# 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} }

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.

#### 22 Citations

Constant mean curvature spheres in homogeneous three-manifolds

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- 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 for… Expand

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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 mean… Expand

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- 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 prove… Expand

On the two-systole of real projective spaces

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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 real… Expand

Min–max theory for constant mean curvature hypersurfaces

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- 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 prove… Expand

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

- Mathematics
- 2016

In this thesis we construct constant mean curvature annuli in homogeneous manifolds. These annuli generalise cylinders and unduloids in Euclidean space.
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The global geometry of surfaces with prescribed mean curvature in ℝ³

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- 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 mean… Expand

On the min-max width of unit volume three-spheres.

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- 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 the… Expand

Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group $\mathsf{H}^1$

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- 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 homogeneous… Expand

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