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.
Minimal surfaces in H^2xR have infinite total curvature under certain conditions.
problem Understanding total curvature concentration in minimal surfaces in H^2xR.
method Analyzing stable minimal surfaces with specific asymptotic boundaries.
result Minimal surfaces in H^2xR have infinite total curvature under certain geometric conditions.
Introduce new boundary mass for asymptotically flat half-manifolds
problem Define boundary mass for asymptotically flat half-manifolds
method Introduce new boundary mass
result Define boundary mass for asymptotically flat half-manifolds
Negative-curvature surfaces uniquely define boundary distances.
problem Defining distances on curved surfaces at infinity.
method Renormalized distance calculation for negatively-curved surfaces.
result Negative-curvature surfaces are boundary distance rigid.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
Study on elasticity with mixed boundary conditions, proving spectral asymptotics.
problem Analyzing spectral asymptotics for linear elasticity with mixed boundary conditions.
method Established two-term spectral asymptotics for linear elasticity on smooth compact manifolds.
result Verification of general formulae through explicit examples in 2D and 3D.
This paper proves a normal form for cornered asymptotically hyperbolic metrics.
problem Cornered asymptotically hyperbolic metrics and their geometric properties.
method Proves a Cartan-Hadamard type theorem for the normal exponential map and constructs a normal form.
result Normal form for cornered asymptotically hyperbolic metrics.
Study asymptotic behavior of translators in hyperbolic product space.
problem Classify asymptotic boundary components of translators in H2imesR. method Inspired by earlier work on minimal and constant mean curvature surfaces, use symmetric translators as barriers.
result Prove classification of asymptotic boundary components under continuity assumptions.
The paper solves two cases of the asymptotically flat scalar-flat Yamabe problem with boundary.
problem Finding asymptotically flat scalar-flat metrics on manifolds with boundary.
method Solving elliptic PDEs with sub- and supersolutions.
result Existence of conformally equivalent asymptotically flat scalar-flat metrics.
New manifolds found without interior conjugate points.
problem Existence of interior conjugate points in hyperbolic manifolds.
method Construction of non-trapping asymptotically hyperbolic manifolds.
result Found manifolds without interior conjugate points.
Proves Penrose inequality for specific asymptotically flat manifolds.
problem Proving Penrose inequality for certain types of manifolds.
method Developed a new approximation scheme for a flow and established monotonicity of a free boundary Hawking mass.
result Proved the Riemannian Penrose inequality for specified manifolds.
In an earlier paper, we proved that, under certain hypotheses, the moduli space of an asymptotically cylindrical special Lagrangian submanifold with fixed boundary of an asymptotically cylindrical Calabi-Yau 3-fold is a smooth manifold. Here we prove the analogous result for an asymptotically cylindrical special Lagran…
Study on hyperbolic manifolds with special boundaries.
problem Characterizing geometric properties of hyperbolic manifolds with specific boundaries.
method Analyzing asymptotically hyperbolic manifolds with a horospherical boundary.
result Derived geometric formulas for the model case of horoballs and their complements.
Study on high-codimensional minimal surfaces in hyperbolic space.
problem Understanding high-codimensional minimal surfaces in hyperbolic space.
method Investigating asymptotic behavior and boundary regularity of area-minimizing currents.
result Established boundary regularity results for high-codimensional minimal surfaces near their asymptotic boundaries.
Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.
problem Constructing solutions to Bogomolny equations with given boundary and asymptotic conditions.
method Using generalized Nahm pole boundary condition and real symmetry breaking condition.
result Solutions analogous to instanton solutions, satisfying different asymptotic conditions.
Study the mass of flat 3-manifolds with boundary using specific methods.
problem Calculate the mass of asymptotically flat 3-manifolds with boundary.
method Use the method of Bray-Kazaras-Khuri-Stern to derive a mass formula.
result Derive sufficient conditions for the positivity of the mass.
Study shows X-ray transform injectivity for hyperbolic manifolds.
problem Recovering metrics from boundary measurements on hyperbolic manifolds.
method Injectivity of X-ray transform in several cases.
result Injectivity of X-ray transform proven for asymptotically hyperbolic manifolds.
Study on heat content for domains with fractal boundaries.
problem Analyzing short-time asymptotics of heat content for domains with fractal boundaries.
method Developing mathematical analysis on de Gennes' hypothesis and exploring fractal curvatures.
result Fractal curvatures and their scaling exponents may emerge in the short-time heat content asymptotics of domains with fractal boundaries.
Proves existence of curved surfaces in hyperbolic space.
problem Finding surfaces with specific curvature and boundary conditions.
method Proves existence using Weingarten curvature and asymptotic boundary conditions.
result Proves existence of locally Lipschitz continuous hypersurfaces.
In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in H2×R. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary C is a Jordan curve homologous to zero in the asymptotic boundary of H2×R, say $\partial_\infty H^2\tim…
Local X-ray transform works well near boundaries in hyperbolic spaces.
problem Injectivity of X-ray transform near boundaries for hyperbolic metrics.
method Local injectivity proof for geodesic X-ray transform on asymptotically hyperbolic manifolds.
result Local injectivity near a boundary point for X-ray transform in dimensions 3 and higher, up to O(ρ5). 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. Study calculates mass for certain hyperbolic spaces with noncompact boundaries.
problem Calculating mass for specific types of hyperbolic spaces.
method Defined a mass-type invariant for asymptotically hyperbolic manifolds with noncompact boundaries.
result Proved a sharp positive mass inequality under suitable conditions.
