The paper studies manifolds with positive weighted Ricci curvature of negative effective dimension.
problem Investigating properties of Riemannian manifolds with specific curvature conditions.
method Analyzing complete Riemannian manifolds with lower weighted Ricci curvature bound and discussing eigenvalues.
result If the minimum first nonzero eigenvalue is attained, the manifold splits off the real line as a warped product of hyperbolic nature.
Study on isoperimetric inequality on weighted Riemannian manifolds with negative effective dimension.
problem When does equality hold in the isoperimetric inequality on weighted Riemannian manifolds with negative effective dimension?
method Analyzes the conditions for equality in the isoperimetric inequality on weighted Riemannian manifolds with Ricci curvature bounded below.
result A weighted Riemannian manifold satisfying the isoperimetric inequality must be a warped product of hyperbolic nature.
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 study bounds torsion homology sizes in negatively curved manifolds.
problem Bounding torsion homology sizes in negatively curved manifolds.
method Effective simplicial thick-thin decomposition.
result Linear upper bounds for torsion homology sizes in all dimensions de3. Let Mn, n∈{4,5,6}, be a compact, simply connected n-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on Mn by a torus Tn−2 is equivariantly diffeomorphic to an isometric action on a normal biqu…
We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric T3 or T2 action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to [2n/3] and the free rank…
Let M0n be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M∈M0n, then M is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…
We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in hep-th/0510044. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponenti…
We obtain an Einstein metric of constant negative curvature given an arbitrary boundary metric in three dimensions, and a conformally flat one given an arbitrary conformally flat boundary metric in other dimensions. In order to compute the on-shell value of the gravitational action for these solutions, we propose to in…
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.
This work shows dimension regularization can replace skip-gram negative sampling for graph embeddings, improving efficiency and performance.
problem Efficiently enforcing dissimilarity among node embeddings in graph learning.
method Dimension regularization as an alternative to skip-gram negative sampling.
result Dimension regularization is a more efficient approach to enforcing dissimilarity in graph embeddings.
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.
Compact manifolds with positive scalar curvature have negative Kodaira dimension.
problem Understanding the properties of compact manifolds with positive scalar curvature.
method Analyzing the canonical bundle and complex structures on compact Riemannian manifolds.
result Compact manifolds with positive scalar curvature have negative Kodaira dimension.
Study uses NMF to reduce cancer microarray data dimensions.
problem High dimensionality of cancer microarray data hinders understanding.
method Used Non-negative Matrix Factorization (NMF) for dimensionality reduction.
result NMF achieves 98% classification accuracy.
Classifies SNC-algebras in 5D, calculating curvature.
problem Classifying SNC-algebras in higher dimensions.
method Defined SNC-algebras and used Lie group properties.
result Classified SNC-algebras in dimension five.
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.
New black hole solutions with positive and negative masses in 4 and 5 dimensions.
problem Constructing static vacuum black hole solutions with signed masses.
method Axisymmetric and bi-axisymmetric solutions in 4 and 5 dimensions, using Weyl-Papapetrou coordinates.
result Signed mass black holes can be superposed, with specific topologies in 5 dimensions.
Gravity derived from thermodynamics via optimal transport.
problem Equivalence between gravity and thermodynamics.
method Linking optimal transport to Raychaudhuri equation in warped-product spacetimes.
result Equivalence between gravity and concavity of entropy under time evolution.
Extends Kanai's result to higher dimensions for negatively curved manifolds.
problem Proving ergodic frame flows on negatively curved manifolds.
method Analyzing frame flows over negatively curved manifolds with specific curvature conditions.
result Proves that under certain conditions, the manifold is homothetic to a real hyperbolic manifold.
Solves Fu-Yau equation for negative slope parameters in arbitrary dimensions.
problem Solving the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
method Solves the Fu-Yau equation for negative slope parameters in arbitrary dimensions.
result First non-trivial solutions of the Fu-Yau equation in any dimension strictly greater than 2.
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorem…
Study finds infinitely many non-radial solutions for negative scalar curvature in higher dimensions.
problem Prescribing scalar and boundary mean curvature in a ball with negative scalar curvature.
method Analyzes the existence of infinitely many non-radial positive solutions in dimensions 5 and above.
result First existence result for negative scalar curvature in higher dimensions.
New compact K-E manifolds with negative curvature found.
problem Constructing compact Kähler-Einstein manifolds with negative curvature.
method Created compact Kähler-Einstein manifolds of dimension n with negative sectional curvature.
result Found compact Kähler-Einstein manifolds of negative curvature not covered by the ball.
