The paper discusses conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
problem Conditions for gluing multiple Alexandrov spaces into an Alexandrov space.
method Proposes a Gluing Conjecture and proves it under certain conditions.
result The Gluing Conjecture is true under specific conditions, generalizing Petrunin's Gluing Theorem.
Proves Borel Conjecture for certain 3D spaces.
problem Characterizing fundamental groups of 3D Alexandrov spaces.
method Analyzes properties of Alexandrov 3-spaces.
result Proves Borel Conjecture for specific types of spaces.
Proves a similar inequality to a conjecture about hyperbolic space hypersurfaces.
problem Proving a conjecture about weighted Alexandrov-Fenchel inequalities for hyperbolic space hypersurfaces.
method Analyzes horospherically convex hypersurfaces in hyperbolic space.
result Proves a similar inequality to the conjectured one, provides a counterexample when applicable.
The paper conjectures bounds on intrinsic volumes for Riemannian manifolds and Alexandrov spaces.
problem Bounding intrinsic volumes for Riemannian manifolds and Alexandrov spaces.
method Conjectures and suggests extensions of intrinsic volumes to Alexandrov spaces.
result Conjectures on bounds for intrinsic volumes and their behavior under Gromov-Hausdorff limits.
We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…
We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.
Estimates eigenvalues on differential forms on Alexandrov spaces.
problem Estimating eigenvalues of differential form Laplacians on Alexandrov spaces.
method Using Alexandrov spaces with curvature bounded below, constructing differential form Laplacians, and applying local biLipschitz assumptions.
result The differential form Laplacian has a compact resolvent under local biLipschitz assumption, and its kernel is identified with an intersection homology group.
Proves Lipschitz regularity of harmonic maps from Alexandrov spaces.
problem Lipschitz regularity of harmonic maps from finite dimensional Alexandrov spaces.
method Extends Huang-Wang's argument to finite dimensional Alexandrov spaces.
result Solves F. H. Lin's conjecture on harmonic maps.
Proves Euler characteristic of collapsing Alexandrov spaces.
problem Euler characteristic of collapsing Alexandrov spaces.
method Analyzes strata and fibers of the limit space.
result Euler characteristic equals sum of products of strata and fiber Euler characteristics.
Given a compact Alexadrov n-space Z with curvature curv ≥κ, and let f:Z→X be a distance non-increasing onto map to another Alexandrov n-space with curv ≥κ. The relative volume rigidity conjecture says that if X achieves the relative maximal volume i.e. vol(Z)=vol(X), then X is isometric to $…
We prove that Alexandrov's conjecture relating the area and diameter of a convex surface holds for the surface of a general ellipsoid. This is a direct consequence of a more general result which estimates the deviation from the optimal conjectured bound in terms of the length of the cut locus of a point on the surface.…
Unified framework for Alexandrov 3-spaces, extending manifold results.
problem Extend manifold topology results to Alexandrov 3-spaces.
method Generalize connected sum, prime decomposition, and Dehn surgery.
result Unified framework for Alexandrov 3-spaces, including non-manifold spaces.
Estimates volume of convex Alexandrov spaces with boundary.
problem Estimating volume of Alexandrov spaces with convex boundaries.
method Gradient flow of semi-concave functions.
result Volume upper bound achieved implies Boundary Conjecture.
Abstract Riemann curvature tensor defined on RCD spaces.
problem Defining Riemann curvature tensor on RCD spaces. method Distributional-like approach to introduce curvature tensor.
result Conjecture that RCD spaces are Alexandrov if curvature is bounded. Classifies 4D spaces with circle symmetry and positive curvature.
problem Classifying spaces with specific geometric properties.
method Equivariant homeomorphism classification, infinitesimal geometry analysis.
result Classification of 4D Alexandrov spaces and orbifolds with circle symmetry.
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
Paper improves eigenvalue bounds for Laplacian in Alexandrov spaces.
problem Upper bounds of eigenvalues of Laplacian in subsets.
method Reduces subsets in Chung-Grigor'yan-Yau's inequality for Alexandrov spaces.
result Shows eigenvalue bounds can be achieved with fewer subsets.
The paper solves a conjecture about spacelike hypersurfaces in de Sitter space.
problem Proving an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
method Investigating the locally constrained inverse curvature flow to establish the inequality.
result Established an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
This paper explores rigid properties of Alexandrov spaces with maximal radius.
problem Rigidity of Alexandrov spaces with maximal radius and specific curvature conditions.
method Analyzes Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \fracπ{2}. Uses rigidity theorems and geometric/topological constraints.
result Shows flexibility and rigidity in Alexandrov spaces with maximal radius under different conditions.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Metric measure boundary vanishes on certain spaces without boundary.
problem Existence of infinite geodesics on Alexandrov spaces without boundary.
method Solving conjecture by showing metric measure boundary vanishes on mRCD(K,N) spaces. result Metric measure boundary vanishes on mRCD(K,N) spaces without boundary. The paper proves properties of boundaries of Riemannian foliations.
problem Proving properties of boundaries of Riemannian foliations.
method Proved boundaries of orbit spaces and leaf spaces are Alexandrov spaces with lower curvature bounds.
result Rigidity theorem for positively curved leaf spaces with maximal boundary volume.
Gromov and Lawson conjectured that a closed spin manifold M of dimension n with fundamental group pi admits a metric with positive scalar curvature if and only if an associated element in KO_n(B pi) vanishes. In this note we present counter examples to the `if' part of this conjecture for groups pi which are torsion fr…
Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, close…
The paper proves uniqueness of immersed spheres in three-manifolds.
problem Proving uniqueness of immersed spheres in three-manifolds.
method Solves the Hopf uniqueness problem for a class of immersed surfaces modeled by elliptic PDEs.
result Any compact immersed surface of genus zero in the class is a candidate sphere.
