Research projects

Geometric Analysis

MTM2017-89677-P
January 1, 2018 — September 30, 2021
Geometric Analysis is a 3-year project funded by MICINN/FEDER programme (MTM2017-89677-P). The principal investigators are Joaquín Pérez and Antonio Alarcón. I participate as an external collaborator.

About the project

This research project is headed by Joaquín Pérez and Antonio Alarcón and it continues the line of the MTM2011-22547 and MTM2014-52368-P projects. The team consists of 17 researchers in different areas of Geometric Analysis working in different centers in Spain and United States. The research topics covered in this project deal with minimal and constant mean curvature surfaces, isoperimetric problem, Riemann surfaces, overdetermined elliptic problems, mean curvature flow and conformal and contact geometry.

More information in the project website.

Contributions

I participate in the project as an external collaborator.

Publications

Peer-reviewed published papers

  • , , 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}.

  • , , 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.

  • , , Rotationally invariant constant Gauss curvature surfaces in the Berger spheres. J. Math. Anal. Appl., 489, , 112.
    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 KK0K \geq K_0 for a positive constant K0K_0, which we determine explicitly and depends on the geometry of the ambient Berger sphere. For values of K0KKPK_0 \leq K \leq K_P, for a specific constant KPK_P, it was not known until now whether complete constant Gauss curvature KK surfaces existed in Berger spheres, so our classification provides the first examples. For K>KPK > K_P, we prove that the rotationally invariant spheres from our classification are the only topological spheres with constant Gauss curvature in Berger spheres.