Maximal diameter theorem for graphs with positive Ricci curvature.
problem Diameter comparison in directed graphs with positive Ricci curvature.
method Introduced a Lin-Lu-Yau type Ricci curvature for directed graphs and investigated rigidity properties for the equality case.
result Concluded a maximal diameter theorem of Cheng type.
The paper extends Bonnet-Myers theorem for manifolds with nonnegative Ricci curvature.
problem Compactness and diameter estimation for manifolds with nonnegative Ricci curvature.
method General curvature conditions for estimating diameter and compactness criteria.
result Established compactness theorems for manifolds with polynomial or exponential Ricci curvature decay.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
The paper studies Finsler manifolds and their integral curvatures.
problem Understanding the curvature of Finsler manifolds.
method Proves Myers type theorems on integral curvatures of Finsler manifolds.
result Establishes diameter estimates based on integral curvature bounds.
New diameter estimate on manifolds with positive Bakry-Émery Ricci tensor.
problem Estimating the diameter of manifolds with specific curvature conditions.
method Using generalized mean curvature comparison on excess function.
result Sharper diameter estimate than previous results.
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
Bonnet-Myers theorem applied to Q-curvature on 4-manifolds.
problem Bounding Q-curvature on 4-manifolds.
method Scalar curvature and Q-curvature bounds.
result Diameter bound for Q-curvature quotient.
The paper calculates graph Ricci curvature and finds properties of specific graph types.
problem Understanding Ricci curvature on irregular graphs.
method Developed a formula for graph Ricci curvature based on optimal bijections.
result Derived structural and theorem results for specific graph types.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.
problem Understanding finite diameter constraints in Lorentzian geometry.
method Constructing Alexandrov-Patchwork and proving Bonnet-Myers theorem for Lorentzian spaces.
result Lorentzian spaces with curvature bounds have finite diameter.
The study proves a theorem for quaternionic contact manifolds, showing they are compact under certain conditions.
problem Understanding the compactness of quaternionic contact manifolds.
method Proving a Bonnet-Myers type theorem for quaternionic contact manifolds with specific Ricci-type bounds.
result Quaternionic contact manifolds of dimension > 7 are compact under given conditions.
The study proves a compactness theorem for manifolds with specific curvature conditions.
problem Proving compactness theorems for manifolds with Bakry-Emery Ricci tensor.
method Using Bakry-Emery Ricci tensor and smooth measure.
result Generalized Myers compactness theorem proved.
The standard Bonnet-Myers theorem says that if the Ricci scalar of a Riemannian manifold is bounded below by a positive number, then the manifold is compact. Moreover, a bound of its diameter is pointed out. The theorem was extended to Finsler manifolds. In this paper we prove that if a certain condition on the average…
The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
problem Analyzing modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.
method Proving Laplacian comparison theorems using modified m-Bakry-Emery Ricci tensors under m≤1.
result Optimal conditions for modified m-Bakry-Emery Ricci tensors under m≤1 are derived.
Graphs with Bakry-Emery curvature have bounded diameter.
problem Bounding the diameter of graphs with positive Ricci curvature.
method Proving diameter bounds using Bakry-Emery curvature and vertex degree.
result First Bonnet-Myers type theorem for unbounded graph Laplacians.
Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two open problems for closed 3 manifolds with positive Ricci curvature since they are b…
New curvature measure defined for graphs, with bounds on diameter and spectral gap.
problem Defining curvature for graphs and proving its properties.
method Solving linear systems to compute curvature; applying minimax theorem.
result Graphs with positive curvature have bounded diameter and spectral gap.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
Rigidity properties of hypercube graphs via curvature methods.
problem Rigidity of hypercube graphs under curvature constraints.
method Semigroup methods and new direct methods translating curvature to combinatorial properties.
result Sharp inequalities for diameter and eigenvalues only hold for hypercubes.
New curvature measure on graphs improves diameter and eigenvalue bounds.
problem Improving curvature bounds on graph structures.
method Hybrid curvature definition on variable neighborhoods.
result Gradient estimates and curvature bounds proven.
The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.
problem Finding diameter bounds for graphs with positive entropic Ricci curvature.
method Using a localized gradient estimate and an equivalent definition of entropic Ricci curvature, the study derives a Bonnet-Myers type diameter bound.
result The derived diameter bound is optimal for arithmetic mean, but not for logarithmic mean.
For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider …
Improved inequalities on Kähler manifolds with curvature bounds.
problem Diameter bounds on Kähler manifolds with positive Ricci curvature.
method Proving new Beckner-Sobolev inequalities.
result Improved diameter bounds compared to Bonnet-Myers bound.
Extends Bonnet-Myers theorem with new generalizations.
problem Generalizing Bonnet-Myers theorem.
method Complementary generalization of existing extensions by Calabi and Cheeger-Gromov-Taylor.
result New theorem extending previous extensions of Bonnet-Myers.
New curvature measure for graphs improves diameter and eigenvalue estimates.
problem Estimating properties of graphs using Ricci curvature.
method Introduced integral Ricci curvature Iκ0 for graphs. result Uniform estimates for diameter, number of vertices, and eigenvalue.
