Study classifies and characterizes translators in hyperbolic static universe.
problem Classifying and characterizing translators in hyperbolic static universe.
method Classified and characterized translators foliated by horospheres and rotationally invariant ones, both space-like and time-like.
result Obtained a characterization of the bowl and certain translators foliated by horospheres.
It is shown that any two-dimensional spacetimes with compact Cauchy surfaces can be causally isomorphically imbedded into the two-dimensional Einstein's static universe. Also, it is shown that any two-dimensional globally hyperbolic spacetimes are conformally equivalent to a subset of the two-dimensional Einstein's sta…
The group of conformal diffeomorphisms and the group of causal automorphisms on two-dimensional globally hyperbolic spacetimes are clarified. It is shown that if spacetimes have non-compact Cauchy surfaces, then the groups are subgroups of that of two-dimensional Minkowski spacetime, and if spacetimes have compact Cauc…
Paper derives Riccati equation for static spaces and proves its applications.
problem Deriving Riccati equation for static spaces.
method Proving splitting theorem and connectivity of conformal boundary.
result Establishes compactness of universal covering for static triples.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
Explicit models for the restricted conformal group of the Einstein static universe of dimension greater than two and for its universal covering group are constructed. Based on these models, as an application we determine all oriented and time-oriented conformal Lorentz manifolds whose restricted conformal group has max…
Paper defines Bartnik mass for hyperbolic extensions and proves staticity.
problem Defining and proving staticity of asymptotically hyperbolic minimal mass extensions.
method Definition of Bartnik mass, construction of metrics, one-parameter family analysis.
result Static potential for asymptotically hyperbolic admissible extensions achieving Bartnik mass.
New metrics found in hyperbolic manifolds as volume-minimizers.
problem Finding critical points of volume-renormalized mass.
method Critical points of the volume-renormalized mass over asymptotically hyperbolic manifolds.
result V-static metrics are critical points of volume-renormalized mass.
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
New static black hole uniqueness theorems for negative cosmological constant.
problem Uniqueness of static black holes in asymptotically locally hyperbolic spaces.
method Inequality relating surface gravity and topology, rigidity of Kottler black holes, monotone quantities under IMCF, regularity theorem for IMCF.
result Static black holes are uniquely determined by their geometry and topology.
Paper finds inequalities for convex domains in hyperbolic space.
problem Finding inequalities for convex domains in hyperbolic space.
method Introducing hyperbolic ellipsoids and using orthogonal projection to establish inequalities.
result Affine isoperimetric inequalities for static convex domains in hyperbolic space characterized by hyperbolic ellipsoids.
The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.
problem Proving a positive mass theorem for non-compact static domains in hyperbolic space.
method Formulating and proving a positive mass theorem under natural dominant energy conditions, using elliptic boundary conditions on spinors.
result Retrieve a sharper version of a recent result by Souam about the rigidity of non-compact static domains.
The paper proves new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
problem Proving new inequalities for convex hypersurfaces in hyperbolic and spherical spaces.
method Locally constrained inverse curvature flows in hyperbolic and spherical spaces.
result Established new Alexandrov-Fenchel and Minkowski inequalities involving general convex weight functions.
We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperb…
The study characterizes quasiperiodic surfaces in pseudo-hyperbolic spaces with curvature conditions.
problem Characterizing quasiperiodic surfaces in pseudo-hyperbolic spaces.
method Curvature conditions, Gromov hyperbolicity, conformal hyperbolicity.
result Limit curves of quasiperiodic surfaces in the Einstein Universe have canonical quasisymmetric parametrizations.
Classifies conformal transformations in spacetimes without observer horizons.
problem Understanding conformal transformations in spacetimes without observer horizons.
method Proves classification of conformal transformations into two types: escaping and non-escaping.
result Conformal transformations of Einstein's static universe are classified.
Paper proves inequalities in sub-static warped product manifolds.
problem Proving inequalities in sub-static warped product manifolds.
method Proved Heintze-Karcher type inequalities involving shifted mean curvature.
result Uniqueness results for hypersurfaces satisfying curvature equations.
Static spacetimes are stable attractors in a flow equation.
problem Stability of static spacetimes with negative cosmological constant.
method New expander entropy for Ricci-harmonic flow.
result Static metrics are stable if and only if a positive mass theorem holds.
We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.
The paper explores centroids and static equilibrium points in non-Euclidean geometries.
problem Investigating centroids and static equilibrium points in spherical, hyperbolic, and normed spaces.
method Extending Gal'perin's work, the paper examines convex bodies in these spaces and analyzes the minimum number of equilibrium points.
result Every plane convex body in any of these spaces has at least four equilibrium points, and there are mono-monostatic convex bodies in 3D spherical, hyperbolic, and certain normed spaces.
The paper finds universal inequalities for eigenvalues on hyperbolic spaces.
problem Eigenvalues of the Dirichlet Laplacian on conformally flat Riemannian manifolds.
method Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
result Establishes universal inequalities for eigenvalues of the Dirichlet Laplacian on hyperbolic spaces.
Study properties of balanced hyperbolic compact complex manifolds.
problem Understanding cohomology and harmonic spaces of balanced hyperbolic manifolds.
method Proved vanishing theorems and Hard Lefschetz-type theorems for balanced hyperbolic compact complex manifolds.
result Non-existence of certain L1 currents on the universal covering space of a balanced hyperbolic manifold. New maps connect universal circles to ideal sphere for hyperbolic manifolds.
problem Understanding universal circles for Anosov foliations with branching.
method Introduced a new type of Cannon--Thurston map for leftmost universal circles.
result Fundamental group acts on leftmost universal circle with pseudo-Anosov dynamics.
