The Øresund Seminar on Analysis, PDE & Related Topics

Abstract: 

In 1978, Elliot Lieb proved the Wehrl entropy conjecture for Glauber coherent states. An analogous result holds for Bloch states, where coherent states correspond to the reproducing kernels of the space of holomorphic polynomials in one complex variable of bounded degree. Coherent states do not just optimize the Wehrl entropy, but also the concentration, understood as the norm restricted to a given geometric set. Moreover, it has been proved that they are the unique optimizers of both quantities. 

In this talk, we will focus on the stability of the previous extremal properties. Namely, if the concentration is close to the maximal one among all sets of a given measure, we will quantify how close the set and the polynomial are to a disc and to a coherent state, respectively.

This is a joint work with Joaquim Ortega-Cerdà (UB-CRM). 

Abstract: 

The talk will present some recent results about the free boundary problem for an incompressible, irrotational liquid drop of nearly spherical shape with capillarity. The first part of the talk will deal with the extension of some classical results from the flat case to the spherical geometry, including the reduction to a problem on the boundary, the Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both by a differential geometry approach and with the method of the 'good unknown of Alinhac'. The second part of the talk will deal with the bifurcation of travelling waves, which are nontrivial fixed profiles rotating with constant angular velocity. Bifurcation from both simple and multiple eigenvalues is considered, where the Hamiltonian and variational structure of the problem, together with its conserved quantities, play a key role.

From works in collaboration with Vesa Julin, Domenico Angelo La Manna and Giuseppe La Scala.

Abstract: 

The classification of stationary black hole solutions of the Einstein field equations, broadly referred to as the "no-hair conjecture", is a challenging and fundamental line of research in general relativity. The problem is more tractable for black hole spacetimes which are static, but even under this stronger assumption the existing results are mostly limited to static black holes with zero or positive cosmological constant. In this talk, I will present a geometric inequality for isolated static vacuum black holes with a negative cosmological constant which has far-reaching implications for their geometry and uniqueness.

The inequality relates the surface gravity, area, and topology of a horizon in a static spacetime to its conformal infinity, and equality is achieved only by the Kottler black holes. From this, we deduce several new static uniqueness theorems for Kottler. Namely, we show: (1) the Kottler black hole over the sphere which minimizes surface gravity is unique, (2) the Kottler black hole over the torus is unique, assuming the horizons have non-spherical topology, and (3) uniqueness for the higher-genus Kottler black holes is equivalent to the Riemannian Penrose inequality. This is based on joint work with Ye-Kai Wang.

Abstract: 

For a dynamical system, modelled by the iterations of a mapping T:X→X, an invariant measure is a measure μ on X such that the measure of a set is the same as the measure of the pre-image under T of the set. For such measures, the distribution of a typical orbit is exactly the measure μ.

It turns out that for many such dynamical systems, the orbit visits a small ball approximately according to a Poisson process (if things are scaled appropriately). Results of this type are known at least since Pitskel' (1991).

I will explain how such results can be proved and also mention a new variant of such results that I have recently obtained in cooperation with Maxim Kirsebom and Philipp Kunde.