# Third Minho Meeting on

Mathematical Physics

### Universidade
do Minho, Braga, Portugal

November 4 2011

### Abstracts

**Simone Calogero (University of Granada, Spain)**

Relativistic diffusion

In this talk I will present two models, a kinetic one and a fluid one, to describe diffusion in general relativity. I will show that diffusion changes drastically the properties of even the simplest cosmological models, namely the flat Robertson-Walker spacetimes.

**João Costa (ISCTE-Lisbon University Institute and Instituto Superior Técnico, Lisbon Portugal)**

Towards the Einstein-scalar field system with positive cosmological constant

With the Einstein-scalar field equations with positive cosmological constant in mind, we employ Christodoulou's framework, developed to study the vanishing cosmological constant case, to spherically symmetric solutions of the linear wave equation in de Sitter spacetime and derive expected properties: boundedness in terms of (characteristic) initial data, and a Price law establishing pointwise exponential decay to a constant. We will also discuss the relation between the linear case and the full non-linear Einstein-scalar field system.

**Anna Fino (University of Torino, Italy)**

Special Hermitian Structures and Symplectic Geometry

Symplectic forms taming complex structures on compact manifolds are strictly related to a special type of Hermitian metrics, known in the literature as "strong Kaehler with torsion" metrics. I will present general results on "strong Kahler with torsion" metrics, their link with symplectic geometry and more in general with generalized complex gometry. Moreover, I will show for certain 4-dimensional non-Kaehler symplectic 4-manifolds some recent results about the Calabi-Yau equation in the context of symplectic geometry.

**José Natário (Instituto Superior Técnico, Lisbon, Portugal)**

Cosmic censorship in spherical symmetry

We will review the existing results concerning the weak and strong cosmic censorship conjectures for spherically symmetric solutions of the Einstein field equations. Previous knowledge General Relativity, although desirable, is not assumed.

**Valeria Ricci (University of Palermo, Italy)**

Particle models for kinetic equations: an introduction and some rigorous results

We shall give an introduction to the validity problem for kinetic equations and we shall review some convergence theorems concerning the derivation from a microscopic dynamics of systems of partial differential equations describing, at the mesoscopic scale, collections of particles interacting through various (collisions and mean field) type of interaction.

**Isabel Salavessa (Instituto Superior Técnico, Lisbon, Portugal)**

Stability of submanifolds with parallel mean curvature
in calibrated geometry

It is well known that m-spheres are the unique smooth solutions for the isoperimetric problem in
R^{m+1}. This can be proved by showing that spheres are the unique stable hypersurfaces with constant
mean curvature for the Area functional, acting on hypersurfaces with a fixed enclosed volume.
This was proved by Barbosa and do Carmo (1980) and extended to geodesic spheres in space forms
in a jonit work of the same autors with Eschenburg (1988). We show how to extend this variational
problem to m-submanifolds in an (m+n)-dimensional Riemannian manifold N possessing a semicalibration
Omega of rank (m+1), by defining an enclosed Omega-volume for one-parameter variations.
The Jacobi operator arising from the second variation is now the usual one with an extra term,
a first-order differential operator depending on the calibration, conditioning the stability.
Geodesic spheres of a totally geodesic fibre of fibrations of R^{m+n} and of H^{m+n} are shown to be stable
for the calibration-fibration, but unstable for the Hopf fibrations of S^{m+n}. If
N= R^{m+n} necessary and sufficient conditions are given on the calibration for m-spheres
to be the unique stable solutions. We study the case of 2-spheres S2 in R7 with the associative
3-calibration coming from the G2-structure, and related to this variational problem,
we prove using spectral theory of S2, that some integral Cauchy-Riemann type inequalities
hold for pairs of functions on S2. We extend such inequality to 4-tuples of functions and
show to be valid only on the L2-complement of a non-zero finite dimensional space of functions,
what proves the instability of S2 . Some other examples coming from special geometries will
be studied. Finally, we propose a forced mean curvature flow related to this problem.

References:

(IS) Bull. Braz. Math .Soc. NS (2010) 41(4), 495-530

(IS) Cauchy-Riemann inequalities on 2-spheres of R^{7} (2011) preprint arXiv:1105.3153