Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.
problem Convex hyperbolic cone-metrics on 3-manifold boundaries and their bent realizations.
method Alexandrov-Weyl-type problem, bent metrics, controllably polyhedral, Lipschitz topology.
result Unique bent realizations for convex hyperbolic cone-metrics on 3-manifold boundaries.
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 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.
Non-convex extremal length found in surface metrics.
problem Extremal length functions on surfaces are not always convex.
method Used harmonic maps to R-trees and minimal surfaces in Rn. result Found measured foliations with non-convex extremal length functions.
The paper examines hyperbolicity in bounded strongly minimally convex domains in R^d.
problem Investigating hyperbolicity in bounded strongly minimally convex domains.
method Analyzing the minimal metric and Hilbert metric in convex domains.
result Every bounded strongly minimally convex domain is Gromov hyperbolic.
Optimal inequalities found between Riemannian and Hilbert metrics in convex projective domains.
problem Finding optimal bounds between Riemannian and Hilbert metrics in convex projective domains.
method Optimal control techniques applied to Riemannian metrics induced by centro-affine hypersurface immersions.
result Optimal inequalities between Riemannian and Hilbert metrics for a class of convex projective domains.
Establishes smooth Ricci flows from convex surfaces in 3D space.
problem Existence and uniqueness of Ricci flow starting from convex surfaces.
method Smooth Ricci flows starting from smooth convex surfaces.
result Uniform convergence of metrics to initial convex surface.
Extends metric to Margulis spacetimes for convex properties.
problem No specific problem stated; extends metric.
method Extends Thurston's asymmetric metric to Margulis spacetimes and proves convex properties.
result Established convex properties of the extended metric.
The paper proves Gromov hyperbolicity of certain metrics using isoperimetric inequalities.
problem Investigating Gromov hyperbolicity of specific metrics.
method Using isoperimetric inequalities to characterize Gromov hyperbolicity.
result Characterization of domains where these metrics are Gromov hyperbolic.
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting met…
Curve shortening in metric-affine plane shrinks convex curves to points.
problem Shortening curves in non-Euclidean spaces.
method Curve shortening flow in metric-affine plane with geometric conditions.
result Closed convex curves in metric-affine plane shrink to points in finite time.
In this paper, we prove that a strongly convex complex Finsler metric F on a domain D⊂Cn is projectively flat (resp. dually flat) if and only if F comes from a strongly convex complex Minkowski metric.
Study shows non-compact convex hulls in certain metric spaces.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
New interpretation of discrete conformality using polyhedral convex hulls.
problem Understanding discrete conformality in 3D.
method Epstein-Penner convex hull construction and induced metrics.
result New bijections and interpretations of discrete conformality.
The paper proves properties of complex Finsler metrics on specific domains.
problem Investigating invariant complex Finsler metrics on complex domains.
method Analyzing holomorphic automorphism groups and constructing metrics.
result Explicitly constructed metrics on polydisks with properties similar to Bergman metric.
Characterizes Kähler-Berwald metrics on complex manifolds.
problem Identifying Kähler-Berwald metrics among strongly convex complex Finsler metrics.
method Geometric characterization using Cartan and Chern-Finsler connections.
result Characterizes Kähler-Berwald metrics in terms of parallelism of the canonical complex structure.
Extends Alexandrov's result to unbounded convex domains in hyperbolic 3-space.
problem Determining unbounded convex domains in hyperbolic 3-space from boundary data.
method Using conformal structure and induced metric on the boundary.
result A wide range of boundary data can be realized on unbounded convex domains in hyperbolic 3-space.
The study finds conditions for certain surfaces to have a specific type of metric.
problem Understanding the geometry of surfaces with specific metrics.
method Analyzes surfaces of revolution and derives conditions for a strongly convex slope metric.
result Necessary and sufficient conditions for surfaces of revolution to admit a strongly convex slope metric are established.
The paper constructs convex subsets in anti-de Sitter space with specific metrics on boundaries.
problem Creating convex subsets with prescribed metrics on boundaries in anti-de Sitter space.
method Using quasi-symmetric maps and properties of hyperbolic metrics, the paper constructs convex subsets with specific metrics on boundaries.
result Existence of globally hyperbolic convex subsets with prescribed metrics on boundaries.
Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
We prove that for every metric on the torus with curvature bounded from below by -1 in the sense of Alexandrov there exists a hyperbolic cusp with convex boundary such that the induced metric on the boundary is the given metric. The proof is by polyhedral approximation. This was the last open case of a general theorem:…
Constructs a convex Finsler metric on vector bundles under specific conditions.
problem Creating a convex Finsler metric on vector bundles with positive curvature.
method Uses the negativity of direct image bundles and Minkowski inequality for norms.
result Shows how to upgrade a Kobayashi positive Finsler metric to a convex one.
