Hola, I’m Francisco Torralbo, PhD in Mathematics and professor at the Universidad de Granada. I am the cosupervisor of the research project CMC and member of IMAG.
My research interests include the theory of surfaces of constant mean or constant Gauss curvature, mainly in three and four-dimensional manifolds.
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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.
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 $\mathbb{H}^2\times\mathbb{R}$.
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 $1/3\leq\tau^2\leq 1$.
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.
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.
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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During the academic year 2021–2021 I am teaching the following courses:
Toda la información relativa al curso puedes encontrarla en la plataforma PRADO
Toda la información relativa al curso puedes encontrarla en la plataforma PRADO
