Publications

Peer-reviewed published papers

• Jesús Castro-Infantes, José M. Manzano, & Francisco Torralbo, Conjugate Plateau constructions in product spaces. New trends in Geometric Analysis, 10, 2023, 43—118.
  10.1007/978-3-031-39916-9_3. arXiv:2203.13162
  Abstract

  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.

• José M. Manzano, Francisco Torralbo, Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$. Camb. J. Math., 10(3), 2022, 657—688.
  10.4310/CJM.2022.v10.n3.a2. arXiv:2007.06882
  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 $\mathbb{H}^2\times\mathbb{R}$.

• Francisco Torralbo, Francisco Urbano, Index of compact minimal submanifolds of the Berger spheres. Calc. Var. Partial Differential Equations, 61(104), 2022, —.
  10.1007/s00526-022-02215-6. arXiv:2110.08027
  Abstract

  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 $1/3\leq\tau^2\leq 1$.

• José M. Manzano, Francisco Torralbo, Compact embedded surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$. Amer. J. Math., 142(6), 2020, 1981—1994.
  10.1353/ajm.2020.0050. arXiv:1802.04070
  Abstract

  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.

• Francisco Torralbo, Joeri Van der Veken, Rotationally invariant constant Gauss curvature surfaces in the Berger spheres. J. Math. Anal. Appl., 489, 2020, 1—12.
  10.1016/j.jmaa.2020.124183. MR ZBL arXiv:1912.02681
  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.
  10.4310/CAG.2016.v24.n2.a7. MR ZBL arXiv:1311.2500
  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.
  10.1016/j.jmaa.2014.10.067. MR ZBL arXiv:1407.5423
  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.

• Francisco Torralbo, Francisco Urbano, Minimal surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. J. Geom. Anal., 25(2), 2015, 1132—1156.
  10.1007/s12220-013-9460-3. MR ZBL arXiv:1301.1580
  Abstract

  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.

• José M. Manzano, Francisco Torralbo, 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.
  10.1307/mmj/1417799222. MR ZBL arXiv:1104.1259
  Abstract

  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.
  10.1090/S0002-9939-2013-11810-1. MR ZBL arXiv:1012.0679
  Abstract

  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.
  10.1090/S0002-9947-2011-05346-8. MR ZBL arXiv:0807.1880v2
  Abstract

  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.
  10.1512/iumj.2012.61.4667. MR ZBL arXiv:0906.1439v1
  Abstract

  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.
  10.1007/s10455-011-9288-7. MR ZBL arXiv:1007.1072v1
  Abstract

  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.
  10.1007/s00209-011-0862-2. MR ZBL arXiv:706.4390
  Abstract

  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.
  10.1016/j.difgeo.2010.04.007. MR ZBL arXiv:0911.5128v1
  Abstract

  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.
  10.1090/S0002-9939-10-10316-5. MR ZBL arXiv:0903.1735v1
  Abstract

  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.