Study asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
problem Asymptotics of Poisson kernel and Green's functions for fractional conformal Laplacian.
method Sharp expansions derived for the Poisson kernel and Green's functions near singularities.
result Sharp expansions of the Green's functions solve the first part of Kim-Musso-Wei's conjecture.
Study short-time existence of conformal Ricci flow on hyperbolic manifolds.
problem Short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds.
method Proved local Shi's type curvature derivative estimate for conformal Ricci flow.
result Short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds.
Study characterizes conformal boundaries of de Sitter spacetimes.
problem Characterize conformal infinity of asymptotically de Sitter spacetimes.
method Derive constraints relating stress-energy tensor to conformal geometric data using higher conformal fundamental forms.
result Constraints on stress-energy tensor relate to conformal geometric data.
Study computes Cheeger constants for specific submanifolds in asymptotically hyperbolic spaces.
problem Computing Cheeger constants for conformally compact asymptotically constant mean curvature submanifolds.
method Analyzes conformally compact asymptotically constant mean curvature submanifolds in asymptotically hyperbolic spaces.
result Identifies conditions for Cheeger constant equality and vanishing mean curvature.
Boundary distances determine conformal metrics
problem Determining conformal metrics from boundary distances
method Comparing renormalized boundary distances
result Metrics are equal if distances match
Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.
problem Analyzing geodesics in conformally compact manifolds with varying curvature.
method Examining asymptotic behavior and regularity of geodesics near boundary.
result Non-trapped geodesics extend to conformal infinity with C1,α regularity, endpoints smooth on initial conditions. New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
Researchers describe the mass of conformal differential operators in terms of their asymptotic expansions.
problem Understanding the mass of conformal differential operators and its invariance under conformal transformations.
method Explicit description of the full asymptotic expansion of the Schwartz kernel of complex powers of m-Laplace type operators. result The mass of conformal differential operators is a conformal invariant in odd dimensions when the kernel is trivial.
Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.
problem Obstructing higher conformal Yang-Mills equations on conformally compact manifolds.
method Formal asymptotics, Dirichlet-to-Neumann maps, higher transverse derivative boundary operators.
result Obstructing current is the variation of a conformally invariant coefficient in the interior Yang-Mills energy expansion.
Develops calculus for conformal hypersurfaces and new Willmore energy functionals.
problem Invariant theory for conformal hypersurfaces.
method Solving singular Yamabe problem, developing calculus of differential operators, computing asymptotics.
result New higher Willmore energy functionals for embedded surfaces.
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We…
Generalizes Simon's theorem to spacetime and hyperbolic manifolds.
problem Proving uniqueness of black holes in various spacetime settings.
method Extending Simon's conformal positive mass theorem to spacetime and hyperbolic manifolds.
result Proves a conformal positive mass theorem on asymptotically hyperbolic manifolds.
Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.
problem Determining the metric on an AH manifold from scattering data.
method Relating eigenvalue problem to Conformal Laplacian and using Guillarmou--Guillopé and Chang--González results.
result Scattering matrix at energy 2n+1 determines jet of the metric on the boundary up to diffeomorphism and conformal factor. Study weakly asymptotically hyperbolic geometries with curvature approaching -1.
problem Investigate geometries with curvature approaching -1 but not necessarily smooth.
method Analyze curvature invariants and establish Fredholm results for geometric elliptic operators.
result Identify an obstruction to higher order decay of curvature invariants.
Study finds necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
problem Understanding the asymptotic behavior of black hole geometries.
method Used hidden symmetry and conformal geometry technology to find necessary conditions.
result Necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
Researchers use conformal infinity to study spacetimes near AdS2×S2.
problem Studying the rigidity of asymptotically AdS2×S2 spacetimes.
method Developed a new approach based on conformal infinity.
result Obtained new results including similar to [4] but for higher dimensions and more than two ends.
Solves Dirichlet problem for harmonic maps to give geodesic insights.
problem Asymptotic Dirichlet problem for harmonic maps.
method Holographic characterization using conformal geodesics.
result Characterizes conformal geodesics on the boundary.
The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.
problem Determining if a spacetime is conformally AdS based on null geodesic travel times.
method Analyzing all null geodesics from a point to its antipodal point, considering various spacetime conditions.
result The spacetime is conformally AdS if and only if all null geodesics from a point refocus at its antipodal point.
We derive necessary conditions for the spinorial Witten-Nester energy to be well-defined for asymptotically locally AdS spacetimes. We find that the conformal boundary should admit a spinor satisfying certain differential conditions and in odd dimensions the boundary metric should be conformally Einstein. We show that …
Researchers find Fenchel-Nielsen coordinates for asymptotically conformal maps on hyperbolic surfaces.
problem Parametrizing the space of asymptotically conformal maps on hyperbolic surfaces.
method Using Fenchel-Nielsen coordinates, the researchers find parametrizations of the little Teichmüller space and its closure in the length spectrum metric.
result The quotients of Teichmüller spaces are contractible, and Wolpert's lemma on geodesic lengths is not sharp.
In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null…
New tractor geometry derived from asymptotically flat spacetimes.
problem Understanding the geometry of spacetimes near their boundaries.
method Derived null-tractor bundle from interior spacetime geometry, proved connections' uniqueness, and expressed results in BMS coordinates.
result Tractor connection encodes mass and angular momentum in 3D, and asymptotic shear in higher dimensions.
The paper proves a new theorem linking mass and electric charge for certain types of manifolds.
problem Proving a new positive mass theorem for manifolds with charge.
method Using conformal relations and scalar curvature, the authors derive a new theorem.
result The sum of mass is not less than the modulus of total electric charge under certain conditions.