We introduce a new family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric plane…
Metric surfaces can be divided into small triangles.
problem Decomposing metric surfaces into triangles.
method Proving any metric space homeomorphic to a surface can be divided into non-overlapping convex triangles of small diameter.
result Metric surfaces can be decomposed into triangles of arbitrarily small diameter.
In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with C∞ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also …
We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
Paper surveys balanced metrics and proves a geodesic convexity result.
problem Understanding balanced metrics and stability in algebraic geometry.
method Survey and proof of geodesic convexity result.
result Geodesically convex function on a complete Riemannian manifold admits a critical point if and only if its asymptotic slope at infinity is positive.
Derives curvature formulas for convex metric sums and conditions for positive average variation.
problem Understanding how the curvature of a convex sum of metrics changes and whether it can increase the average curvature.
method Explicit formulae for curvature of convex sums of Riemannian metrics, studying total geodesic flat torus.
result Necessary and sufficient conditions for positive average variation of curvature of \(g_t\).
New conical metrics found on toric varieties with convex cones.
problem Finding conical metrics on toric affine varieties.
method Existence result for inhomogeneous Monge-Ampere equation, transversal a priori estimates.
result Existence of conical Ricci flat Kahler metrics on Q-Gorenstein affine toric varieties.
The paper characterizes complex Finsler metrics invariant under U(n) and their properties.
problem Characterizing U(n)-invariant strongly convex complex Finsler metrics. method Analyzing conditions for strong convexity and proving theorems about these metrics.
result A U(n)-invariant strongly convex complex Finsler metric is a real Berwald metric if and only if it comes from a Hermitian metric. The study finds at least 2 free-boundary minimal disks in convex 3-balls.
problem Finding minimal disks in convex 3-balls.
method Combining mean curvature flow, min-max theory, and degree theory.
result Existence of at least 2 free-boundary minimal disks in convex 3-balls for generic metrics.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
We survey some basic geometric properties of the Funk metric of a convex set in Rn. In particular, we study its geodesics, its topology, its metric balls, its convexity properties, its perpendicularity theory and its isometries. The Hilbert metric is a symmetrization of the Funk metric, and we show some pro…
We prove that the space of convex real projective structures on a surface of genus g≥2 admits a mapping class group invariant Kähler metric where Teichmüller space with Weil-Petersson metric is a totally geodesic complex submanifold.
Let (M,∂M) be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>−1, and that the th…
Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.
problem Identifying compact Fuchsian manifolds with convex boundaries.
method Proving uniqueness based on the induced path metric on the boundary.
result Compact Fuchsian manifolds with convex boundaries are uniquely determined by the induced path metric on the boundary.
Anti-de Sitter spacetimes with convex boundaries are uniquely determined by their boundary metrics.
problem Identifying anti-de Sitter spacetimes based on boundary properties.
method Proving rigidity for spacetimes with holonomy close to Fuchsian and convex boundaries.
result Globally hyperbolic compact anti-de Sitter spacetimes are determined by their boundary metrics.
Study of hyperbolic directions in convex projective geometry.
problem Understanding properties of quasi-geodesics in convex projective geometry.
method Three perspectives: Hilbert metric, boundary projective geometry, and automorphisms.
result Relationship between different definitions of Morse and regular quasi-geodesics.
We show that the volume entropy of the Hilbert metric on a closed convex projective surface tends to zero as the corresponding Pick differential tends to infinity. The proof is based on the theorem, due to Benoist and Hulin, that the Hilbert metric and Blaschke metric are comparable.
We prove that a 3-dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by the metric induced on its boundary. Furthemore, any hyperbolic metric on the torus with cone singularities of positive curvature can be realized as the induced metric on the boundary of a convex polyhedral cusp. The …
The Hessian of the renormalized volume of geometrically finite hyperbolic 3-manifolds without rank-1 cusps, computed at the hyperbolic metric g with totally geodesic boundary of the convex core, is shown to be a strictly positive bilinear form on the tangent space to Teichmüller space. The metric g is known fro…
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
We prove that the Teichmüller space of surfaces of genus g with p punctures contains balls which are not convex in the Teichmüller metric whenever 3g−3+p>1.
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.
Study Gromov hyperbolic domains in Minkowski space, proving equivalence to boundary properties.
problem Investigate Gromov hyperbolic domains in Minkowski space.
method Explicit comparisons between metrics, dynamical arguments, and quasi-hyperbolic metric.
result Gromov hyperbolicity of convex, future complete domains is equivalent to stable acausality of the boundary.
The Funk metric connects billiards, projective geometry, and convex geometry.
problem Exploring the Funk metric's invariants and inequalities.
method Using the Funk metric, extending results from projective geometry and convex geometry.
result General affine inequalities and volume maximizers in Funk geometry.
The paper shows that g-convex functions on manifolds are sparse.
problem Characterizing and understanding the sparseness of g-convex functions.
method Established criteria for g-convexity and used them to prove sparseness results.
result Most g-convex functions on compact manifolds have few critical points.