The paper proves positive mass theorems for initial data sets with noncompact boundaries.
problem Proving positive mass theorems for initial data sets with noncompact boundaries.
method Defining an energy-momentum vector at spatial infinity and proving positive mass inequalities under DECs.
result Proves positive mass inequalities for initial data sets with noncompact boundaries under suitable DECs.
Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.
problem Understanding the behavior of solutions to the Allen-Cahn equation on manifolds with boundary.
method Analyzing the asymptotic behavior of Dirichlet minimizers, relating Neumann data to boundary geometry, and using invertibility of the linearized Allen-Cahn operator.
result Computed expansions of the solution to high order and established a projection theorem about Allen-Cahn solutions near minimal surfaces.
Proves existence of convex surfaces with specific curvature in hyperbolic space.
problem Existence of smooth convex surfaces with prescribed curvature and boundary.
method Proves existence under strictly locally convex subsolution assumption.
result Smooth complete strictly locally convex hypersurface existence proved.
Two masses on surfaces with boundary converge to ADM mass.
problem Evaluating quasi-local masses on surfaces with boundaries.
method Hawking mass and Huisken's isoperimetric mass on surfaces with boundary, convergence to ADM mass.
result Convergence of Hawking and Huisken's masses to ADM mass.
Geodesic lines with specific boundaries found on a special type of manifold.
problem Existence of geodesic lines with prescribed asymptotic boundaries.
method Proper exponential map assumption, solution to the asymptotic Plateau problem.
result Existence of geodesic lines with Morse index ≤ n-1.
Study critical points of metrics on compact manifolds with boundary.
problem Analyzing mass functionals on specific types of manifolds.
method Examining two functionals of Riemannian metrics on compact manifolds with boundary.
result Motivated by mass functionals, the study reveals insights into critical points of metrics.
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 boundary structure of gauge fields on AdS spaces.
problem Understanding boundary conditions of gauge fields on asymptotically AdS backgrounds.
method Employing gauge PDE approach to incorporate boundary-defining function.
result Construction of efficient boundary calculus for gauge fields on AdS backgrounds.
The paper improves estimates for asymptotically hyperbolic Einstein manifolds in even dimensions.
problem Estimating the boundary regularity of asymptotically hyperbolic Einstein manifolds.
method Analyzing the (n−3)-th derivative of scalar curvature and using Hölder continuity. result The AHE metric is Cm,α conformally compact under certain conditions. Mass calculation extended to manifolds with noncompact boundaries.
problem Computing mass in asymptotically flat and hyperbolic manifolds with boundaries.
method Coordinate-free approach using Einstein and Newton tensors.
result Mass can be computed as surface integrals involving these tensors.
Proves stable smooth minimal hypersurface for trapped region boundary in large dimensional manifolds.
problem Boundary stability of trapped region in large dimensional manifolds.
method Analyzes asymptotically Euclidean Riemannian manifolds of dimension at least 3.
result Boundary is a stable smooth minimal hypersurface except for a singular set of codimension at least 8.
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.
problem Projective compactness for torsion-free linear connections on a manifold.
method Introduce and study a weakening of projective compactness for torsion-free linear connections on a manifold.
result Induces projective structure on the boundary and relates to asymptotic forms in GR.
The spectral asymptotics for linear elasticity with mixed boundary conditions are shown to be old results.
problem Analyzing the spectral asymptotics for linear elasticity with mixed boundary conditions.
method Demonstrating that the results are essentially old well-known results by other authors.
result The spectral asymptotics results for linear elasticity with mixed boundary conditions are shown to be old results by other authors.
In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the int…
The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
problem Uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
method Analyzes criteria for uniqueness and constructs examples of non-uniqueness.
result Uniqueness of minimal surfaces is equivalent to uniqueness in a smaller class of stable minimal disks.
The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.
problem Analyzing Bergman kernels on complex manifolds with boundary and their asymptotic behavior.
method Establishing asymptotic expansions of partial Bergman kernels for high-frequency Fourier modes on R-symmetric complex manifolds with boundary. result Established R-equivariant extension results for biholomorphic maps between weakly pseudoconvex domains. Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Proves density and mass theorems for specific initial data sets.
problem Initial data sets with boundary in spacetime.
method Harmonic asymptotics and dominant energy condition.
result Spacetime positive mass theorem for initial data sets with apparent horizon boundary.
Study investigates non-existence of bounded solutions on curved spaces.
problem Non-existence of bounded solutions to semi-linear elliptic equations on Cartan-Hadamard manifolds.
method Novel comparison technique using convex hypersurfaces.
result Extends previous results to curved spaces, highlighting curvature's role.
Static potentials on flat manifolds imply zero boundary values or non-compact area minimizers.
problem Characterizing static potentials on asymptotically flat manifolds.
method Analyzing the properties of static potentials and area minimizing hypersurfaces.
result Asymptotically flat manifolds with static potentials either have zero boundary values or contain non-compact area minimizers.
Researchers derive asymptotic expansions for thermoelastic operators on manifolds.
problem Determining precise geometric information from thermoelastic spectra.
method Asymptotic expansions with Dirichlet and Neumann boundary conditions.
result Explicit calculation of first two coefficients for volumes.
The study of Einstein metrics with corners and boundaries.
problem Formal study of Einstein metrics with corners and boundaries.
method Generalization and extension of previous work; formal expansion and existence demonstration.
result Existence of Einstein metrics up to a certain order in a cornered asymptotically hyperbolic normal form.
We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Ise…
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.