Marcelo M. Disconzi

On the Penrose inequality for charged black holes

Bray and Khuri (2011 Asian J. Math. 15 557–610; 2010 Discrete Continuous Dyn. Syst. A 27 741766) outlined an approach to prove the Penrose inequality for general initial data sets of the Einstein equations. In this paper we extend this approach so that it may be applied to a charged version of the Penrose inequality. Moreover, assuming that the initial data are time-symmetric, we prove the rigidity statement in the case of equality for the charged Penrose inequality, a result which seems to be absent from the literature. A new quasi-local mass, tailored to charged initial data sets is also introduced, and used in the proof.

On the Limit of Large Surface Tension for a Fluid Motion with Free Boundary

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary has constant curvature, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

Some remarks on uniformly regular Riemannian manifolds

We establish the equivalence between the family of closed uniformly regular Rie- mannian manifolds and the class of complete manifolds with bounded geometry. In 2012, H. Amann introduced a class of (possibly noncompact) manifolds, called uniformly reg- ular Riemannian manifolds. Roughly speaking, an m-dimensional Riemannian manifold (M,g) is uniformly regular if its differentiable structure is induced by an atlas such that all its local patches are of approximately the same size, all derivatives of the transition maps are bounded, and the pull-back metric of g in every local coordinate is comparable to the Euclidean metric gm. The precise definition of uniformly regular Riemannian manifolds will be presented in Section 2 below. In this article, the concept of closed manifolds refers to manifolds without boundary, not necessar- ily compact. The main objective of this short note is to prove that the family of closed uniformly regular Riemannian manifolds coincides with the class of complete manifolds with bounded ge- ometry in the following sense. A closed uniformly regular Riemannian manifold is geodesically complete, has positive injectivity radius, and all covariant derivatives of the curvature tensor are bounded. The second and third conditions are usually referred to as of bounded geometry. The precise definitions of positive injectivity radius and bounded geometry will be given later in this introductory section.

The free boundary Euler equations with large surface tension

Abstract We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.

On the well-posedness of relativistic viscous fluids with non-zero vorticity

We study the problem of coupling Einsteins equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given an appropriate relativistic interpretation, and show that the solutions enjoy the finite propagation speed property.

Remarks on Positive Energy Vacua via Effective Potentials in String Theory

We study warped compactifications of string/M theory with the help of effective potentials, continuing previous work of the last two authors and Michael R. Douglas presented in (On the boundedness of effective potentials arising from string compactifications. Communications in Mathematical Physics 325(3):847–878, 2014). Under physically reasonable assumptions, we provide a mathematically rigorous proof of the existence of positive local minima of a large class of effective potentials. The dynamics of the conformal factor of the internal metric, which is responsible for instabilities in these constructions, is explored, and such instabilities are investigated in the context of de Sitter vacua. We prove existence results for the equations of motion in the case of a slowly varying warp factor, and the stability of such solutions is also addressed. These solutions are a family of meta-stable de Sitter vacua from type IIB string theory in a general non-supersymmetric setup.

MOTION OF SLIGHTLY COMPRESSIBLE FLUIDS IN A BOUNDED DOMAIN. II

We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is near the analogous solution for an incompressible fluid provided the initial conditions for the two problems are close. In particular, the divergence of the initial velocity of the compressible flow at time zero is assumed to be small. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for this type of non-linear equations.

On the Choquet-Bruhat–York–Friedrich formulation of the Einstein–Euler equations

Short-time existence for the Einstein-Euler and the vacuum Einstein equations is proven using a Friedrich inspired formulation due to Choquet-Bruhat and York, where the system is cast into a symmetric hyperbolic form and the Riemann tensor is treated as one of the fundamental unknowns of the problem. The reduced system of Choquet-Bruhat and York, along with the preservation of the gauge, is shown to imply the full Einstein equations.

On the Boundedness of Effective Potentials Arising from String Compactifications

We study effective potentials coming from compactifications of string theory. We show that, under mild assumptions, such potentials are bounded from below in four dimensions, giving an affirmative answer to a conjecture proposed by the second author in Douglas (JHEP 3:71, 2010). We also derive some sufficient conditions for the existence of critical points. All proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem.

A NOTE ON QUANTIZATION IN THE PRESENCE OF GRAVITATIONAL SHOCK WAVES

We study the quantization of a free scalar field when the background metric satisfies Einsteins equations and develops gravitational shock waves.