Study shows mass-capacity inequality for specific geometric manifolds.
problem Establishing mass-capacity inequality for certain geometric manifolds.
method Using conformally flat manifolds with nonnegative scalar curvature.
result Equality implies harmonically conformal to a specific subset of Euclidean space.
The hyperbolic positive energy theorem links causal properties to energy-momentum vectors in asymptotically hyperbolic spaces.
problem Establishing the causal-future-directed character of energy-momentum vectors in hyperbolic spaces.
method Analyzing n-dimensional asymptotically hyperbolic Riemannian manifolds with spherical conformal infinity, focusing on the dominant energy condition. result The causal-future-directed character of the energy-momentum vector can be traced back to that of asymptotically Euclidean initial data sets.
New bounds on efficiency for conformalized regression methods.
problem Efficiency of conformal prediction in regression models.
method Non-asymptotic bounds on prediction set length for conformalized quantile and median regression.
result Identifies phase transitions in convergence rates across different regimes of miscoverage level.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…
Study Cheeger constant and Yamabe type for ALH manifolds.
problem Understanding the Cheeger constant and Yamabe type of ALH manifolds.
method Analyzes the Cheeger constant and Yamabe type of asymptotically locally hyperbolic manifolds with conformal compactification.
result Establishes a relationship between the Cheeger constant and the Yamabe type of the conformal infinity.
Conformally compact asymptotically hyperbolic metrics have been intensively studied. The goal of this note is to understand what intrinsic conditions on a complete Riemannian manifold (M,g) will ensure that g is asymptotically hyperbolic in this sense. We use the geodesic compactification by asymptotic geodesic rays to…
An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetri…
New formula for volume in 4D hyperbolic manifolds.
problem Volume calculation for specific geometric manifolds.
method Derive new renormalized volume formula.
result Generalizes existing formulas for Poincare-Einstein manifolds.
The main purpose of this monograph is to give an elementary and self-contained account of the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinities sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. The proof is based on an …
The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum gravity. In work of Andersson, Chrusciel, and Friedrich, the same invariant arises as…
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.
The paper solves fractional scalar curvature problems on conformal infinities.
problem Prescribed fractional scalar curvature on conformal infinities.
method Introduced and solved the fractional scalar curvature problem on conformal infinities.
result Existence of smooth solutions to the fractional Yamabe problem in the endpoint case.
Proves conditions for solving Einstein constraint equations on Euclidean manifolds.
problem Finding solutions to Einstein constraint equations on asymptotically Euclidean manifolds.
method Conformal method, Lichnerowicz equation, global supersolutions, limit equation criterion.
result Characterizes the Yamabe classes and solves the prescribed scalar curvature problem for nonpositive scalar curvatures.
The paper examines the smoothness of hyperbolic metrics near boundaries.
problem Analyzing the regularity of asymptotically hyperbolic metrics near boundaries.
method Following Michael Anderson's method, the paper studies Cm,α conformally compact Riemannian metrics with Einstein equation. result The conformal compactifications of these metrics are Cm+2,α up to the boundary when Weyl curvature is in Cm,α and the boundary metric is in Cm+2,α. Sharp eigenvalue estimates for submanifolds of asymptotically hyperbolic spaces.
problem Estimating eigenvalues of the p-Laplacian on submanifolds of asymptotically hyperbolic manifolds.
method Sharp upper and lower bounds derived using conformal techniques and properties of submanifolds.
result Lower bounds on the first eigenvalue for minimal and bounded mean curvature submanifolds.
Constructs constant-mean-curvature hyperboloidal data sets satisfying the shear-free condition.
problem Ensuring spacetime development has a regular conformal boundary at future null infinity.
method Using the conformal method and traceless Hessian, parametrize and construct initial data sets.
result Constructs constant-mean-curvature hyperboloidal initial data sets satisfying the shear-free condition.
In the first part of this article we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version of the positive mass theorem on conformal spin manifolds.
We present a set of global invariants, called "mass integrals", which can be defined for a large class of asymptotically hyperbolic Riemannian manifolds. When the "boundary at infinity" has spherical topology one single invariant is obtained, called the mass; we show positivity thereof. We apply the definition to confo…
Study of 0-instantons on hyperbolic manifolds, proving invariants and energy formulas.
problem Understanding 0-instantons on hyperbolic manifolds and their properties.
method Analyzing asymptotic expansions and using Fefferman-Graham expansion for Poincaré-Einstein metrics.
result The 0-instanton obstruction tensor is a conformal invariant related to Weyl curvature, vanishing for smooth 0-instantons.
In this paper, we study short-time existence of static flow on complete noncompact asymptotically static manifolds from the point of view that the stationary points of the evolution equations can be interpreted as static solutions of the Einstein vacuum equations with negative cosmological constant. For a static vacuum…
We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.
The paper characterizes ambient metrics using conformal completion and null infinity properties.
problem Characterizing ambient metrics from a conformal perspective.
method Proving conformal completion and analyzing null infinity properties.
result Identifying conformally covariant conditions to characterize ambient metrics.
In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold (Mn+1,g), with a pole p and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.
In this paper we study the extent to which conformally compact asymptotically hyperbolic metrics may be characterized intrinsically. Building on the work of the first author, we prove that decay of sectional curvature to -1 and decay of covariant derivatives of curvature outside an appropriate compact set yield Hölder …
Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.
problem Boundary behavior of the singular Yamabe problem near singular boundaries.
method Analysis of asymptotic behaviors and derivation of optimal estimates for background metrics.
result Solutions are well approximated by solutions in tangent cones at singular points.