Complete Finsler spaces with negative Ricci curvature are reversible.
problem Characterizing Finsler spaces with constant negative Ricci curvature.
method Utilizing projectively invariant pseudo-distance and Schwarzian derivative.
result Every connected complete Finsler space with constant negative Ricci scalar is reversible.
Study finds solutions for spacetimes with negative cosmological constant.
problem Existence of spacetimes with negative cosmological constant.
method Proved existence of solutions for Einstein-complex scalar field equations.
result Found large families of solutions with negative cosmological constant.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
New mass definition for negative cosmological constant spacetimes.
problem Defining quasilocal mass for spacetimes with negative cosmological constant.
method Spinorial approach based on previous work for vanishing cosmological constant.
result Non-negative mass, equal to Misner-Sharp mass in spherical symmetry, zero for AdS.
Researchers found infinite families of non-singular static spacetimes with negative cosmological constant.
problem Finding non-singular static spacetimes with negative cosmological constant.
method Constructing infinite-dimensional families of solutions to the Einstein-Maxwell equations.
result Infinite-dimensional families of non-singular static space times with negative cosmological constant.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Method constructs discrete surfaces with negative curvature.
problem Creating discrete surfaces with negative Gaussian curvature.
method Birkhoff decomposition and nonlinear d'Alembert formula.
result Simple algorithm for Birkhoff decomposition.
New non-singular spacetimes found with negative cosmological constant.
problem Finding non-singular spacetimes with a negative cosmological constant.
method Constructing infinite-dimensional families of solutions to complex equations.
result Infinite-dimensional families of non-singular stationary space-times with negative cosmological constant.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
Constructs periodic solutions for wave equations, including Einstein's, with negative cosmological constant.
problem Finding periodic solutions for nonlinear wave equations, especially Einstein's equations with negative cosmological constant.
method Analytic continuation to construct periodic solutions.
result Infinite-dimensional families of time-periodic solutions for vacuum and Einstein-Maxwell-dilaton-scalar fields.
Graphs with non-negative Ollivier curvature have constant bounded harmonic functions.
problem Liouville property for graphs with non-negative Ollivier curvature.
method Proving Liouville property and improving concentration results.
result Every bounded harmonic function on graphs with non-negative Ollivier curvature is constant.
Study constructs infinite families of non-singular black hole solutions with a negative cosmological constant.
problem Constructing non-singular black hole solutions with a negative cosmological constant.
method Using an elliptic system of equations with complex coefficients for complex-valued tensor fields.
result Infinite-dimensional families of non-singular stationary black hole solutions with a negative cosmological constant.
New solutions found with negative mass in general relativity.
problem Finding metrics with negative mass in general relativity.
method Constructing families of metrics with specific properties.
result Obtained new classes of solutions with negative mass.
Constructed static vacuum metrics in 5D with negative cosmological constant.
problem Finding static vacuum solutions in 5D with negative cosmological constant.
method Numerically constructed two families of metrics with squashed conformal infinity.
result Constructed metrics with squashed conformal infinity conformal to any squashed 3D sphere.
The paper defines new Schur-constant models for non-negative variables.
problem Modeling equilibrium distributions for non-negative variables.
method Introduces Schur-constant equilibrium distribution models for arithmetic non-negative random variables.
result Derived properties include implicit correlation and sum distribution.
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.
problem Proving isoperimetric inequalities in manifolds with small negative Ricci curvature.
method Expanding on the ABP method, the paper uses the elliptic Kato constant to control the non-negativity of the Ricci-tensor and applies techniques from Li-Tam and Kasue.
result Sharp isoperimetric inequalities in the limit are proven in the presence of small negative curvature.
Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in R3. We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into t…
The study proves Lieb-Thirring inequalities on hyperbolic manifolds.
problem Proving Lieb-Thirring inequalities on manifolds with negative constant curvature.
method Analytical proof of inequalities on hyperbolic manifolds.
result Discrete spectrum below the continuous spectrum (d−1)2/4,∞). Authors construct hypertori with constant negative mean curvature in a sphere.
problem Constructing constant mean curvature hypertori in a sphere.
method Constructing two different constant mean curvature (2n−1)-dimensional hypertori in a 2n-dimensional sphere. result Two different constant mean curvature (2n−1)-dimensional hypertori with negative mean curvature in a 2n-dimensional sphere. New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.
problem Constructing new non-rotational discrete pseudospherical surfaces.
method Explicit parametrizations and Bäcklund transformations for discrete constant negative Gaussian curvature surfaces of revolution.
result Conditions for Bäcklund transformations to preserve periodicity are provided.
