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Most recent published papers

  • , , & , Conjugate Plateau constructions in product spaces. New trends in Geometric Analysis, 10, , 43118.
    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 S2×R\mathbb{S}^2\times \mathbb{R}, H2×R\mathbb{H}^2\times \mathbb{R} and R3\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 S2×R\mathbb{S}^2\times \mathbb{R} and H2×R\mathbb{H}^2\times \mathbb{R}. Camb. J. Math., 10(3), , 657688.
    Abstract

    We obtain a 11-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in S2×R\mathbb{S}^2\times\mathbb{R} and H2×R\mathbb{H}^2\times\mathbb{R}, being the mean curvature larger than 12\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 H2×R\mathbb H^2\times\mathbb{R}. We also find (among unduloids) families of embedded constant mean curvature tori in S2×R\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 S2×R\mathbb{S}^2\times\mathbb{R}, which have constant mean curvature H>12H>\frac12. Finally, we prove that there are no properly immersed surface with constant mean curvature H12H\leq\frac{1}{2} at bounded distance from a horizontal geodesic in H2×R\mathbb{H}^2\times\mathbb{R}.

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

    The stability and the index of compact minimal submanifolds of the Berger spheres Sτ2n+1\mathbb{S}^{2n+1}_{\tau}, 0<τ10<\tau\leq 1, are studied. Unlike the case of the standard sphere (τ=1\tau=1), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if τ21/2\tau^2\leq 1/2. Moreover, there are no stable compact minimal dd-dimensional submanifolds of Sτ2n+1\mathbb{S}^{2n+1}_\tau when 1/(d+1)<τ211 / (d+1) < \tau^2 \leq 1 and the stable ones are classified for τ2=1/(d+1)\tau^2=1 / (d+1) when the submanifold is embedded. Finally, the compact orientable minimal surfaces of Sτ3\mathbb{S}^3_{\tau} with index one are classified for 1/3τ211/3\leq\tau^2\leq 1.

  • , , Compact embedded surfaces with constant mean curvature in S2×R\mathbb{S}^2 \times \mathbb{R}. Amer. J. Math., 142(6), , 19811994.
    Abstract

    We obtain compact orientable embedded surfaces with constant mean curvature 0<H120 < H \leq \frac{1}{2} and arbitrary genus in S2×R\mathbb{S}^2\times\mathbb{R}. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature 12\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 S2×R\mathbb{S}^2 \times \mathbb{R}, H2×R\mathbb{H}^2 \times \mathbb{R} hand R3\mathbb{R}^3 with bounded height and enjoying the symmetries of certain tessellations of S2\mathbb{S}^2, H2\mathbb{H}^2 and R2\mathbb{R}^2 by regular polygons.

Most recent proceedings

  • , , & , Parallel mean curvature surfaces in four-dimensional homogeneous spaces. In Proceedings Book of International Workshop on Theory of Submanifolds, (pp. 5778). .
    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.

  • , Minimal Lagrangian immersions in RH2×RH2\mathbb{RH}^2\times \mathbb{RH}^2. In Symposium on the differential geometry of submanifolds, (pp. 217220). 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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  • ftorralbo@ugr.es
  • (+34) 958 2 43279
  • Dpto. Geometría y Topología. Facultad de Ciencias. Universidad de Granada C/ Fuentenueva s/n, 18071 – Granada – SPAIN
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