Study on moduli spaces of negatively curved metrics on surfaces.
problem Understanding the structure of moduli spaces of uniformly negatively curved metrics on surfaces.
method Construction of locally constant functionals based on geodesic string counts.
result Moduli space of metrics on RimesS1 is disconnected. In this paper we prove that for all n=4k−2, k≥2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that π1(T<0(M)) is non-trivial. T<0(M) denotes the Teichmüller space…
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of S7-bundles over S8 and quotients of Milnor and Shimada spheres. result The moduli space of metrics has infinitely many path components.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
Kernel k-Groups uses Hartigan's method for clustering in metric spaces of negative type.
problem Clustering in metric spaces of negative type.
method Weighted energy statistics, quadratically constrained quadratic program, kernel k-groups, Hartigan's method.
result Improved performance in higher dimensions compared to spectral clustering and kernel k-means.
Proves existence of smooth isometric immersions for certain curved surfaces.
problem Existence of smooth isometric immersions for negatively curved surfaces.
method Compensated compactness and invariant regions in hyperbolic conservation laws.
result Proves existence of C1,1 isometric immersions for specific metrics. The study of Kähler metrics on domains restricts their boundary geometry.
problem Understanding the geometry of domains with negatively pinched Kähler metrics.
method Analyzing the existence and properties of negatively pinched Kähler metrics on domains.
result The boundary of a convex domain without complex subvarieties of positive domain if it admits a complete Kähler metric with pinched negative holomorphic bisectional curvature.
We prove that n-dimensional (n⩾3) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure n-space (i.e., the Euclidean metric n-space).
The paper classifies metrics with constant negative Q-curvature in Euclidean spaces.
problem Classifying metrics with constant negative Q-curvature in Euclidean spaces.
method Variational techniques and finite volume conditions.
result Existence and classification of singular and nonsingular metrics with constant negative Q-curvature.
In the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of th…
Constructs metrics with negative curvature on specific manifold types.
problem Creating negatively curved metrics on locally conformally flat manifolds.
method Using Morse functions to construct conformal metrics.
result Successfully constructs conformal metrics with negative sectional curvature.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…
Study on moduli spaces of non-negative curvature metrics on manifolds.
problem Understanding the topology of moduli spaces of non-negative curvature metrics.
method Construction of manifolds with specific curvature properties and analysis of their moduli spaces.
result First classes of manifolds with non-trivial rational homotopy, homology, and cohomology groups for moduli spaces of non-negative sectional curvature.
The Wu-Yau theorem is verified for negative curvature, and new examples of Kähler-Einstein metrics are found.
problem The Wu-Yau theorem and its positive analog.
method Examples and conjectures to verify the Wu-Yau theorem and its positive analog.
result New examples of Kähler-Einstein metrics without negative holomorphic sectional curvature.
Constructs Lie groups with negative Ricci curvature.
problem Finding Lie groups with negative Ricci curvature.
method Constructs Lie groups with specific algebraic structures and representations.
result Proves existence of Lie groups with negative Ricci curvature for various Levi factors.
The Hessian of Ricci Calabi functional is non-negative at Kähler Einstein metrics.
problem Understanding the geometry of Kähler metrics on Fano manifolds.
method Analyzing the Hessian of the Ricci Calabi functional.
result The Hessian of the Ricci Calabi functional is non-negative at generalized Kähler Einstein metrics.
Given a negatively curved geodesic metric space M, we study the statistical asymptotic penetration behavior of (locally) geodesic lines of M in small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of M. We prove Khintchine-type and logarithme law-type results for the s…
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
problem Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
method Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations using specific curvature conditions.
result Explicit construction of Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
The Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible.
problem Proving the non-contractibility of Teichmüller space for complex hyperbolic manifolds.
method Analyzing the Teichmüller space of negatively curved metrics on complex hyperbolic manifolds.
result The Teichmüller space of negatively curved metrics on complex hyperbolic manifolds is not contractible.
Study on flag curvatures of specific metrics on Lie groups.
problem Investigate flag curvatures of left invariant (α,β)-metrics and Randers metrics. method Explicit formulas and necessary/sufficient conditions for Berwald and Douglas types.
result Flag curvatures of (α,β)-metrics and Randers metrics on Lie groups can be zero, positive, or negative. New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. New metric with negative curvature found near positive-curvature spaces.
problem Understanding spaces with positive curvature bounds.
method Analyzing metrics in relation to curvature bounds.
result Near any point in a positive-curvature space, there exists a metric with negative upper curvature.
