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Search Results: 1 - 10 of 186090 matches for " Levi Lopes de Lima "
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A Feynman-Kac formula for differential forms on manifolds with boundary and applications
Levi Lopes de Lima
Mathematics , 2015,
Abstract: We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct $L^2$ harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity property expressed in terms of a certain stochastic average of the Weitzenb\"ock operator $R_p$ acting on $p$-forms and the second fundamental form of the boundary. This extends previous work by Elworthy-Li-Rosenberg on closed manifolds to this setting. As an application we find a geometric obstruction to the existence of metrics with 2-convex boundary and positive $R_2$ in this stochastic sense. We also discuss a version of the Feynman-Kac formula for spinors under suitable boundary conditions.
Infinite connected sums, K-area and positive scalar curvature
Levi Lopes de Lima
Mathematics , 2004,
Abstract: Recently, Whyte used the index theory of Dirac operators and the Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a generalization of the $\hat{A}$-class to obstruct such metrics. In this note we prove a variant of Whyte's result where infinite $K$-area in the sense of Gromov is used to obstruct metrics with positive scalar curvature.
The ADM mass of asymptotically flat hypersurfaces
Levi Lopes de Lima,Frederico Gir?o
Mathematics , 2011,
Abstract: We provide integral formulae for the ADM mass of asymptotically flat hypersurfaces in Riemannian manifolds with a certain warped product structure in a neighborhood of infinity, thus extending Lam's recent results on Euclidean graphs to this broader context. As applications we exhibit, in any dimension, new classes of manifolds for which versions of the Positive Mass and Riemannian Penrose inequalities hold and discuss a notion of quasi-local mass in this setting. The proof explores a novel connection between the co-vector defining the ADM mass of a hypersurface as above and the Newton tensor associated to its shape operator, which takes place in the presence of an ambient Killing field.
An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality
Levi Lopes de Lima,Frederico Gir?o
Mathematics , 2012,
Abstract: We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. As an application we establish, in any dimension, an optimal Penrose inequality for asymptotically hyperbolic graphs carrying a minimal horizon, with the equality occurring if and only if the graph is an anti-de Sitter-Schwarzschild solution. This sharpens previous results by Dahl-Gicquaud-Sakovich and settles, for this class of initial data sets, the conjectured Penrose inequality for time-symmetric space-times with negative cosmological constant.
A Penrose inequality for asymptotically locally hyperbolic graphs
Levi Lopes de Lima,Frederico Gir?o
Mathematics , 2013,
Abstract: We use the inverse mean curvature flow to prove a sharp Alexandrov-Fenchel-type inequality for a class of hypersurfaces in certain locally hyperbolic manifolds. As an application we derive an optimal Penrose inequality for asymptotically locally hyperbolic graphs in any dimension $n\geq 3$. When the horizon has the topology of a compact surface of genus at least one, this provides an affirmative answer, for this class of initial data sets, to a question posed by Gibbons, Chru\'sciel and Simon on the validity of a Penrose-type inequality for black hole solutions carrying a higher genus horizon.
On the index of constant mean curvature 1 surfaces in hyperbolic space
Levi Lopes de Lima,Wayne Rossman
Mathematics , 2008,
Abstract: We show that the index of a constant mean curvature 1 surface in hyperbolic 3-space is completely determined by the compact Riemann surface and secondary Gauss map that represent it in Bryant's Weierstrass representation. We give three applications of this observation. Firstly, it allows us to explicitly compute the index of the catenoid cousins and some other examples. Secondly, it allows us to be able to apply a method similar to that of Choe (using Killing vector fields on minimal surfaces in Euclidean 3-space) to our case as well, resulting in lower bounds of index for other examples. And thirdly, it allows us to give a more direct proof of the result by do Carmo and Silveira that if a constant mean curvature 1 surface in hyperbolic 3-space has finite total curvature, then it has finite index. Finally, we show that for any constant mean curvature 1 surface in hyperbolic 3-space that has been constructed via a correspondence to a minimal surface in Euclidean 3-space, we can take advantage of this correspondence to find a lower bound for its index.
A rigidity result for the graph case of the Penrose inequality
Levi Lopes de Lima,Frederico Gir?o
Mathematics , 2012,
Abstract: In this note we prove a global rigidity result for asymptotically flat, scalar flat Euclidean hypersurfaces with a minimal horizon lying in a hyperplane, under a natural ellipticity condition. As a consequence we obtain, in the context of the Riemannian Penrose conjecture, a local rigidity result for the family of exterior Schwarzschild solutions (viewed as graphs in Euclidean space).
CMC-1 Surfaces in Hyperbolic 3-space using the Bianchi-Calo method
Levi Lopes de Lima,Pedro Roitman
Mathematics , 2001,
Abstract: In this note we present a method for constructing CMC-1 surfaces in hyperbolic 3-space $\bfH^3(-1)$ in terms of holomorphic data first introduced in Bianchi's Lezioni di Geometria Differenziale of 1927, therefore predating by many years the modern approaches due to Bryant, Small and others. Besides its obvious historical interest, this note aims to complement Bianchi's analysis by deriving explicit formulae for CMC-1 surfaces and comparing the various approaches encountered in the literature.
Deformations of 2k-Einstein structures
Levi Lopes de Lima,Newton Luis Santos
Mathematics , 2010, DOI: 10.1016/j.geomphys.2009.12.017
Abstract: It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.
The Yamabe problem for Gauss-Bonnet curvatures: a local result around space forms
Levi Lopes de Lima,Newton Luis Santos
Mathematics , 2010,
Abstract: It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k
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