Combination theorem for geodesic coarsely convex group pairs.
problem Understanding properties of groups relative to subgroups.
method Definitions of weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex group pairs; combination theorem.
result Combination theorem for geodesic coarsely convex group pairs.
Convexity properties of Weil-Petersson geodesics on the Teichmüller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considera…
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the…
As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic φ-convex function and deduce some basic properties of φ-convex function and geodesic φ-convex function. We also introduce the concept of geodesic φ-convex set and φ-epigraph and in…
New statistical convex-cocompactness found for non-orientable surfaces.
problem Understanding the dynamics of mapping class groups on non-orientable surfaces.
method Using Teichmüller space and complexity length, showing geodesics leave compact regions with exponentially low probabilities.
result Statistical convex-cocompactness of mapping class groups on non-orientable surfaces.
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgro…
Analyzes convex structures in Teichmüller space unit tangent spheres.
problem Characterize faces and extreme points of unit tangent spheres in Teichmüller space.
method Analyzes Finsler infinitesimal balls of Thurston metric, characterizes faces, exposed faces, and extreme points.
result Characterizes faces and extreme points of unit tangent spheres in Teichmüller space.
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
Stable nets on convex hypersurfaces maintain their shape under small perturbations.
problem Maintaining the shape of nets on convex surfaces under slight changes.
method Constructing stable geodesic nets on convex hypersurfaces.
result Stable geodesic nets on convex hypersurfaces do not change shape under small perturbations.
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
Given any K and N we show that there exists a compact geodesic metric measure space satisfying locally the CD(0,4) condition but failing CD(K,N) globally. The space with this property is a suitable non convex subset of R^2 equipped with the l^\infty-norm and the Lebesgue measure. Combining many such spaces gives a (non…
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
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.
This paper explores the relation between convex functions and the geometry of space-times and semi-Riemannian manifolds (an investigation initiated by Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give geometric-topological proofs of geodesic connectedness for classes of space-times to which kno…
The study finds at least two short, simple geodesic chords on a disk with convex boundary.
problem Existence of short, simple geodesic chords on a 2-disk with convex boundary.
method Proof of existence using Riemannian geometry and bounds on lengths.
result Existence of at least two short, simple orthogonal geodesic chords on a 2-disk with convex boundary.
Asymptotic geodesics in convex polygons are convex for large distances.
problem Understanding convexity of geodesics in Hilbert geometry.
method Analyzing the distance function between asymptotic geodesics for large t.
result The distance function between asymptotic geodesics is convex for sufficiently large t.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. Geodesically convex functions are continuous on Riemannian manifolds.
problem Continuity of geodesically convex functions on Riemannian manifolds.
method Proof of continuity using geodesic convexity and addressing a gap in existing proof.
result All geodesically convex functions are continuous in the interior of their domain on Riemannian manifolds.
The paper proves the existence and properties of geodesics on convex surfaces.
problem Existence and properties of geodesics on convex surfaces with free boundaries.
method Free boundary curve shortening flow on closed surfaces with strictly convex boundary.
result Existence of two free boundary embedded geodesics and geodesics with Morse Index 1 and 2.
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…
We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
New constructions show stable geodesics and figure-eights in convex hypersurfaces.
problem Constructing stable geodesics and figure-eights in convex hypersurfaces.
method Explicit billiard trajectories with controlled parallel transport in convex polytopes.
result Construction of stable figure-eights and index-zero geodesics in convex hypersurfaces.
Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …
Billiard trajectories and geodesics are closely related geometrically.
problem Understanding the relationship between billiard trajectories and geodesics on surfaces.
method Establishing mutual approximation results for billiard trajectories and geodesic segments on surfaces.
result For Riemannian billiard tables, there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory.
In this paper we have generalized the notion of λ-radial contraction in complete Riemannian manifold and developed the concept of pλ-convex function. We have also given a counter example proving the fact that in general λ-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some r…
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension ≥3 with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …
Strong geodesic convex function and strong monotone vector field of order m on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order m for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.
New study shows acceleration in hyperbolic spaces is impossible for strongly geodesically convex functions.
problem Acceleration in hyperbolic spaces for strongly geodesically convex functions is impossible.
method Perturbing hard functions with sums of bump functions chosen by a resisting oracle.
result Acceleration is unachievable for any deterministic algorithm in hyperbolic spaces for strongly geodesically convex functions.
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
Geodesic balls are isoperimetric in hyperbolic spaces with certain densities.
problem Proving isoperimetric properties in hyperbolic spaces with specific densities.
method Using geodesic balls and radial, strictly log-convex densities.
result Geodesic balls are isoperimetric in real hyperbolic space HRn. Paper analyzes convergence of proximal algorithm in metric spaces without geodesic convexity.
problem Analyzing convergence of proximal algorithm in general metric spaces.
method Analysis of the Wasserstein proximal algorithm without geodesic convexity assumption.
result Establishes unbiased and linear convergence rate for proximal algorithm under natural Wasserstein inequality.
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
Study coning totally geodesic boundaries of hyperbolic manifolds.
problem Understanding metrics on coned-off spaces of hyperbolic manifolds.
method Analyzing the geometric and group-theoretic properties of coned-off spaces.
result Explicit conditions for negatively curved metrics and locally convex subsets.
We show that for every simple closed curve α, the extremal length and the hyperbolic length of αare quasi-convex functions along any Teichmuller geodesic. As a corollary, we conclude that, in Teichmuller space equipped with the Teichmuller metric, balls are quasi- convex.
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 geodesics on flat tori, focusing on convex bodies.
problem Analyze geodesics orthogonal to convex subsets on flat tori.
method Define anisotropic Sobolev spaces and study properties of geodesics.
result Compute residues of geometric Epstein function in terms of intrinsic volumes.