Disproves Fedotov's conjecture on higher-order Shephard inequalities.
problem Fedotov's conjecture on higher-order Shephard inequalities.
method Using Hodge-Riemann relations for simple convex polytopes.
result Fedotov's conjecture is disproved.
The paper studies asymptotic dimensions of manifolds and spaces, proving key results about their geometric decompositions.
problem Understanding the asymptotic dimensions of manifolds and spaces with geometric decompositions.
method Analyzing the fundamental groups and using geometric decompositions to derive asymptotic dimension bounds.
result Asymptotic dimensions of certain manifolds and spaces are bounded and equal to specific values.
This note explores comparison geometry concepts and theorems.
problem Exploring various comparison theorems in geometry.
method Analyzes Rauch and Toponogov theorems and their applications.
result Introduction of Gromov-Hausdorff convergence and Alexandrov Spaces.
The paper proves a conjecture about manifold limits and characterizes their structure.
problem Characterizing limits of manifolds with a uniform contractibility function.
method Short proof using Gromov-Hausdorff distance and ANR properties.
result Obstruction vanishes if and only if the manifold can be approximated by PL-manifolds.
The paper measures and limits the extent of non-smooth points in Alexandrov spaces.
problem Understanding the extent of non-smooth points in Alexandrov spaces.
method Defining a non-negative function K(x) to measure the extent of non-smoothness and quantitatively estimating its distribution. result The Hausdorff dimension estimate and quantitative Hausdorff measure estimate for the set of C2-singular points. We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with …
Study confirms C1 regularity for convex functionals with bounded degeneracy set.
problem Confirming C1 regularity for minimizers of convex functionals with small degeneracy set. method Building on previous work, confirms C1 regularity when D2F is positive and bounded away from finitely many points. Constructs a counterexample in R4 where F is strictly convex but D2F degenerates on a Simons cone intersection. result Confirms C1 regularity for minimizers of convex functionals with small degeneracy set. This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to …
Embeds finite Alexandrov spaces into Euclidean products.
problem Embedding Alexandrov spaces into Euclidean products.
method Canonical embedding into a product of Alexandrov and Euclidean spaces.
result Affine functions on X arise from Euclidean functions.
Establishes equivalence between models of derived stacks.
problem Tackles the equivalence between different models of derived geometry.
method Uses Quillen equivalence to show categories of higher derived stacks are equivalent.
result Shows equivalence among models of derived manifolds, Carchedi-Roytenberg, Behrend-Liao-Xu, and Alexandrov-Kontsevich-Schwarz-Zaboronsky.
Study on 3D Alexandrov spaces with Ricci curvature bounds.
problem Characterizing 3D Alexandrov spaces with specific Ricci curvature bounds.
method Analysis of Alexandrov spaces with CD∗(K,N) sense, focusing on positive or nonnegative Ricci curvature. result Classification of closed 3D Alexandrov spaces with CD∗(0,3) and CD∗(2,3). Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
Paper proves BGW tau-function can be represented as Q-polynomials.
problem Enumerative geometric interpretations of BGW tau-function.
method Proves BGW tau-functions are hypergeometric tau functions of BKP hierarchy.
result Original BGW tau-function can be represented as a linear combination of Schur Q-polynomials.
Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
The paper extends an Alexandrov theorem to Minkowski spacetime.
problem Generalizing Alexandrov's theorem to Minkowski spacetime.
method Adapting conditions for closed codimension-two spacelike submanifolds in Minkowski spacetime.
result A generalized Alexandrov theorem for Minkowski spacetime.
Classifies 4D Alexandrov spaces with torus actions.
problem Classifying Alexandrov spaces with torus actions.
method Equivariant classification and homeomorphism analysis.
result Alexandrov spaces are homeomorphic to Riemannian orbifolds.
Generalizes Toponogov theorem to Alexandrov spaces.
problem Estimating curve length in non-Euclidean spaces.
method Generalization of Toponogov theorem.
result Proved the length of a curve in two-dimensional Alexandrov spaces.
Generalizes a soul-bound for noncompact Alexandrov spaces.
problem Finding a lower bound for injectivity radius in Alexandrov spaces.
method Introduces the soul of Alexandrov spaces and applies a generalized bound.
result Injectivity radius is at least πK⁻¹/² if not equal to the soul's.
The paper improves collapsing Alexandrov spaces results using good coverings.
problem Collapsing Alexandrov spaces with no proper extremal subsets.
method Application of good coverings of Alexandrov spaces.
result Construction of an infinitely long exact sequence of homotopy groups and a spectral sequence of cohomology groups.
Graph comparison ties to Alexandrov's theorems.
problem Graph comparison conditions on metric spaces.
method Proof of Alexandrov's implications from graph comparisons.
result Complete description of graphs with trivial comparisons.
Introduces Alexandrov spaces with curvature below, covering various theorems.
problem Understanding spaces with curvature constraints.
method Explains comparison conditions, globalization, tangent spaces, etc.
result Globalization theorem and other theorems established for Alexandrov spaces.
Proves Alexandrov theorem for weighted surfaces.
problem Proving Alexandrov theorem for surfaces with constant weighted mean curvature.
method Using Heintze--Karcher type inequality for log-convex weights in Euclidean and Hyperbolic spaces.
result Obtains Alexandrov theorem for λ-self expanders.