Study on uniqueness of ad-invariant metrics in Lie algebras.
problem Uniqueness of ad-invariant metrics in Lie algebras up to automorphisms.
method Analysis of Lie algebras, cotangent Lie algebras, and specific conditions for uniqueness.
result Uniqueness of ad-invariant metric on T∗g implies solvability of g, but not conversely. Paper proves uniqueness of Type II Yamabe metrics on manifolds.
problem Uniqueness of Type II Yamabe metrics on compact manifolds.
method Investigates sufficient conditions for metric uniqueness and proves corresponding theorems.
result Establishes sufficient condition for a metric to be the unique Type II Yamabe metric.
Explain convexity of K-energy leading to unique metrics.
problem Uniqueness of constant scalar curvature Kahler metrics and extremal metrics.
method Convexity of K-energy along weak geodesics in Kahler potentials.
result Uniqueness of extremal metrics up to automorphisms.
Unique extremal Kähler metric found near a divisor.
problem Uniqueness of extremal Kähler metric near a smooth divisor.
method Analyzes Poincaré type extremal Kähler metric with cusp singularity.
result Uniqueness of extremal Kähler metric up to holomorphic transformations.
Curve shortening flow is not unique on certain metrics.
problem Non-uniqueness of curve shortening flow on specific metrics.
method Formulated a uniqueness conjecture and constructed a non-static solution.
result Curve shortening flow is not unique on a non-flat metric on the plane.
This paper studies several aspects of asymptotically hyperbolic Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness …
Proves existence and uniqueness of weighted metrics for smooth spaces.
problem Existence and uniqueness of weighted metrics for smooth metric measure spaces.
method Proves existence and uniqueness using weighted ambient metrics and Poincaré metrics.
result Existence and uniqueness of weighted metrics for smooth metric measure spaces.
The purpose of this paper is to prove the uniqueness of conical Kähler-Einstein metrics, under the condition that the twisted Ding-functional is proper. This is a generalization of the author's previous work, and we shall first investigate the uniqueness of twisted Kähler-Einstein metrics, and then use these smooth p…
Paper proves uniqueness of Einstein metrics on balls.
problem Uniqueness of conformally compact Einstein metrics on balls.
method Compactness result for conformally compact Einstein metrics.
result Global uniqueness of Graham-Lee metrics on balls.
Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.
problem Proving uniqueness of Poincaré type cscK metric with singularity at a smooth divisor.
method Holomorphic transformations, asymptotic behavior analysis, fixed point problem.
result Unique Poincaré type cscK metric with singularity at a smooth divisor is unique up to holomorphic transformations.
We prove that generically (positive) Yamabe metrics are unique in their conformal class, and describe some sufficient conditions which imply that a Yamabe metric of locally maximal scalar curvature is an Einstein metric.
Uniqueness of weighted extremal metrics on Kähler manifolds proven.
problem Uniqueness of weighted extremal Kähler metrics on compact Kähler manifolds.
method Proof of uniqueness using modified Mabuchi energy and weighted K-semistability.
result Uniqueness of weighted extremal Kähler metrics up to automorphisms.
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
Graph Laplace operators uniquely identify metrics and densities on manifolds.
problem Identifying Riemannian metrics and sampling densities from graph Laplace operators.
method Analyzing intrinsic and extrinsic graph Laplace operators on compact Riemannian manifolds.
result Graph Laplace operators uniquely determine metrics and densities under certain conditions.
Proves uniqueness of barycenters on manifolds without restrictions.
problem Finding unique barycenters on complex geometric spaces.
method Introduces new disintegrated Monge-Kantorovich metrics for barycenter problems.
result Uniqueness of barycenters on connected, complete Riemannian manifolds.
We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.
Geodesics on polygons in a unit disk are studied with unique metric properties.
problem Characterizing geodesics on polygons within a unit disk.
method Defining a metric on polygons such that geodesics are curves in the family C.
result The constructed metric space is not isometric to any convex domain in R^2.
Unique tangent cones found for Kahler-Einstein metrics on singular varieties.
problem Finding unique tangent cones for Kahler-Einstein metrics on singular varieties.
method Analyzing unique Ricci flat currents with local bounded potential.
result Local tangent cones of the unique Kahler-Einstein metric are unique.
Unique circle patterns on spheres found for spherical conical metrics.
problem Non-uniqueness in circle packing for spherical metrics.
method Prescribed geodesic total curvature instead of cone angles.
result Unique existence of circle patterns for spherical conical metrics.
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
Study on heat equation and eigenfunctions on RCD spaces, proving unique continuation.
problem Unique continuation for caloric functions and eigenfunctions on RCD spaces.
method Establish weak unique continuation theorem for caloric functions and eigenfunctions on compact RCD(K,2) spaces.
result Existence of non-trivial eigenfunctions and caloric solutions vanishing up to infinite order at one point.