"Ends of hyperbolic 3-manifolds should support canonical Wick Rotations, so they realize effective interactions of their ending globally hyperbolic spacetimes of constant curvature." We develop a consistent sector of WR-rescaling theory in 3D gravity, that, in particular, concretizes the above guess for many geometrica…
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proo…
The study proves geometric inequalities for static convex domains in static rotationally symmetric spaces.
problem Proving geometric inequalities for static convex domains in static rotationally symmetric spaces.
method Locally constrained curvature flow in a static rotationally symmetric space Nn+1, proving graphical solutions and static convexity preservation. result Proves weighted geometric inequalities for static convex domains close to a slice of Nn+1. 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.
To deal with changing environments, a new performance measure -- adaptive regret, defined as the maximum static regret over any interval, was proposed in online learning. Under the setting of online convex optimization, several algorithms have been successfully developed to minimize the adaptive regret. However, existi…
Unified framework for fixed-income pricing and liability replication.
problem Static arbitrage and discount curve construction.
method Model-free framework for static fixed-income pricing and liability replication.
result Existence of strictly positive discount curves reproducing market prices and least-cost super-replicating portfolios.
The study shows conditions for larger volumes in the universal cover of a manifold.
problem Conditions for larger volumes in the universal cover of a manifold.
method Analyzes the relationship between the volume of a manifold and the volume of its universal cover.
result Guarantees the existence of balls with greater-than-hyperbolic volumes in the universal cover.
The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.
problem Understanding the geometric significance of the universal Liouville action.
method Analyzing the Weil-Petersson universal Teichmüller space and its relation to hyperbolic 3-manifolds.
result The gradient flow of the universal Liouville action converges to the origin, providing a bound on Weil-Petersson distance.
Researchers prove Fredholm property for Dirac operator on specific spacetimes.
problem Proving Fredholm property for Dirac operator on asymptotically static spacetimes.
method Combining time-dependent scattering theory and Egorov's theorem for pseudo-differential hyperbolic systems.
result The Dirac operator is Fredholm under Atiyah-Patodi-Singer boundary conditions.
Uniform comparison of hyperbolic ball volumes on universal cover.
problem Comparing hyperbolic ball volumes on universal cover.
method Proving a constant δ_n exists such that for any metric g on M, if the volume ratio is less than δ_n, the volume of hyperbolic balls is at least as large as in hyperbolic space.
result Every Riemannian metric g on M with a specific volume ratio satisfies the volume of hyperbolic balls on the universal cover is at least as large as in hyperbolic space.
A measured laminations on the universal hyperbolic solenoid § is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid § is uniquely determined by a measured lamination on §; it is a leafwise earthquake with…
In this article, we first establish the main tool - an integral formula for Riemannian manifolds with multiple boundary components (or without boundary). This formula generalizes Reilly's original formula from \cite{Re2} and the recent result from \cite{QX}. It provides a robust tool for sub-static manifolds regardless…
In the framework of standard static space times, we state a family of sufficient or necessary conditions for a set of physically reasonable energy and convergence conditions in relativity and related theories. We concentrate our study on questions about the sub-harmonicity of the warping function, the scalar curvature …
Improved bounds linking entropy and volume in hyperbolic 3-manifolds.
problem Establishing bounds between entropy and volume in hyperbolic 3-manifolds.
method Heegaard Floer homology and hyperbolic geometry.
result Entropy is bounded by hyperbolic volume with logarithmic factor.
In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidi…
Proves uniqueness of certain spacetime solutions with extremal horizons.
problem Proving uniqueness of extremal Schwarzschild de Sitter spacetime solutions.
method Analytic proof in four and higher dimensions, spectral problem for hyperbolic surfaces.
result Proves extremal Schwarzschild de Sitter solutions are unique up to identifications.
Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.
problem Understanding the behavior of Laplacian determinants on random hyperbolic surfaces.
method Investigated various models of random hyperbolic surfaces and their Laplacian determinants as genus increases.
result For all popular models, the determinant grows exponentially with a universal exponent as the genus goes to infinity.
Simple model uses time series momentum to outperform benchmarks in equity and bond markets.
problem Finding systematic excess returns in various markets.
method Time series momentum applied to multiple investable indices without complex parameter estimation.
result Significant outperformance in equity and bond markets, nearly doubling returns.
Proves a sharp inequality for toroidal surfaces in Horowitz-Myers geon.
problem Extending Minkowski inequality to non-static spacetimes.
method Proves a sharp inequality for toroidal hypersurfaces in Horowitz-Myers geon.
result Sharp inequality for toroidal surfaces in asymptotically hyperbolic manifold.
Given a uniform foliation by Gromov hyperbolic leaves on a 3-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are R-covered and we giv…
Trajectories of light rays in a static spacetime are described by unparametrised geodesics of the Riemannian optical metric associated with the Lorentzian spacetime metric. We investigate the uniqueness of this structure and demonstrate that two different observers, moving relative to one another, who both see the univ…
We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+1)-manifold or its universal mod 2 Abelian cover can.
Constructs a universal Cannon-Thurston map for a new curve complex.
problem Mapping class groups and their boundaries.
method Using Birman exact sequence, proves hyperbolicity, constructs map.
result Universal Cannon-Thurston map to surviving curve complex boundary.
Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set PH is contractible for every finite subgroups H of G. This is the main ingredient for proving that P is a finite model for the universal space e.g. of proper actions. As a corollar…
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.