The study examines the flexibility of entropies for negatively curved surfaces.
problem The study investigates the flexibility of topological and metric entropies for negatively curved surfaces.
method The authors compare different metrics on surfaces of negative curvature and analyze their topological and metric entropies.
result The study proves that the topological and metric entropies for metrics of negative curvature are flexible and only equal in the case of constant negative curvature.
A Steiner deltoid maintains constant area across all boundary points of an ellipse.
problem Finding curves associated with ellipses with constant area.
method Negative Pedal Curve of the Ellipse with respect to a boundary point M.
result The Steiner deltoid has constant area over all boundary points.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…
We construct a large class of new singularity-free static Lorentzian four-dimensional solutions of the vacuum Einstein equations with a negative cosmological constant. The new families of metrics contain space-times with, or without, black hole regions. Two uniqueness results are also established.
The paper classifies metrics with constant negative Q-curvature in Euclidean spaces.
problem Classifying metrics with constant negative Q-curvature in Euclidean spaces.
method Variational techniques and finite volume conditions.
result Existence and classification of singular and nonsingular metrics with constant negative Q-curvature.
New results on tori restrict sectional curvature when Ricci curvature is negative and bounded.
problem Restricting sectional curvature on tori with mixed Ricci bounds.
method Using Lohkamp's theorem and explicit constants.
result Explicit constants show sectional curvature is positive in some directions.
Inverts operator on hyperbolic surfaces, constructing invariant distributions.
problem Constructing explicit inversion formula for X-ray normal operator.
method First, inversion formula for attenuated normal operator on Poincaré disk and closed hyperbolic surfaces. Then, explicit construction of invariant distributions.
result Explicit construction of invariant distributions with prescribed pushforward.
New findings on static near horizon geometries and quasi-Einstein manifolds, including rigidity results for negative cosmological constant.
problem Rigidity of quasi-Einstein manifolds under different cosmological constant conditions.
method Analysis of quasi-Einstein equations on closed manifolds, focusing on static vacuum solutions and their properties.
result For negative cosmological constant, rigidity holds under specific conditions on the 1-form \(X\), including incompressibility, constant norm, and nontrivial cohomology.
The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.
problem Understanding Markov partitions for general hyperbolic flows.
method Rigorous construction of Markov partitions for geodesic flow on Riemann surfaces of constant negative curvature.
result Explicit forms of rectangles and local cross sections provided for the geodesic flow.
We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds wit…
Proves unique Kähler-Einstein metric on certain manifolds.
problem Negative curvature Kähler manifolds.
method Holomorphic curvature bounds, uniqueness proof.
result Uniform equivalence of metrics on manifolds.
In this paper we consider a domain in a space of negative constant sectional curvature. Such assumption about the sectional curvature let us develop a new technique and improve existing lower bounds of eigenvalues from Dirichlet eigenvalue problem, obtained by Alessandro Savo in 2009.
Study shows how brain completes missing parts of curves and constructs surfaces of negative curvature.
problem How the brain completes missing parts of curves and constructs surfaces of negative curvature.
method Solving variational problems to find sub-Riemannian geodesics and constructing surfaces of constant negative curvature.
result There is a one-to-one correspondence between sub-Riemannian geodesics used by the brain and rotational surfaces of constant negative curvature.
The study proves constant-curvature analogues of hot spots conjecture for triangles.
problem Proving the hot spots conjecture in constant curvature domains.
method Analyzing geodesic triangles of constant negative curvature and using Killing fields.
result First mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points in constant curvature triangles.
The paper extends rigidity results for special surfaces in general initial data sets.
problem Rigidity of marginally outer trapped surfaces with negative σ-constant.
method Generalization of previous results to higher genus and dimensions.
result A splitting result for the ambient manifold when it contains a stable closed MOTS.
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
The sectional curvature of the Weil-Petersson metric on Teichmuller space is known to be negative. We show that this Weil-Petersson sectional curvature is not pinched from above by any negative constants, i.e., there is no negative upper bound.
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
Improved score matching for non-negative data models.
problem Estimating parameters of non-negative probability density functions.
method Generalized score matching method for non-negative data.
result Improved estimation efficiency and theoretical guarantees.
New complete hypersurfaces found in affine geometry.
problem Finding new complete hypersurfaces in affine geometry.
method Classifying special Calabi hypersurfaces and solving equations.
result Found a class of new Euclidean and Calabi complete affine hypersurfaces.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…
To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as …
We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with co…