there exists a constant mean curvature $H$-sphere in the space that is unique up to an ambient isometry.

#### Figures and Tables from this paper

Constant mean curvature spheres in homogeneous three-manifolds
• Mathematics
• 2017
We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature forExpand
Existence of constant mean curvature 2-spheres in Riemannian 3-spheres
• Mathematics
• 2020
We prove the existence of branched immersed constant mean curvature 2spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed meanExpand
Min-max theory for constant mean curvature hypersurfaces
• Mathematics
• 2017
In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we proveExpand
On the two-systole of real projective spaces
• Mathematics
• 2018
We establish an integral-geometric formula for minimal two-spheres inside homogeneous three-spheres, and use it to provide a characterisation of each homogeneous metric on the three-dimensional realExpand
Min–max theory for constant mean curvature hypersurfaces
• Mathematics
• Inventiones mathematicae
• 2019
In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we proveExpand
A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov
• Mathematics
• 2015
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A. D.Expand
Constant Mean Curvature Annuli in Homogeneous Manifolds
In this thesis we construct constant mean curvature annuli in homogeneous manifolds. These annuli generalise cylinders and unduloids in Euclidean space. In the first part we show existence ofExpand
The global geometry of surfaces with prescribed mean curvature in ℝ³
• Mathematics
• 2018
We develop a global theory for complete hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant meanExpand
On the min-max width of unit volume three-spheres.
• Mathematics
• 2018
How large can be the width of Riemannian three-spheres of the same volume in the same conformal class? If a maximum value is attained, how does a maximising metric look like? What happens as theExpand
Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\mathsf{H}^1$
• Mathematics
• 2015
In this paper we prove that isoperimetric sets in three-dimensional homogeneous spaces diffeomorphic to $\mathbb{R}^3$ are topological balls. We also prove that in three-dimensional homogeneousExpand

#### References

SHOWING 1-10 OF 35 REFERENCES
Compact stable constant mean curvature surfaces in homogeneous 3-manifolds
• Mathematics
• 2012
We classify the stable constant mean curvature spheres in the homogeneous Riemannian 3-manifolds: the Berger spheres, the special linear group and the Heisenberg group. We show that all of them areExpand
Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds
Abstract We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl ( 2 , R ) . In particular, all constantExpand
Existence of regular neighborhoods for H-surfaces
• Mathematics
• 2010
In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces andExpand
Compact stable constant mean curvature surfaces in the Berger spheres
• Mathematics
• 2009
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (aExpand
A Hopf differential for constant mean curvature surfaces inS2×R andH2×R
• Mathematics
• 2004
A basic tool in the theory of constant mean curvature (cmc) surfaces Σ in space forms is the holomorphic quadratic differential discovered by H. Hopf. In this paper we generalize this differential toExpand
Compact minimal surfaces in the Berger spheres
In this article, we construct compact, arbitrary Euler characteristic, orientable and non-orientable minimal surfaces in the Berger spheres. Also, we show an interesting family of surfaces that areExpand
Stable constant mean curvature surfaces
• Mathematics
• 2008
We study relationships between stability of constant mean curvature surfaces in a Riemannian three-manifold N and the geometry of leaves of laminations and foliations of N by surfaces of possiblyExpand
Existence and uniqueness of constant mean curvature spheres in Sol_3
• Mathematics
• 2008
We study the classification of immersed constant mean curvature (CMC) spheres in the homogeneous Riemannian 3-manifold Sol_3, i.e., the only Thurston 3-dimensional geometry where this problem remainsExpand
A theorem of Hopf and the Cauchy-Riemann inequality
• Mathematics
• 2007
In 1951, Hopf [9] published a theorem in a seminal paper on surfaces of constant mean curvature which can be stated as follows. Let a genus zero compact surface M be immersed in $${\mathbb{R}^3}$$Expand
Lower Bounds for Morse Index of Constant Mean Curvature Tori
We give three lower bounds for the Morse index of a constant mean curvature torus in Euclidean 3-space in terms of its spectral genus g. The first two lower bounds grow linearly in g and are strongerExpand