Submanifolds and differential equations in geometry

Venue

Programme

• 09:30 - 10:30 Francisco Torralbo (Universidad de Granada)
  Stability and index of compact minimal submanifolds

  A submanifold of a Riemannian manifold is called minimal if its mean curvature vector vanish everywhere. These manifolds appear as critical points of the volume functional. From this point of view, we will introduce the second variation of the volume and study those compact minimal submanifolds with positive semi-definite second variation, that are called stable. We will show that the sphere does not admit compact stable minimal submanifolds and we will characterize those compact minimal submanifolds which fail short from being stable (the lowest index examples).

  If time allows, we will also discuss some generalizations of these results to the Berger spheres (aka spheres endowed with a deformation of the round metric in the direction of the Hopf fibers).

• 10:30 - 11:00 Sandro Caeiro Oliveira (CITMAga - USC)
  Curvature homogeneous critical metrics modeled on a symmetric space

  A pseudo-Riemannian manifold (M, g) is k-curvature homogeneous if, for each pair of points p, q ∈ M, there exists a linear isometry ɸ_pq: T_pM → T_qM satisfying ɸ_pq^*∇ⁱ R_q=∇ⁱ R_p for all 0 ≤ i ≤ k. If k = 0, we say that the manifold is curvature homogeneous. Clearly, homogeneous manifolds are curvature homogeneous but the converse is not true.

  Einstein manifolds are critical metrics for the Hilbert-Einstein functional constrained to constant volume variations. Since any Einstein metric is of constant sectional curvature in dimension three, massive gravity theories were introduced by means of actions which involve higher order scalar invariants.

  In this talk, we classify three-dimensional curvature homogeneous manifolds modeled on symmetric spaces that are critical for some quadratic curvature functional and those that provide solutions in massive gravity theories. A remarkable fact is that the underlying structure in all solutions is either a Brinkmann wave or a Kundt spacetime, which provide non-Einstein metrics which are critical for all quadratic curvature invariants.

• 11:00 - 11:30 Diego Mojón Álvarez (CITMAga - USC - UDC)
  Weighted Einstein field equations in smooth metric measure spaces

  Semi-Riemannian manifolds can be generalized by introducing a density function f (equivalently, a distinguished measure), giving rise to smooth metric measure spaces. The geometric features of these spaces may be studied through so-called weighted tensors, which suitably generalize the usual curvature-related tensors of a semi-Riemannian manifold while including information on the density function. One of these weighted tensors is the Bakry-Émery Ricci tensor.

  In this talk, we consider the Lorentzian setting and define, in terms of this tensor, a weighted Einstein tensor which is symmetric, divergence-free, and concomitant of the metric and the density function. The solutions of the resulting vacuum Einstein field equation, in the isotropic case (∇f lightlike), have nilpotent Ricci operator. Moreover, they belong to two families: either Brinkmann waves or Kundt spacetimes, depending on whether their Ricci operator is 2-step or 3-step nilpotent, respectively. In particular, in dimension 3, all isotropic solutions can be given in local coordinates as plane waves or Kundt spacetimes.

• 11:30 - 12:00 Break
• 12:00 - 12:30 Alberto Rodríguez Vázquez (CITMAga - USC)
  Totally geodesic submanifolds in Hopf-Berger spheres

  Every homogeneous metric in a sphere can be constructed by rescaling the metric of the total space of the complex, quaternionic or octonionic Hopf fibration in the direction of the fibers. The problem of classifying totally geodesic submanifolds in a given Riemannian manifold is a classical topic in the area of submanifold geometry. In the setting of Riemannian symmetric spaces, this problem has been extensively studied. In the more general setting of Riemannian homogeneous spaces, the classification of totally geodesic submanifolds is a much more complex problem and we lack general techniques to approach it.

  In this talk, I will report on an ongoing work with Carlos Olmos (Universidad Nacional de Córdoba), where we give a classification of totally geodesic submanifolds in the total space of the complex, quaternionic, or octonionic Hopf fibration where the metric of the fibers is rescaled by a positive factor.

• 12:30 - 13:30 José Miguel Manzano (Universidad de Jaén)
  Constant mean curvature surfaces in three-manifolds carrying a Killing vector field

  A Killing submersion is a Riemannian submersion, from an orientable three-manifold to an orientable surface, whose fibers are the integral curves of a Killing vector field. Indeed, any three-manifold with a Killing vector field can be locally endowed with a Killing submersion structure away from zeroes of the vector field. The first part of this talk will be devoted to discuss local and global classification results for Killing submersions by prescribing an arbitrary base surface and two arbitrary geometric functions, namely the bundle curvature and the length of the Killing vector field. In this discussion, we will assume that the total space is either Riemannian or Lorentzian. Moreover, we will show that the only Killing submersions with Killing vector field of constant length in which the total space is homogeneous are the Bianchi--Cartan--Vranceanu spaces, i.e., the so-called E(κ,τ)-spaces. We will also discuss a geometric duality for graphical spacelike surfaces between Riemannian and Lorentzian Killing submersions swapping the mean curvature of the surface and the bundle curvature of the ambient space, and obtain some consequences. This talk is partially based on joint works with Hojoo Lee and Ana M. Lerma.

Organizers

• José Carlos Díaz Ramos, CITMAga - Universidade de Santiago de Compostela.
• Miguel Domínguez Vázquez, CITMAga - Universidade de Santiago de Compostela.
• Eduardo García Río, CITMAga - Universidade de Santiago de Compostela.