Study disproves conjecture about metric completion of curve spaces.
problem Completeness properties of spaces of immersed curves with reparametrization-invariant metrics.
method Examined Sobolev-type metrics on real-valued immersed curves, demonstrating multiple distinct limit points.
result Metric completion of spaces of immersed open curves includes multiple distinct limit points, not a single point as previously conjectured.
Completeness of Sobolev metrics on curve spaces proven.
problem Proving completeness of Sobolev metrics on curve spaces.
method Analyzing Sobolev metrics with nonconstant coefficients on curve spaces.
result Necessary and sufficient conditions for metric completeness provided.
Study shows how nonstandard hulls can contain metric completions.
problem Characterizing the Heine-Borel property in metric spaces.
method Using nonstandard analysis and metric completions.
result Characterization of the Heine-Borel property in terms of inapproachable finite points.
Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
Complete negative Kähler-Einstein metric found on Stein manifolds.
problem Existence of complete Kähler-Einstein metrics on Stein manifolds with negative curvature.
method Normalized Kähler-Ricci flow to deform metrics to complete negative Kähler-Einstein metric.
result Existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature.
Completeness of surface metrics established for Sobolev spaces.
problem Ensuring completeness of reparametrization-invariant Sobolev metrics on surface spaces.
method Recasting completeness criteria for infinite-dimensional Riemannian manifolds and applying geometric estimates based on the Michael--Simon--Sobolev inequality.
result Established metric and geodesic completeness for specific Sobolev metrics on immersed surfaces, validating Mumford's conjecture.
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for q>3/2. Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Complete Calabi-Yau metrics found on higher-dimensional complex space.
problem Finding complete Calabi-Yau metrics in higher dimensions.
method Constructing metrics with specific properties on Cn. result Examples of complete Calabi-Yau metrics with Euclidean volume growth.
Study Kähler-Einstein metrics on singular varieties, proving metric completion properties.
problem Kähler-Einstein metrics on singular projective varieties.
method Approximation with constant scalar curvature metrics, RCD space analysis.
result Metric completion of smooth part is non-collapsed RCD space and homeomorphic to original variety.
For a complete Riemannian metric, a pointwise conformal transformation may lead to a complete or incomplete transformed Riemannian metric, depending on the behavior of the conformal factor. We establish conditions on the growth of the conformal factor towards the infinity of the Riemannian metric, such that the conform…
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.
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order n≥2. result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.
problem Extending the concept of metric spaces to Lorentzian spaces and proving their c-completion.
method Revisiting Lorentzian metric spaces, constructing c-completion, proving feasibility and endowing with Lorentzian metric space structure.
result The c-completion of Lorentzian metric spaces is feasible and well-suited, completing the original space in a precise sense.
We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finite-dimensional, orientable manifold with respect to a natural metric called the L2 metric. The primary motivation for studying this problem comes from Teichmueller theory, where similar considerati…
Complete classification of metric fibrations in Euclidean space.
problem Classifying metric fibrations in Euclidean space.
method Completed a minor gap in Gromoll and Walschap's classification.
result Completed the classification of Riemannian foliations on Euclidean spaces.
Complete left-invariant metrics on Lie groups with specific properties.
problem Completeness of left-invariant semi-Riemannian metrics on Lie groups.
method Introducing bi-Lipschitz Riemannian Clairaut metrics and proving completeness conditions.
result All left-invariant metrics are complete for certain Lie groups.
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
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.
Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
The paper studies geodesic completeness for Lie groups and their metrics.
problem Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups.
method Euler-Arnold formalism, detailed study of geodesics, classification of metrics.
result Full classification of geodesic completeness for the Lie group SL(2, C).
We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g is geodesically rigid: if any other complete metric gˉ has the same (unparametrized) geodesics with g, then the Levi-Civita connections of g and gˉ coincide.
The study proves the non-existence of certain Kähler metrics with specific curvature properties.
problem Non-existence of complete Kähler metrics with negatively pinched holomorphic sectional curvature.
method Construction of a Kähler metric with negatively pinched holomorphic sectional curvature and application of equivalence of invariant metrics.
result The dichotomy of completeness and non-existence of Kähler metrics with negatively pinched holomorphic sectional curvature.
A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. Characterizes metrics with finite total Q-curvature and introduces new volume entropy.
problem Understanding metrics with finite total Q-curvature and their geometric properties.
method Characterization of metrics through total Q-curvature and introduction of new volume entropy.
result Controlled volume growth for complete metrics with finite total Q-curvature and bounded scalar curvature.
Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.
problem Understanding the space of all locally elliptic and bounded Riemannian metrics on manifolds.
method Introduced an extended metric space and proved its properties.
result Proved the space of rough Riemannian metrics is complete and connected.