We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n≥10, is highly non-connected.
The paper finds many negatively curved Kähler metrics on complex manifolds.
problem Finding Kähler metrics with negative curvature on complex manifolds.
method Analyzes vector bundles and proves dimension estimates and Liouville theorems.
result Proves existence of complete Kähler metrics with negative curvature on certain total spaces.
We show that on Kahler manifolds with negative first Chern class, the sequence of algebraic metrics introduced by H. Tsuji converges uniformly to the Kahler-Einstein metric. For algebraic surfaces of general type and orbifolds with isolated singularities, we prove a convergence result for a modified version of Tsuji's …
New proof shows no negative curvature Einstein metrics in specific dimensions.
problem Proving nonexistence of certain Einstein metrics in 9 and 10 dimensions.
method Cohomogeneity-one approach to show nonexistence of negative curvature Einstein metrics.
result Noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions.
Study shows nontrivial homotopy groups for negatively curved metrics on hyperbolic manifolds.
problem Understanding the homotopy groups of Teichmüller spaces for negatively curved metrics.
method Analyzes Teichmüller spaces of negatively curved metrics on hyperbolic manifolds, proving nontrivial homotopy groups for some dimensions.
result Proves existence of nontrivial rational homotopy groups and elements of infinite order in π_i B Diff(M).
The paper proves conjectures and classifies metrics on 3D manifolds.
problem Proving conjectures and classifying metrics on 3D manifolds with specific curvature conditions.
method Analytical proofs and classification theorems.
result Critical metrics on 3D manifolds are isometric to geodesic balls in space forms.
We study the moduli space of negatively curved metrics of a hyperbolic manifold.
New Einstein metrics found in curved spaces.
problem Finding Einstein metrics in curved spaces.
method Analyzing almost-Einstein metrics to find genuine Einstein metrics.
result Negative curvature preserved in Einstein metrics.
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with singularities conjugate to ADE-type is proved. In 1988, Claude Lebrun gave examples of scalar-flat Kähler ALE metrics with negative mass, on the total space of the bundle O(−n) over S2. A corollary of this index …
We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. …
We show that the space of nonpositively curved metrics of a negatively curved manifold is highly non connected.
The paper proves properties of manifolds with negative holomorphic sectional curvature.
problem Negative holomorphic sectional curvature properties of manifolds.
method Analyzing irreducible subvarieties and extending results to quasi-negative curvature.
result Quasi-projective manifolds with negative holomorphic sectional curvature are of log general type.
This is a survey on known results and open problems about Smooth and PL-Rigidity Problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved me…
Researchers prove unique embedding of curved surfaces into Minkowski spacetime.
problem Embedding curved surfaces into Minkowski spacetime.
method GHMC Minkowski spacetime construction.
result Unique GHMC Minkowski spacetime for given curved surfaces.
In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n>2. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler …
Embeds persistence diagrams into Hilbert spaces to use kernel methods.
problem No inner product structure on persistence diagrams.
method Shows non-embeddability of persistence diagrams into Hilbert spaces.
result Persistence diagrams with bottleneck distance do not coarse embed into Hilbert spaces.
The sectional curvature of the Weil-Petersson metric on Teichmuller space is known to be negative. We show that this Weil-Petersson sectional curvature is not pinched from above by any negative constants, i.e., there is no negative upper bound.
Investigates maps and properties in spaces with negative dimensions and curvature.
problem Existence of transport maps and local-to-global property in spaces with negative dimensions and bounded Ricci curvature.
method Examines metric measure spaces with negative curvature dimensions and applies reduced curvature-dimension conditions.
result Establishes the existence of transport maps and proves the local-to-global property.
The paper proves conditions for Kähler manifolds with negative curvature.
problem Conditions for existence of Kähler-Einstein metrics and holomorphic curves.
method Analyzes compact Kähler manifolds homotopic to negatively curved Riemannian manifolds.
result Compact Kähler manifolds with negative curvature admit Kähler-Einstein metrics of general type.
We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on th…
Study stability of curvature-dimension condition for negative dimensions.
problem Stability of curvature-dimension condition with negative dimension parameters.
method Introduced CD(K, N)-condition for N < 0, defined distance d_{\mathsf{iKRW}}, proved convergence stability.
result Limit structure of converging metric measure spaces remains CD(K, N) for N < 0.
We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations of some classical results concerning moduli spaces of positive scalar curvature m…