# Stability of constant mean curvature surfaces

- Geometric Analysis Doc-Course
- IMUS (Sevilla, Spain).

Hola, I’m Francisco Torralbo, PhD in Mathematics and currently Research Fellow at the Universidad de Granada under the research project CURVCONST.

My research interests include the theory of surfaces of constant mean or constant Gauss curvature, mainly in three and four-dimensional manifolds.

I am the cosupervisor of the research project CMC and I am also a external collaborator of the Geometric Analysis project MTM2017-89677-P.

Previously, I was a postdoctoral researcher at KU Leuven and assistant professor at Centro Universitario de la Defensa and Universidad de Cádiz.

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- José M. Manzano,
Francisco Torralbo,
*Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$.*arXiv:2007.06882 [math.DG]## Abstract

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 $\mathbb{H}^2\times\mathbb{R}$.

- José M. Manzano,
Francisco Torralbo,
*Compact embedded surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$*. To appear in American Journal of Mathematics.## Abstract

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 $\mathbb{S}^2 \times \mathbb{R}$, $\mathbb{H}^2 \times \mathbb{R}$ 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.

- Francisco Torralbo,
Joeri Van der Veken,
*Rotationally invariant constant Gauss curvature surfaces in the Berger spheres*. J. Math. Anal. Appl.,**489**, 2020, ??—??.## Abstract

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant $K_0$, which we determine explicitly and depends on the geometry of the ambient Berger sphere. For values of $K_0 \leq K \leq K_P$, for a specific constant $K_P$, it was not known until now whether complete constant Gauss curvature $K$ surfaces existed in Berger spheres, so our classification provides the first examples. For $K > K_P$, we prove that the rotationally invariant spheres from our classification are the only topological spheres with constant Gauss curvature in Berger spheres.

- 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.## Abstract

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.## Abstract

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.

- José M. Manzano,
Francisco Torralbo,
&
Joeri Van der Veken,
*Parallel mean curvature surfaces in four-dimensional homogeneous spaces*. In Proceedings Book of International Workshop on Theory of Submanifolds, 2016 (pp. 57—78). .## Abstract

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.

- Francisco Torralbo,
*Minimal Lagrangian immersions in $\mathbb{RH}^2\times \mathbb{RH}^2$*. In Symposium on the differential geometry of submanifolds, 2007 (pp. 217—220). Valenciennes (France). ISBN: 029507. Publisher Lulu.## Abstract

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.

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- Geometric Analysis Doc-Course
- IMUS (Sevilla, Spain).

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During the academic year 2021–2021 I am teaching the following courses:

- Matemáticas I, Grado en Química. Semestre 1, 2020-2021.
Toda la información relativa al curso puedes encontrarla en la plataforma PRADO

- Actualización científica en Matemáticas, Máster Universitario en Matemáticas. Semestre 2, 2020-2021.

- (+34) 958 2 41000 # 20041
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