Research Papers

Minimal Surfaces

  1. (with Ivaldo Nunes) Rigidity of free-boundary minimal surfaces.
    In preparation.

  2. A survey of geometric inequalities for conformally flat manifolds.
    Submitted.

Geometric Flows

  1. (with Alexandre Freire) Mass-capacity inequalities for conformally flat manifolds with boundary. To appear, Comm. PDE. Available online at arXiv.org/1107.1407

    In this paper we prove a mass-capacity inequality and a volumetric Penrose inequality for conformally flat manifolds, in arbitrary dimensions. We also give new, more general proofs of the Polya-Szego and Aleksandrov-Fenchel inequalities for hypersufaces of Euclidean space. We show that these hold for mean-convex Euclidean domains in all dimensions. (The original proofs held for convex domains of 3-dimensional Euclidean space.) For each inequality, the case of equality is fully characterized.

  2. (with Esther Cabezas-Rivas) On the Shrink-wrap Principle. In preparation.

  3. Hawking mass can decrease along high-dimensional inverse mean curvature flow.
    In preparation.

Mathematical Relativity

  1. A volumetric Penrose inequality for conformally flat manifolds, Ann. Henri Poincaré 12 (2011), 67-76.

    We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to Rn-&Omega,n&ge 3, and so that their boundary is a minimal hypersurface. (Here, &Omega is an open bounded subset of Rn with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by (V/&betan)(n-2)/n, where V is the Euclidean volume of &Omega and &betan is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga. Surprisingly, we do not require the boundary to be outermost.

  2. Existence of outermost apparent horizons with product of spheres topology, Comm. Anal. Geom., 16 (2008) 799-817.

    In this paper we construct the first examples of (n+m+2)-dimensional asymptotically flat Riemannian manifolds with non-negative scalar curvature that have outermost minimal hypersurfaces with non-spherical topology for n, m ≥ 1. The outermost minimal hypersurfaces are, topologically, SnxSm+1. In the context of general relativity these hypersurfaces correspond to outermost apparent horizons of black holes.

Nonlinear PDE

  1. A note on the Yamabe constant of an outermost minimal hypersurface, Proc. Amer. Math. Soc. 138 (2010), 4103-4107.

    Using an elementary argument we find an upper bound on the Yamabe constant of the outermost minimal hypersurface of an asymptotically flat manifold with nonnegative scalar curvature that satisfies the Riemannian Penrose Inequality. Provided the manifold satisfies the Riemannian Penrose Inequality with rigidity, we show that equality holds in the inequality if and only if the manifold is the Riemannian Schwarzschild manifold.

  2. The zero scalar curvature Yamabe problem on noncompact manifolds with boundary, Indiana Univ. Math. J., 55 (2006) 1449-1459.

    Let (Mn,g), n ≥ 3 be a noncompact complete Riemannian manifold with compact boundary and f a smooth function on ∂M. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of g that is complete, has zero scalar curvature on M and has mean curvature f on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.

Topology

  1. Monotonicity of the Yamabe invariant under connect sum over the boundary, Ann. Global Anal. Geom., 35 (2009) 115-131.

    We show that the Yamabe invariant of manifolds with boundary satisfies a monotonicity property with respect to connected sums along the boundary, similar to the one in the closed case. A consequence of our result is that handlebodies have maximal invariant.

Symbolic Dynamics

  1. (with P. Dartnell and A. Maass) Combinatorial constructions associated to the dynamics of one-sided cellular automata, Theoret. Comput. Sci., 304 (2003) 485-497.

    In this paper we study combinatorial constructions which lead to one-sided invertible cellular automata with different dynamical behavior: equicontinuity, existence of equicontinuous points but not equicontinuity, sensitivity and expansivity. In particular, we provide a simple characterization of the class of equicontinuous invertible one-sided cellular automata and we construct families of expansive one-sided cellular automata.