# Francisco Torralbo

## Tesis doctoral

Defendí mi tesis doctoral "Superficies de curvatura media paralela en $\mathbb{S}^2 \times \mathbb{S}^2$ y $\mathbb{H}^2 \times \mathbb{H}^2$ y superficies de curvatura media constante en espacios homogéneos" (ISBN: 978-84-693-1309-1, Editada por la Universidad de Granada) el 25 de febrero de 2010 obteniendo con ella el grado de doctor por la Universidad de Granada.

## Preprints

José M. Manzano , Francisco Torralbo , Compact embedded surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$. arXiv:1802.04070 [math.DG].

We obtain compact orientable embedded surfaces with constant mean curvature $0\leq H\leq frac{1}{2}$ 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 $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$, and $\mathbb{R}^3$ with bounded height and enjoying the symmetries of certain tessellations of $\mathbb{S}^2$, $\mathbb{H}^2$, and $\mathbb{R}^2$ by regular polygons.

José M. Manzano , Francisco Torralbo , Joeri Van der Veken , Parallel mean curvature surfaces in four-dimensional homogeneous spaces. Proc. International Workshop on Theory of Submanifolds 1 (2016), 57–78.
DOI: 10.24064/iwts2016.2017.8

We survey different classification results for surfaces with parallel mean curvature immersed into some Riemannian homogeneous four-manifolds, including real and complex space forms, and product spaces. We provide a common framework for this problem, with special attention to the existence of holomorphic quadratic differentials on such surfaces. The case of spheres with parallel mean curvature is also explained in detail, as well as the state-of-the-art advances in the general problem.

José M. Manzano , Julia Plehnert , Francisco Torralbo , Compact embedded minimal surfaces in $\mathbb{S}^2\times \mathbb{S}^1$. Comm. Anal. Geom. 24 (2) (2016), 409–429.
DOI: 10.4310/CAG.2016.v24.n2.a7

We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times \mathbb{R}$ via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous 3-manifolds.

Francisco Torralbo , A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in $\mathbb{H}^2\times \mathbb{R}$. J. Math. Anal. Appl. 423 (2) (2015), 1660–1670.
DOI: 10.1016/j.jmaa.2014.10.067

A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter space-time are obtained in order to illustrate this correspondence.

Francisco Torralbo , Francisco Urbano , Minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. J. Geom. Anal. 25 (2) (2015), 1132–1156.
DOI: 10.1007/s12220-013-9460-3

A general study of minimal surfaces of the Riemannian product of two spheres $\mathbb{S}^2\times\mathbb{S}^2$ is tackled. We stablish a local correspondence between (non-complex) minimal surfaces of $\mathbb{S}^2 \times \mathbb{S}^2$ and certain pair of minimal surfaces of the sphere $\mathbb{S}^3$. This correspondence also allows us to link minimal surfaces in $\mathbb{S}^3$ and in the Riemannian product $\mathbb{S}^2 \times \mathbb{R}$. Some rigidity results for compact minimal surfaces are also obtained.

Francisco Torralbo , José M. Manzano , New examples of constant mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$. Michigan Math. J. 63 (4) (2014), 701–723.
DOI: 10.1307/mmj/1417799222

We construct non-zero constant mean curvature H surfaces in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translations. In $\mathbb{S}^2 \times \mathbb{R}$ (for any $H > 0$) or $\mathbb{H}^2 \times \mathbb{R}$ (for $H > 1/2$), a 1-parameter family of unduloid-type surfaces is obtained, some of which are shown to be compact in $\mathbb{S}^2 \times \mathbb{R}$. Finally, in the case of $H = 1/2$ in $\mathbb{H}^2 \times \mathbb{R}$, the constructed examples have the symmetries of a tessellation of $\mathbb{H}^2$ by regular polygons.

Francisco Torralbo , Francisco Urbano , On stable compact minimal submanifolds. Proc. Amer. Math. Soc. 142 (2) (2014), 651–658.
DOI: 10.1090/S0002-9939-2013-11810-1

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable compact minimal surfaces of the product of a 2-sphere and any Riemann surface are the complex ones.

Francisco Torralbo , Francisco Urbano , Surfaces with parallel mean curvature vector in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times \mathbb{H}^2$. Trans. Amer. Soc. 364 (2) (2012), 785–813.
DOI: 10.1090/S0002-9947-2011-05346-8

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times \mathbb{H}^2$ are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ is proved.

Francisco Torralbo , Francisco Urbano , Compact stable constant mean curvature surfaces in homogeneous 3-manifolds. Indiana U. Math. J. 61 (3) (2012), 1129–1156.
DOI: 10.1512/iumj.2012.61.4667

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 are stable in the last two cases while in some Berger spheres there are unstable ones. Also, we classify the stable compact orientable constant mean curvature surfaces in a certain subfamily of the Berger spheres. This allows to solve the isoperimetric problem in some Berger spheres.

Francisco Torralbo , Compact minimal surfaces in the Berger spheres. Ann. Global Anal. Geom. 41 (4) (2012), 391–405.
DOI: 10.1007/s10455-011-9288-7

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 are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{R}$, $\mathbb{H}^2 \times \mathbb{R}$ and the Heisenberg group with many symmetries.

Ildefonso Castro , Francisco Torralbo , Francisco Urbano , On Hamiltonian stationary Lagrangian spheres in non-Einstein Kähler surfaces. Math. Z. 271 (1-2) (2012), 257–270.
DOI: 10.1007/s00209-011-0862-2

Hamiltonian stationary Lagrangian spheres in Kähler-Einstein surfaces are minimal. We prove that in the family of non-Einstein Kähler surfaces given by the product $\Sigma_1 \times \Sigma_2$ of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces $\Sigma_1$ and $\Sigma_2$ are spheres, is unstable.

Francisco Torralbo , Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds. Diff. Geo. Appl. 28 (5) (2010), 593–607.
DOI: 10.1016/j.difgeo.2010.04.007

We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group $Sl(2,\mathbb{R})$. In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Delaunay-type surfaces are obtained. Finally the relation between the area and volume of these spheres in the Berger spheres is studied, showing that, in some cases, they are not solution to the isoperimetric problem.

Francisco Torralbo , Francisco Urbano , On the Gauss curvature of compact surfaces in homogeneous 3-manifolds. Proc. Amer. Math. Soc. 138 (2) (2010), 2561–2567.
DOI: 10.1090/S0002-9939-10-10316-5

Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.

## Proceedings

Francisco Torralbo , Minimal Lagrangian immersions in $\mathbb{RH}^2\times \mathbb{RH}^2$. Symposium on the differential geometry of submanifolds , 3– 7 de julio de 2007, Valenciennes (Francia).

A relation, via the Gauss map, between the maximal spacelike surfaces in anti De-Sitter space and minimal Lagrangian immersions in the product of two hyperbolic planes is presented. Using this connection new examples of minimal surfaces invariant under the action of one-parameter groups of isometries are constructed.