Research projects

Constant mean curvature surfaces in homogeneous 3-manifolds

June 1, 2020 — May 31, 2023
CMC is a 3-year project funded by the Plan Estatal de Investigación Científica y Técnica y de Innovación: Generación de conocimiento in the young researchers category (type A). The principal investigators are José M. Manzano and Francisco Torralbo.

About the project

This project finished last May 31, 2023. For the new research project see CMC

This project focuses on the study of constant mean curvature surfaces in homogeneous Riemannian manifolds of dimension three and four. The main goal of this research proposal is to delve into the role played by the mean curvature in the properties of a surface from different points of view. The posed problems deal with the construction and characterization of examples, as well as the study of global geometric properties related to open problems that have guided the research in this area in the last decades.

The main goals of CMC are the following:

  • Move forward in our understanding of minimal and H-surfaces in the homogeneous Riemannian 3-manifolds Ε(κ, τ) by means of construction and characterization of examples and the study of admissible topological types.
  • Develop the theory of parallel mean curvature surfaces in Thurston 4-dimensional geometries in order to get classification results about them.


CMC brings together researchers from the Dpt. Geometry and Topology of the University of Granada, the Dpt. of Mathematics of the University of Jaén, the Geometry Section of the Mathematics Department of KU Leuven , and the Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica of the Università dell’Aquila

Principal investigators




  • , , Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times \mathbb{R}$. To appear in Journal of the Institute of Mathematics of Jussieu. arXiv:2001.07028 [math.DG]

    For each $k \geq 3$, we construct a 1-parameter family of complete Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times \mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb{H}^2\times \mathbb{R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi$. Finally, we provide examples of complete properly embedded minimal surfaces with infinitely many ends, each of them asymptotic to a vertical plane and with finite total curvature.

Peer-reviewed published papers

  • , , & , A construction of constant mean curvature surfaces in $\mathbb{H}^2\times \mathbb{R}$ and the Krust property. To appear in Int. Math. Res. Not., , , .

    We show the existence of a 2-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0<H\leq 2$ in $\mathbb{H}^2\times \mathbb{R}$. They are symmetric with respect to a horizontal slice and a $k$ vertical planes disposed symmetrically, and extend the so called minimal saddle towers and k-noids. We show that the orientation plays a fundamental role when the mean curvature is positive via the sister conjugate minimal surface from $\widetilde\mathrm{SL}_2(\mathbb{R})$ or $\mathrm{Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders from the convex side, often giving rise to non-embedded examples. In the discussion of embeddedness, we prove that the Krust property does not hold for any $H>0$, i.e., there are minimal graphs over convex domains in $\widetilde\mathrm{SL}_2(\mathbb{R})$, $\mathrm{Nil}_3$ or the Berger spheres, whose sister conjugate surfaces with constant mean curvature $H$ in $\mathbb{H}^2\times \mathbb{R}$ are not graphs.

  • , , Isoparametric surfaces in $\mathbb{E}(\kappa,\tau)$-spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci., XXII(1), , 269285.

    We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometrygroup: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential

  • , , & , Conjugate Plateau constructions in product spaces. New trends in Geometric Analysis, 10, , 43118.

    This survey paper investigates, from a purely geometric point of view, Daniel’s isometric conjugation between minimal and constant mean curvature surfaces immersed in homogeneous Riemannian three-manifolds with isometry group of dimension four. On the one hand, we collect the results and strategies in the literature that have been developed so far to deal with the analysis of conjugate surfaces and their embeddedness. On the other hand, we revisit some constructions of constant mean curvature surfaces in the homogeneous product spaces $\mathbb{S}^2\times \mathbb{R}$, $\mathbb{H}^2\times \mathbb{R}$ and $\mathbb{R}^3$ having different topologies and geometric properties depending on the value of the mean curvature. Finally, we also provide some numerical pictures using Surface Evolver.

  • , , Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$. Camb. J. Math., 10(3), , 657688.

    We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, being the mean curvature larger than $\frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $\mathbb H^2\times\mathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $\mathbb{S}^2\times\mathbb{R}$, which have constant mean curvature $H>\frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $H\leq\frac{1}{2}$ at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times\mathbb{R}$.

  • , , Index of compact minimal submanifolds of the Berger spheres. Calc. Var. Partial Differential Equations, 61(104), , .

    The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_{\tau}$, $0<\tau\leq 1$, are studied. Unlike the case of the standard sphere ($\tau=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $\tau^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_\tau$ when $1 / (d+1) < \tau^2 \leq 1$ and the stable ones are classified for $\tau^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_{\tau}$ with index one are classified for $1/3\leq\tau^2\leq 1$.

  • , , Compact embedded surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$. Amer. J. Math., 142(6), , 19811994.

    We obtain compact orientable embedded surfaces with constant mean curvature $0 < H \leq \frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$, $\mathbb{H}^2 \times \mathbb{R}$ hand $\mathbb{R}^3$ with bounded height and enjoying the symmetries of certain tessellations of $\mathbb{S}^2$, $\mathbb{H}^2$ and $\mathbb{R}^2$ by regular polygons.