Teichmüller geodesics can have flexible limit sets.
problem Teichmüller geodesics may not converge to a single point in the boundary.
method Analyzing the topology of the hierarchically hyperbolic space boundary.
result Limit sets of Teichmüller geodesics can be almost anything allowed by the topology.
Infinite-dimensional geometry: completeness and geodesics in Hilbert manifolds.
problem Failure of Hopf--Rinow theorem in Hilbert manifolds.
method Investigates conformal flexibility and completeness properties in infinite-dimensional settings.
result Conformal class of metrics on Hilbert manifolds contains complete representatives.
Two flexible, degenerate constructions related to Thurston's theorem.
problem Understanding the structure and local non-rigidity of Teichmüller spaces and their representations.
method Constructing geodesic segments and open sets in Teichmüller spaces with specific properties.
result Geodesic segments and open sets with degenerate properties in Teichmüller spaces.
Geodesic curves improve flexibility in covariance estimation.
problem Inflexible covariance families limit spatiotemporal modeling.
method Use geodesic curves to build more flexible covariance families.
result Natural projection minimizes geodesic distance to sample covariance.
New metric on cohomology space shows flexibility and rigidity.
problem Metric geometry of big cohomology classes.
method Introduced a new metric d1 on finite energy space E1(X,θ). result Geodesic rays can be constructed in the space.
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.
Study on flexibility of metric entropy and geodesic flow constraints on surfaces.
problem Flexibility and constraints of metrics and flows on surfaces.
method Analysis of geodesic flow, entropy, and metrics in a fixed conformal class.
result Additional restrictions on topological entropy and new geometric properties.
New method for comparing curves with flexible matching constraints.
problem Comparing plane curves with general elastic metrics.
method Combining transform for elastic metrics with parametrization-invariant fidelity metrics.
result Simple optimization problem for discretized curves.
The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
problem Semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
method Intertwining results from dynamics, algebraic geometry, geometric group theory, and Teichmüller theory.
result Each component of the boundary is a product of simple factors, each behaving like a diagonal embedding.
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…
Neural solver computes Wasserstein geodesics and velocity fields efficiently.
problem Computing Wasserstein geodesics and velocity fields efficiently.
method Sample-based neural network approach to solve the minimax problem.
result Directly samples from target distribution and estimates velocity field.
The study examines the flexibility of geometric and dynamical properties of metrics on surfaces.
problem Examining the flexibility of geometric and dynamical properties of metrics on surfaces.
method Analysis of metrics conformally equivalent to a fixed hyperbolic metric.
result Established bounds for the diameter and Laplace spectrum, and demonstrated flexibility in metric entropy.
New data-driven Cartan connection tracks complex vascular structures.
problem Tracking complex vascular structures in multi-orientation images.
method Formulated a data-driven Cartan connection on M2 for geodesic tracking. result Improved geodesic tracking of vascular trees with globally optimal curves.
RNGI model bridges two probability densities on Riemannian manifolds efficiently.
problem Limited applicability of Euclidean stochastic interpolants to Riemannian manifolds.
method Introduces RNGI model interpolating between Riemannian manifold probability densities along geodesics.
result Proves temporal marginal density solves transport equation on Riemannian manifold.
New curvature definitions for networks simplify complex computations.
problem Complex curvature calculations for networks.
method Introducing new curvature definitions based on Menger and Haantjes curvatures.
result Simplified and faster computation of network curvatures.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
This paper studies spectral properties of spheres with one equator.
problem Spectral rigidity and flexibility of spheres with one equator.
method Defined marked length spectrum, proved isospectrality, and classified contact forms.
result Marked length spectrum determines the metric up to Z2-symmetry. 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.
Improved Bayesian inference using power priors with historical data.
problem Improving Bayesian inference with historical data.
method Generalized power priors that adapt to the α parameter of Amari's α-divergence. result Improved performance through appropriate choices of the α parameter. A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
Constructs examples of domains divided by groups in dimensions 3 and above.
problem Dividing convex sets with properly embedded cones.
method Uses Zariski dense relatively hyperbolic groups and properly embedded cones.
result Answers a question of Benoist and provides a topological criterion for convex projective structures.
Unified PCA framework on flag manifolds for robust data analysis.
problem Outliers and manifold data in PCA.
method Generalization of PCA to flag manifolds, optimization problems, and tangent-PCA integration.
result Novel robust and dual geodesic PCA variations.
