Detalles de Evento

Conferencia 1 (Hora - 16.00h)

Speaker: Rodrigo Avalos

Title: A Q-curvature positive energy theorem and rigidity of Q-singular asymptotically Euclidean manifolds

Abstract: In this talk we will present recent results related to a notion of energy, which is associated to fourth-order gravitational theories, where it plays an analogous role to that of the classical ADM energy in the context of general relativity. We shall show that this quantity obeys a positive energy theorem with natural rigidity in the critical case of zero energy. Furthermore, we will comment on how the resulting notion of energy is deeply connected to Q-curvature, underlying positive energy theorems for the Paneitz operator as well as several rigidity phenomena associated to Q-curvature analysis. In particular, AE Q-singular manifolds turn out to be highly rigid as a consequence of the rigidity of our positive energy theorem put together with elliptic theory on appropriate weighted spaces. As a bypass, one can show how a certain fourth order analogue of the Ricci tensor retains optimal control of the decay of an AE metric.

Conferencia 2 (Hora - 17.00h)

Speaker: Fidel Fernandez

Title: Geometric analysis techniques for a variational problem in (pseudo-)Finsler geometry (joint with M. Á. Javaloyes and M. Sánchez)

Abstract: We will start by motivating the problem of extending general relativity by means of (Lorentz-)Finsler geometry, both from the physical and purely mathematical viewpoints. The search for analogues to the Einstein field equation leads to a natural functional which we will vary with respect to a pseudo-Finsler metric \(L\) and a nonlinear connection \(N\), thus extending the Palatini formalism. After reducing the equations to the case in which \(N\) is torsion-free, we will illustrate how (under a globality hypothesis on each \(T_pM\)) various geometric analysis tools allow one to establish properties of their solutions:

(1) In indefinite signature, a recursive argument shows that \(N\) is unique if it is required to be analytic on each \(T_pM\). Its lightlike
geodesics coincide with those of \(L\).

(2) In Lorentzian signature, the classical maximum principle implies that \(N\) must be Ricci-flat. If a certain tensor invariant of \(L\)
vanishes, the analyticity condition of (1) can be removed.

(3) For Riemannian metrics, the knowledge of the eigenvalues of the Laplacian on the standard sphere implies that \(N\) must be the
Levi-Civita connection.