Paper studies Hausdorff dimensions of specific limit sets for groups on curved spaces.
problem Determining Hausdorff dimensions of non-conical and Myrberg limit sets.
method Developed techniques to calculate Hausdorff dimensions for groups acting on negatively curved spaces.
result Established maximality of Hausdorff dimension for various cases.
We classify manifolds of small dimension that admit both, a Riemannian metric of non-negative scalar curvature, and a -- a priori different -- metric for which all wedge products of harmonic forms are harmonic. For manifolds whose first Betti numbers are sufficiently large, this classification extends to higher dimensi…
Ricci flow deforms metrics with positive curvature to include negative curvature.
problem Preserving positive sectional curvature under Ricci flow in dimension four.
method Evolved cohomogeneity one metrics on S4 and CP2 via Ricci flow. result Metrics with positive sectional curvature lose this property under Ricci flow.
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing so, we define a new invar…
It is proved that a compact Kahler manifold whose Ricci tensor has two distinct, constant, non-negative eigenvalues is locally the product of two Kahler-Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Irreducible Kahler manifolds with two distinct, constant eigenvalues of the Ricci…
The study examines non-negative curvature and conullity of curvature tensors.
problem Analyzing conullity conditions for curvature tensors and their compatibility with non-negative sectional curvature.
method Examined conullity two and three in specific dimensions, and used manifold properties to derive conditions.
result Finite volume manifolds with conullity 3 are locally products, and conditions are compatible with non-negative curvature only for specific manifold types.
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.
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theore…
Constructs Kahler-Einstein metrics near isolated log canonical singularities.
problem Constructing metrics near singularities in complex geometry.
method Constructs Kahler-Einstein metrics with negative scalar curvature near isolated log canonical singularities.
result Metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. New method for inference on covariates in NMF with random effects.
problem Formal inference for covariate effects in NMF with non-negativity constraints.
method NMF-RE model with random effects, ridge updates, df-based cap, asymptotic linearization, wild bootstrap.
result Valid inference on covariates with non-negativity constraint, avoiding degeneracy.
Kernel methods identify treatment effects with unobserved confounding using negative controls.
problem Learning causal relationships with unmeasured confounding.
method Kernel ridge regression algorithms for nonparametric treatment effects.
result Uniform consistency and finite sample rates of convergence proved.
We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space Wloc2,n/2 for manifolds of dimension less than or equal to 7 or spin-manifolds of any dimension. More generally, we give a (negative) lower bound on the ADM mass of metrics for which the scalar curvat…
Conditions ensure constant curvature in negatively curved manifolds.
problem Ensuring constant curvature in negatively curved manifolds.
method Intrinsic conditions on horospheres' geometry.
result Sectional curvature is constant under given conditions.
Negative interest rates stabilize economies, but physical cash limits their effectiveness.
problem Lack of effectiveness of negative interest rates in stabilizing economies.
method Simplified stock-flow consistent model, simulation evidence, discussion of alternative solutions.
result Negative interest rates can stabilize economies, but physical cash limits their effectiveness.
New proof for certain groups in higher dimensions.
problem Properties of discrete subgroups in higher dimensions.
method Proving convex-cocompactness for specific groups.
result Finitely generated Kleinian groups with small critical exponent are convex-cocompact.
We study the isoperimetric, functional and concentration properties of n-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N is negative, and more generally, is in the range N∈(−∞,1), extending the scope from the traditional range $N \i…
We classify closed, simply connected n-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions 2≤n≤6. In dimensions 3k, k=1,2 there is only one such manifold and it is diffeomorphic to the product of k copies of the 3-sphere.
In this paper we announce the following result: ``Every manifold of dimension ≥3 admits a complete negatively Ricci curved metric.'' Furthermore we describe some sharper results and sketch proofs.
We study noncompact, complete, finite volume, negatively curved manifolds M. We construct M with infinitely generated fundamental groups in all dimensions n≥2. We construct M whose cusp cross sections are compact hyperbolic manifolds in all dimension n≥3. In contrast we show that if sectional curvatu…
The paper extends rigidity results for special surfaces in general initial data sets.
problem Rigidity of marginally outer trapped surfaces with negative σ-constant.
method Generalization of previous results to higher genus and dimensions.
result A splitting result for the ambient manifold when it contains a stable closed MOTS.
New Einstein metrics found on complex manifolds.
problem Locally symmetric metrics on complex manifolds.
method Construction of manifolds with specific curvature properties.
result Infinitely many manifolds with negatively curved Einstein metrics but no locally symmetric metrics.
Constructs 7-manifolds with SO(3) invariant non-negative curvature.
problem Finding exotic spheres with non-negative curvature.
method Six-parameter family of highly connected 7-manifolds constructed with invariant metrics.
result All exotic spheres in 7 dimensions have non-negative curvature.
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wi…
We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …