arXiv subjects
Publications and source records attributed to Levi Lopes de Lima.
On an asymptotically Schwarzschild smooth metric measure space modelled on the Euclidean half-space, we define a mass-type quantity similar to the classical ADM mass in General Relativity and show its geometric invariance properties. Such quantity has a close relation with the fractional Yamabe problem and the relevant Green's function.
We begin our journey by recalling the fundamentals of Probability Theory that underlie one of its most significant applications to real-world problems: Parametric Estimation. Throughout the text, we systematically develop this theme by presenting and discussing the main tools it encompasses (concentration inequalities, limit theorems, confidence intervals, maximum likelihood, least squares, and hypothesis testing) always with an eye toward both their theoretical underpinnings and practical relevance. While our approach follows the broad contours of conventional expositions, we depart from tradition by consistently exploring the geometric aspects of probability, particularly the invariance properties of normally distributed random vectors. This geometric perspective is taken further in an extended appendix, where we introduce the rudiments of Brownian motion and the corresponding stochastic calculus, culminating in Itô's celebrated change-of-variables formula. To highlight its scope and elegance, we present some of its most striking applications: the sharp Gaussian concentration inequality (a central example of the "concentration of measure phenomenon"), the Feynman-Kac formula (used to derive a path integral representation for the Laplacian heat kernel), and, as a concluding delicacy, the Black-Scholes strategy in Finance.
We start by revisiting the derivation of the variational formulae for the functional assigning to a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains in certain weighted manifolds. This is further extended to other functionals defined by certain Dirichlet energy integrals, with a Morse index formula for the corresponding critical domains being established. We complement these infinitesimal results by proving a couple of global rigidity theorems for (possibly critical) domains in Gaussian half-space, including an Alexandrov-type soap bubble theorem. Although we provide direct proofs of these latter results, we find it worthwhile to point out that the main tools employed (specifically, certain Pohozhaev and Reilly identities) can be formally understood as limits (when the dimension goes to infinity) of tools previously established by Ciarolo-Vezzoni and Qiu-Xia to handle similar problems in round hemispheres, with the notion of "convergence" of weighted manifolds being loosely inspired by the celebrated Poincaré's limit theorem in the theory of Gaussian random vectors.
We study the deformation theory of Einstein-Yang-Mills fields over conformally compact, asymptotically locally hyperbolic manifolds. We prove that if an Einstein-Yang-Mills field $(g_0,ω_0)$ is trivial (which means that $g_0$ is Poincaré-Einstein and $ω_0$ is a flat connection on a principal bundle over the underlying manifold) and non-degenerate in the appropriate sense then any sufficiently small perturbation of its boundary data at infinity may be realized as the boundary data of some Einstein-Yang-Mills field. This result is obtained as an application of the $0$-calculus of Mazzeo and Melrose and may be viewed as a natural extension of previous results by Graham-Lee, Lee and Usula.
We provide a harmonic level set proof of the positive mass theorem for asymptotically flat $3$-manifolds with a non-compact boundary first established by Almaraz-Barbosa-de Lima.
We study the scalar curvature of incomplete wedge metrics in certain stratified spaces with a single singular stratum (wedge spaces). Building upon several well established technical tools for this category of spaces (the corresponding Yamabe, elliptic and index theories) we provide existence and obstruction results for such metrics under suitable positivity assumptions on the underlying geometry. This is meant to be a follow-up to a previous paper of ours (AGAG, 2022), where the case of spaces with an isolated conical singularity was considered.
We single out a notion of staticity which applies to any domain in hyperbolic space whose boundary is a non-compact totally umbilical hypersurface. For (time-symmetric) initial data sets modeled at infinity on any of these latter examples, we formulate and prove a positive mass theorem in the spin category under natural dominant energy conditions (both in the interior and along the boundary) whose rigidity statement retrieves, among other things, a sharper version of a recent result by Souam to the effect that no such hypersurface admits a compactly supported deformation keeping the original lower bound on the mean curvature. A key ingredient in our approach is the consideration of a family of elliptic boundary conditions on spinors interpolating between chirality and MIT bag boundary conditions.