The authors characterize flexibility in power and energy markets considering time, spatiality, resource, and risk.
problem Evaluating and maximizing flexibility in power systems and markets.
method Characterization of flexibility dimensions (time, spatiality, resource, risk) and their interrelations with flexibility assets, products, and services.
result Flexibility should be evaluated based on multiple dimensions for efficient power systems and markets.
In this paper we define and study flexible links and flexible isotopy in projective space. Flexible links are meant to capture the topological properties of real algebraic links. We classify all flexible links up to flexible isotopy using Ekholms interpretation of Viros encomplexed writhe.
Flexible models of non-flexible polyhedra explained.
problem Non-flexible Siamese dipyramids behave like flexible ones.
method Simple mathematical method to explain model flexibility.
result Physical models of Siamese dipyramids are flexible.
New inequality for odd-degree flexible curves using surface doubling.
problem Bounding the number of non-empty ovals of odd-degree flexible curves.
method Defining an Arnold surface for odd-degree flexible curves and using it to derive a Viro--Zvonilov-type inequality.
result Upper bound on the number of non-empty ovals of odd-degree flexible curves.
Automates detection of fast-ramped flexibility events for DSOs.
problem Monitoring and supervising flexibility activations in power systems.
method Unsupervised detection and open-set classification.
result Automatically identifies critical flexibility activations for early intervention.
Flexible surfaces found in complex projective and product spaces.
problem Finding flexible surfaces in complex projective and product spaces.
method Constructing flexible surfaces within prescribed homology classes.
result Flexible surfaces exist in both CP2 and S2imesS2. Round spheres are uniquely characterized by half-geodesics.
problem Characterizing round spheres in Riemannian geometry.
method Establishing that Riemannian spheres with specific geodesic properties are round.
result Riemannian spheres with all geodesics closed and many half-geodesics are round.
In this paper we study infinitesimal and finite flexibility for generic semidiscrete surfaces. We prove that generic 2-ribbon semidiscrete surfaces have one degree of infinitesimal and finite flexibility. In particular we write down a system of differential equations describing isometric deformations in the case of exi…
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
In this paper we study geometric, algebraic, and computational aspects of flexibility and infinitesimal flexibility of Kokotsakis meshes. A Kokotsakis mesh is a mesh that consists of a face in the middle and a certain band of faces attached to the middle face by its perimeter. In particular any 3x3-mesh made of quadran…
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.
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.
Locally extremal geodesic loops are closed geodesics in Riemannian manifolds.
problem Characterizing geodesic loops in Riemannian manifolds.
method Analyzing properties of geodesic loops and their conjugate points.
result Locally extremal non-self-conjugate geodesic loops are closed geodesics.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
The paper introduces geodesic φ-convex functions and their properties.
problem Generalizing geodesic functions to φ-convex functions.
method Introducing geodesic φ-convex functions and investigating their properties.
result Characterization of geodesic φ-convex functions via their φ-epigraphs.
Defines new geodesic semilocal E-preinvex functions and studies their properties.
problem Defines new functions to generalize existing convex and preinvex concepts.
method Introduces geodesic semilocal E-preinvex functions and proves their properties.
result Establishes sufficient optimality conditions for nonlinear fractional multiobjective programming.
Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than 2π is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…
New quasi-geodesics for Stiefel manifold simplify complex computations.
problem Efficiently solving geodesic endpoint problem on Stiefel manifold.
method Derived new representations of quasi-geodesics for large-scale computations.
result New quasi-geodesics are closer to Riemannian geodesics.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.
problem Investigate geodesics and F-geodesics on tangent bundles.
method Investigate geodesics and F-geodesics on tangent bundles and φ-unit tangent bundles equipped with φ-Sasaki metric over para-Kähler-Norden manifolds.
result Investigate and analyze geodesics and F-geodesics on tangent bundles.
Conformal geodesics can't spiral in Riemannian manifolds.
problem Existence of spiral conformal geodesics on Riemannian manifolds.
method Analyzing properties of conformal geodesics on Riemannian manifolds.
result No conformal geodesic can become trapped in every neighborhood of a point.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
New geodesics found that are not tight but still have useful properties.
problem Understanding geodesics in curve complexes and Teichmüller spaces.
method Introducing and studying weak tight geodesics with canonical constructions.
result Found examples of weak tight geodesics with gaps between them.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…