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
Study provides obstructions for Q-curvature on complete metrics in n-space.
problem Obstructing the Q-curvature prescription for complete conformal metrics.
method Analysis of decay rates and application of Bonnet-Mayer theorem.
result Found obstructions related to decay rates and Q-curvature properties.
Characterizes geodesic completeness for landmark spaces.
problem Ensuring geodesics exist for all times in landmark spaces.
method Integrability criterion based on cometric kernel behavior.
result Full characterization of geodesic completeness for smooth Riemannian metrics.
The paper constructs Einstein metrics on holomorphic bundles.
problem Finding complete conformally Kähler Einstein metrics on holomorphic bundles.
method Explicit momentum construction via ODE methods and Calabi ansatz.
result Non-trivial complete conformally Kähler Einstein metrics on certain holomorphic bundles are found.
We improve Riemannian metrics for constrained systems control.
problem Controlling mechanical systems with configuration constraints.
method Constructing complete Riemannian metrics by modifying incomplete ones.
result A controller can be found to satisfy a design criterion.
Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.
problem Equivalence of invariant metrics on noncompact Kähler manifolds with bounded curvature.
method Use boundedness of ratio between Bergman kernel and wedge product of metric in fundamental domain.
result Equivalence of Bergman metric and Kähler-Einstein metric under specific conditions.
In Finsler geometry the complete lift vector fields have distinguished geometric significance. For example a vector field on a Finsler manifold is said to be conformal if its complete lift is conformal in usual sense. In this work we define a new Riemannian or Pseudo-Riemannian metric on TM derived from a Finsler metri…
Completing segments of a real tree doesn't yield a complete space.
problem Completing segments of a real tree.
method Analyzing the field of real Puiseux series and the tree defined by Brumfiel.
result Completing all segments of the tree does not result in a complete metric space.
Study shows no complete positive scalar curvature for certain 3-manifolds.
problem Existence of complete metrics with positive scalar curvature on specific 3-manifolds.
method Analyzing Whitehead manifold and genus one 3-manifolds.
result No contractible genus one 3-manifold admits a complete metric of positive scalar curvature.
The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.
problem Proving the existence of complete Kähler metrics with negative holomorphic bisectional curvature in certain domains.
method Analyzing bounded domains in Cn with specific curvature properties. result Strictly pseudoconvex bounded domains and domains with squeezing function tending to 1 at boundary points admit complete Kähler metrics with negative holomorphic bisectional curvature everywhere.
Study constructs complete Kahler-Einstein metrics near isolated log-canonical singularities.
problem Constructing complete Kahler-Einstein metrics near isolated log-canonical singularities.
method Two approaches: uniformization by a complex ball and Calabi Ansatz.
result Two metrics are shown to be the same, providing a local model.
Negative curvature proven in Sasaki manifold space completion.
problem Curvature of Sasaki manifold completion.
method Mabuchi metric on Sasaki potentials space.
result Metric completion negatively curved in Alexandrov sense.
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥2 are metrically complete on the space In(S1,Rd) of Sobolev immersions of the same regularity and that any two curves i…
We prove the following statement: Let g be a light-line-complete pseudo-Riemannian Einstein metric of indefinite signature on a connected (n>2)-dimensional manifold M. Assume that a conformally equivalent metric is also Einstein. Then, the metrics are proportional with a constant coefficient. If in addition the manifol…
New method constructs G2-manifolds from Calabi-Yau 3-folds.
problem Creating complete non-compact G2-manifolds.
method Starting with a Calabi-Yau 3-fold, constructs a 1-parameter family of circle-invariant complete G2-metrics.
result Produces infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds.
Study critical metrics on manifolds, proving specific isometries.
problem Investigating critical metrics on complete manifolds.
method Analyzing volume functional and proving isometries.
result Critical metrics on specific manifolds are isometric to standard models.
Study on metric properties near boundary of tube domains.
problem Behavior of complete Kahler-Einstein metric near boundary.
method Estimates of metric and holomorphic bisectional curvatures.
result Obtained estimates near weakly pseudoconvex boundary points.
Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.
problem Metric on space of Hermitian metrics on complex vector bundles.
method Compute metric spray, geodesics, curvature, and use Nash-Moser theorem.
result Metric completion of Hermitian metrics space is L2 integrable singular Hermitian metrics.
The paper examines geodesic completeness in Lie groups with specific vector fields.
problem Investigating geodesic completeness in Lie groups with special vector fields.
method Analyzing left-invariant Lorentzian metrics on simple Lie groups with Killing vector fields.
result Conditions for geodesic completeness in Lie groups with specific vector fields.