In this largely expository note, we discuss the mapping properties of the Laplacian (and other geometric elliptic operators) in spaces with an isolated conical singularity following the approach developed by B.-W. Schulze and collaborators. Our presentation aims at illustrating the versatility of these results by describing how certain representative (and seemingly disparate) applications in Geometric Analysis follow from a common setup.
We first show that existence results due to Kazdan-Warner and Cruz-Vitório can be extended to the category of manifolds with an isolated conical singularity. More precisely, we check that, under suitable conditions on the link manifold, any bounded and smooth function which is negative somewhere is the scalar curvature of some conical metric (with the boundary being minimal whenever it is non-empty). By way of comparison, we complement this analysis by indicating how index theory, as developed by Albin-Gell-Redman, may be used to transfer to this conical setting some of the classical obstructions to the existence of metrics with positive scalar curvature in the spin context. In particular, we use a version of the notion of infinite $K$-area to obstruct such metrics.
We define a notion of extrinsic black hole in pure Lovelock gravity of degree $k$ which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null $2k$-mean curvature in Euclidean space $\mathbb R^{n+1}$, $2\leq 2k\leq n-1$. We then combine a regularity argument with a rigidity result by Araújo-Leite to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge-Wang-Wu, a local rigidity result for the Lovelock-Schwarzschild solutions.
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).
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, Chruściel and Simon on the validity of a Penrose-type inequality for black hole solutions carrying a higher genus horizon.
We prove a sharp Alexandrov-Fenchel-type inequality for star-shaped, strictly mean convex hypersurfaces in hyperbolic $n$-space, $n\geq 3$. The argument uses two new monotone quantities along the inverse mean curvature flow. 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. We also explain how our methods can be easily adapted to 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, Chrusciel and Simon on the validity of a Penrose-type inequality for exotic black holes.
Whyte used the index theory of Dirac operators and 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 coarse version of the $\hat{A}$-class to obstruct such metrics. In this note we prove a version of Whyte's result where a variant of the notion of infinite $K$-area, originally due to Gromov, is used to obstruct metrics with positive scalar curvature.
It is well-known that considerations of symmetry lead to the definition of a host of conserved quantities (energy, linear momentum, center of mass, etc.) for an asymptotically flat initial data set, and a great deal of progress in Mathematical Relativity in recent decades essentially amounts to establishing fundamental properties for such quantities (positive mass theorems, Penrose inequalities, geometric representation of the center of mass by means of isoperimetric foliations at infinity, etc.) under suitable energy conditions. In this article I first review certain aspects of this classical theory and then describe how they can be (partially) extended to the setting in which the initial data set carries a non-compact boundary. In this case, lower bounds for the scalar curvature in the interior and for the mean curvature along the boundary both play a key role. Our presentation aims to highlight various rigidity/flexibility phenomena coming from the validity, or lack thereof, of the corresponding positive mass theorems and/or Penrose inequalities.
We revisit the interplay between the mass, the center of mass and the large scale behavior of certain isoperimetric quotients in the setting of asymptotically flat $3$-manifolds (both without and with a non-compact boundary). In the boundaryless case, we first check that the isoperimetric deficits involving the total mean curvature recover the ADM mass in the asymptotic limit, thus extending a classical result due to G. Huisken. Next, under a Schwarzschild asymptotics and assuming that the mass is positive we indicate how the implicit function method pioneered by R. Ye and refined by L.-H. Huang may be adapted to establish the existence of a foliation of a neighborhood of infinity satisfying the corresponding curvature conditions. Recovering the mass as the asymptotic limit of the corresponding relative isoperimetric deficit also holds true in the presence of a non-compact boundary, where we additionally obtain, again under a Schwarzschild asymptotics, a foliation at infinity by free boundary constant mean curvature hemispheres, which are shown to be the unique relative isoperimetric surfaces for all sufficiently large enclosed volume, thus extending to this setting a celebrated result by M. Eichmair and J. Metzger. Also, in each case treated here we relate the geometric center of the foliation to the center of mass of the manifold as defined by Hamiltonian methods.
In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is lightlike. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.
We define a mass-type invariant for asymptotically hyperbolic manifolds with a noncompact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and has a mean convex boundary is isometric to the hyperbolic half-space.