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Conference on submanifolds and spin geometry

May 13-14-15, 2013
Institut Élie Cartan de Lorraine
Nancy, France

Organizing committee:

  • Benoît Daniel (Université de Lorraine - Nancy, France)
  • Oussama Hijazi (Université de Lorraine - Nancy, France)
  • Sebastián Montiel (Universidad de Granada, Spain)
  • The talks will take place in Salle de Conférences (2nd floor), except those of Tuesday morning (in Salle Döblin, 4th floor).

    Monday, May 13

    Harold Rosenberg (IMPA - Rio de Janeiro, Brazil)
    Properly immersed minimal surfaces in hyperbolic 3-manifolds of finite volume, and in M × S1, M a hyperbolic complete surface of finite area

    I will discuss work done with Pascal Collin and Laurent Hauswirth. We prove a minimal surface of finite topology as described in the title, has finite total curvature equal to 2π times the Euler characteristic. We also describe the asymptotic geometry of the ends of the surface. Examples will be discussed; in particular, in the complement of the figure eight knot with a hyperbolic structure.
    Andrei Moroianu (CNRS - Université de Versailles, France)
    Generalized Killing spinors on Einstein manifolds

    In this talk (based on joint work with Uwe Semmelmann) I will discuss the issue of generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkähler, G2 and Spin(7) manifolds.
    Josef Dorfmeister (Technische Universität München, Germany)
    Minimal surfaces in Nil3 via loop groups

    We will discuss surfaces in Nil3 in the spirit of the work of Taimanov and Berdinskii. Starting from Berdinskii's system of equations for conformal immersions into Nil3 we chracaterize constant mean curvature surfaces by flat connections and equivalently by the harmonicity of some map. In this context we clarify what t means for an arbitrary conformal surface to have a holomorphic Abresch-Rosenberg differential. It turns out that the Berdinskii system produces "associate families" of surfaces, but, in general, only for a few values of the family parameter the corresponding surface can have constant mean curvature. If the mean curvature H vanishes, however, a complete loop group method is avialble. In particular, every minimal surface in Nil3 can be constructed starting from some holomorphic (and unconstrained) data.

    Tuesday, May 14

    09:15 (Salle Döblin)
    Nicolas Ginoux (Universität Regensburg, Germany)
    Dirac-harmonic maps from index theory

    Joint work with Bernd Ammann. Dirac-harmonic maps are critical points of some kind of energy functional coupling a map between Riemannian manifolds and a section of a twisted spinor bundle. Their characterizing equation involves a twisted Dirac operator. In this talk, I will explain how the non-vanishing of its index allows to produce Dirac-harmonic maps out of harmonic ones.
    10:30 (Salle Döblin)
    Simon Raulot (Université de Rouen, France)
    The Dirac operator on untrapped surfaces

    In this talk, we establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for surfaces with constant mean curvature vector field in Minkowski space.
    Andrea Mondino (Scuola Normale Superiore - Pisa, Italy)
    Willmore spheres in Riemannian manifolds

    Given an immersion f of the 2-sphere in a Riemannian manifold (M,g) we study quadratic curvature functionals of the type
    f(S2) H2,      f(S2) |A|2,      f(S2) |Ao|2,
    where H is the mean curvature, A is the second fundamental form, and Ao is the tracefree second fundamental form. Minimizers, and more generally critical points of such functionals can be seen respectively as GENERALIZED minimal, totally geodesic and totally umbilical immersions. In the seminar I will review some results (obtained in collaboration with Kuwert, Rivière and Schygulla) regarding the existence and the regularity of minimizers of such functionals. An interesting observation regarding the results obtained with Rivière is that the theory of Willmore surfaces can be usesful to complete the theory of minimal surfaces (in particular in relation to the existence of canonical smooth representatives in homotopy classes, a classical program started by Sacks and Uhlenbeck).
    Laurent Mazet (CNRS - Université Paris-Est Créteil, France)
    Cyllindrically bounded cmc annuli in H2 × R

    In this talk, I will recall some known results about the classification of constant mean curvature annuli in R3. Then I will explain how similar results can be proved in H2 × R with "less symmetries".

    Wednesday, May 15

    Laurent Hauswirth (Université Paris-Est Marne-la-Vallée, France)
    The Lawson conjecture via integrable systems

    In 2012, Brendle and Andrews-Li proved with analytic argument that embedded constant mean curvature tori in the 3-dimensional sphere are rotational surfaces. We prove this theorem using algebraic method. We consider the space moduli of CMC Alexandrov embedded annuli covering CMC embedded torus. The path connected property and isolated property reformulate via integrable systems in an algebraic problem in the space moduli of hyperelliptic Riemann surfaces with an abelian differential form.
    Joaquin Pérez (Universidad de Granada, Spain)
    Isoperimetric domains with large volume in simply connected homogeneous 3-manifolds

    We will study the relationship between three natural notions related to constant mean curvature surfaces in a non-compact, simply connected homogeneous 3-manifold X: The Cheeger constant Ch(X), the critical mean curvature H(X) and the geometry of isoperimetric domains. We will see that the first two numbers are essentially the same, and that if Ch(X)>0, then there exists a product foliation F of X by pairwise congruent leaves with constant mean curvature H(X) such that the boundaries of isoperimetric domains of large volume are well-approximated in a natural sense by the leaves of F.

    Contact : Benoît Daniel