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
We show that Wang's proof of uniqueness of Anti-de Sitter spacetime can be adapted to provide uniqueness results for strictly static asymptotically locally hyperbolic vacuum metrics with toroidal infinity, and to prove negativity of the free energy E−TS of asymptotically AdS black holes with higher-genus horizons.
Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not un…
In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.
Let S be a compact Sasakian manifold which does not admit non-trivial Hamiltonian holomorphic vector fields. If there exists an Einstein-Sasakian metric on S, then it is unique.
We study the Euler-Lagrange equation for several natural functionals defined on a conformal class of almost Hermitian metrics, whose expression involves the Lee form θ of the metric. We show that the Gauduchon metrics are the unique extremal metrics of the functional corresponding to the norm of the codifferential of…
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval (0,1]. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
Unique conformal metrics found on certain manifolds.
problem Finding unique conformal metrics with constant Q-curvature.
method Proving uniqueness on manifolds with positive scalar curvature.
result Only metrics of the form λg with λ>0 are constant Q-curvature.
Study fixed angle inverse scattering with Riemannian metrics, proving uniqueness and stability.
problem Inverse scattering problem with Riemannian metrics.
method Direct problem via progressing wave expansion; inverse problem using symmetry assumptions.
result Uniqueness and stability results for inverse scattering with Riemannian metrics.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
Unique AdS spacetime found with prescribed metric on a convex surface.
problem Finding an AdS spacetime with a specific metric on a convex surface.
method Constructing a quasifuchsian AdS spacetime with a past-convex Cauchy surface.
result Existence and uniqueness of a quasifuchsian AdS spacetime with the specified properties.
Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.
problem Initial boundary value problem for vacuum Einstein equations.
method Formulated IBVP, solved simultaneously in local harmonic coordinates, constructed unique maximal globally hyperbolic solution.
result Vacuum spacetimes satisfying fixed initial-boundary conditions and corner conditions are geometrically unique near the initial surface.
The aim of this note is to give an explicit description of quasi-Einstein metrics on Hn×R. We shall construct two examples of quasi-Einstein metrics on this manifold and then we shall prove the uniqueness of these examples. Finally, we shall describe the closed relation between quasi-Einstein met…
Researchers prove spectral uniqueness of complex/quaternionic structures on manifolds.
problem Spectral uniqueness of complex/quaternionic structures on manifolds.
method Explicit expression for smallest positive eigenvalue of Laplace-Beltrami operator.
result Irreducible symmetric spaces are spectrally unique within families of homogeneous metrics.
Let E→M be a holomorphic vector bundle over a compact Kaehler manifold (M,ω). We prove that if E admits a ω-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of ω-balanced metrics of certain dir…
Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.
problem Existence and uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds.
method Analyzing properties of Ricci curvature and Lie algebra constraints.
result At most one complete Tn-invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold. Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.
problem Existence of constant scalar curvature Kähler metrics on polarized manifolds.
method Direct proof using microscopic stability thresholds and conditions on the limit.
result Existence of a unique constant scalar curvature Kähler metric under specific conditions.
In this paper we show that for a Berger metric g^ on S3, the non-positively curved conformally compact Einstein metric on the 4-ball B1(0) with (S3,[g^]) as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \ci…
In this paper we show that for a generalized Berger metric g^ on S3 close to the round metric, the conformally compact Einstein (CCE) manifold (M,g) with (S3,[g^]) as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if g^ is an SU(k+1)-…
Stokes equations help uniquely identify manifold metrics from boundary data.
problem Determining Riemannian metric from boundary Cauchy data.
method Proving uniqueness of metric from Stokes equations Cauchy data.
result Partial derivatives of all orders of the metric on the boundary are uniquely determined.
Study uniquely determines Riemannian metric derivatives from boundary data.
problem Determining Riemannian metric derivatives from boundary data.
method Computing the full symbol of the elastic Dirichlet-to-Neumann map.
result The elastic Dirichlet-to-Neumann map uniquely determines all partial derivatives of the Riemannian metric on the boundary.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. Sharp threshold found for metric uniqueness in Riemannian Calderón-type problems.
problem Determining metrics uniquely from Dirichlet-to-Neumann maps in Riemannian Schrödinger problems.
method Adaptation of Lassas-Uhlmann reconstruction theorem and novel Gevrey space techniques.
result Analytic metrics uniquely determine the metric up to boundary-preserving diffeomorphisms, but non-analytic metrics are not uniquely determined.
In this short note, we prove that a bi-invariant Riemannian metric on Sp(n) is uniquely determined by the spectrum of its Laplace-Beltrami operator within the class of left-invariant metrics on Sp(n). In other words, on any of these compact simple Lie groups, every left-invariant metric which is n…
Abreu-Sena-Dias have constructed two distinct families of scalar-flat Kähler non-compact toric metrics using Donaldson's rephrasing of Joyce's construction in action-angle coordinates. In this paper and using the same set-up, we show that these are the only J-complete scalar-flat Kähler metrics on any given strictly un…