The paper extends Bonnet-Myers theorems using Bakry-Emery Ricci curvature.
problem Proving extensions of Bonnet-Myers theorems with a new curvature measure.
method Using Bakry-Emery Ricci curvature to extend Bonnet-Myers theorems.
result Proves extensions of Bonnet-Myers theorems.
Proves a local version of Myers-Steenrod theorem for specific manifolds.
problem Generalization of Myers-Steenrod theorem to local topological groups.
method Proof for local topological groups of isometries acting on specific manifolds.
result New regularity result for locally homogeneous Riemannian metrics.
Generalizes Bonnet-Myers theorem with small Kato constant.
problem Bounding volume of manifolds with small Kato constant.
method Analyzes Ricci curvature in Kato sense and applies to volume bounds.
result Generalizes Bonnet-Myers theorem with new conditions.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
The Myers-Steenrod theorem is extended to Finsler manifolds with low regularity.
problem Extending the Myers-Steenrod theorem to Finsler manifolds with minimal regularity.
method Reduction of Finslerian problems to Riemannian ones using the Binet-Legendre metric.
result Isometries between Ck,α-smooth Finsler metrics are diffeomorphisms of class Ck+1,α. The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
The study compares spectral volumes of manifolds with weakly convex boundaries.
problem Establishing volume comparison theorems for manifolds with weakly convex boundaries.
method Using spectral methods and Ricci tensor eigenvalues, the study compares volumes and diameters of manifolds.
result Sharp upper bounds for the volume and diameter of manifolds with weakly convex boundaries.
The paper proves a compactness theorem for spaces with Bakry-Emery Ricci tensor.
problem Compactness in spaces with Bakry-Emery Ricci tensor.
method Proving f-mean curvature comparison and defining a Myers-type compactness theorem. result Improves a result from Soylu by using a weaker condition on f′(t). In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultima…
The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.
problem Extending the Bakry-Émery-Ricci tensor and proving comparison theorems.
method Generalizations of the drifted Laplacian and Bakry-Émery-Ricci tensor, mean curvature comparison theorem, Myers-type theorem, Cheeger-Gromoll splitting theorem.
result Proved a version of the mean curvature comparison theorem and its consequences.
Study on Kähler Finsler manifolds with curvature bounds, proving theorems.
problem Understanding Kähler Finsler manifolds with curvature constraints.
method Analyzing partial parallelism of complex structure, proving theorems.
result Generalized comparison theorem for positively curved Kähler Finsler manifolds.
The paper proves compactness theorems for specific types of tensors.
problem Proving compactness theorems for Riemannian manifolds with specific tensors.
method Using h-almost Ricci tensors and generalized quasi-Einstein tensors, the paper extends previous theorems. result Theorems are extended to cases where h has at most linear growth. The study proves a new positive energy theorem for manifolds with specific curvature properties.
problem Proving a new positive energy theorem for manifolds with specific curvature properties.
method Establishing a systolic inequality relating boundary mean curvature to the systole of the boundary.
result Obtaining a new positive energy theorem, with equality for Horowitz-Myers metrics.
In this paper,we prove the following Myers-type theorem: if (Mn,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥εRg>0, where ε>0 is an uniform constant, then Mn must be compact.
Study compares H-type sub-Riemannian manifolds using uniform metrics.
problem Comparing H-type sub-Riemannian manifolds with Riemannian metrics.
method Establishes sub-Hessian and sub-Laplacian comparison theorems for a family of approximating Riemannian metrics.
result Proves a sharp sub-Riemannian Bonnet-Myers theorem.
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
In this paper we study the behavior of solutions of a second order differential equation. The existence of a zero and its localization allow us to get some compactness results. In particular we obtain a Myers' type theorem even in the presence of an amount of negative curvature. The technique we use also applies to the…
New geometric approach to manifolds with density, proving rigidity results.
problem Study of Riemannian manifolds with density.
method Introduce a new connection and use it to motivate volume and Laplacian comparison theorems.
result Prove new generalizations of Myers' and Cheng's theorems.
The abstract develops weighted Ricci curvature in Lorentz-Finsler geometry and extends singularity theorems.
problem Extending singularity theorems in weighted Lorentz-Finsler geometry.
method Generalizing Jacobi, Riccati, and Raychaudhuri equations; applying generalized Bishop inequality.
result Weighted Lorentz-Finsler singularity theorems extended.
Harmonic coordinates for Finsler manifolds prove a theorem but not optimal regularity.
problem Proving the Myers--Steenrod theorem for Finsler manifolds.
method Existence of harmonic coordinates for nonlinear Finsler Laplacian.
result Partial results on optimal regularity for Berwald metrics.
We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…
The universe's shape and size are determined in general cosmological models.
problem Determining the shape and size of the universe in general cosmological models.
method Using differential geometry and extensions of the Bonnet-Myers theorem, the researchers derived conditions for a finite universe and provided a list of possible topologies.
result The spatial sections of the universe can be either S1imesS2, S1ildeimesS2, S1imesRP2, RP3#RP3, or covered by the sphere S3 or torus T3. By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality CDE′(n,0), which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative cur…