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 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). 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 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. 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.
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.
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…
We prove a Bonnet-Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifo…
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.
Paper proves Horowitz-Myers conjecture in 3-7 dimensions.
problem Proving Horowitz-Myers conjecture in specific dimensions.
method Alternative proof and local isometry demonstration.
result Metrics achieving conjecture equality locally match Horowitz-Myers metrics.
Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci sol…
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…
Compact Riemannian manifolds with mostly positive curvature have finite fundamental groups.
problem Understanding conditions for finite fundamental groups in compact Riemannian manifolds.
method Using Bismut-Witten Laplacian and Ricci-Hessian inequalities.
result New conditions for finite fundamental groups in manifolds with mostly positive curvature.
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.
Proves a sharp inequality for toroidal surfaces in Horowitz-Myers geon.
problem Extending Minkowski inequality to non-static spacetimes.
method Proves a sharp inequality for toroidal hypersurfaces in Horowitz-Myers geon.
result Sharp inequality for toroidal surfaces in asymptotically hyperbolic manifold.
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 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.
We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs J(2n,n), the Gosset graph and suitable Cartesian …
This paper completes the classification of S1-symmetric static vacuum black holes.
problem Identifying all S1-symmetric static vacuum black hole solutions.
method Analyzing and constructing known solutions and proving their completeness.
result Proves that the Schwarzschild, Boost, and Weyl-Korotkin-Nicolai families exhaust all S1-symmetric static vacuum black hole solutions.
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.
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.
We give a complementary generalization of the extensions of Bonnet-Myers theorem obtained by Calabi and also Cheeger-Gromov-Taylor.
In this paper we prove a new Myers' type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry-Émery Ricci tensor has a positive lower bound. The result is sharper than previous Myers' type results. The proof uses the generalized mean curvature comparison …
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.
We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…
The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we …
Proves closure for specific spacetimes with certain conditions.
problem Proving closure for globally hyperbolic spacetimes.
method Using a Bonnet-Myers type result.
result Proves closure for spacetimes with specific conditions.
Higher-dimensional spacetimes have well-behaved boundaries.
problem Understanding boundaries of higher-dimensional spacetimes.
method Analyzing (n+1)-dimensional Myers-Perry metrics at spacelike infinity. result Optimal conformal completion at spacelike infinity for Cn−3,1 differentiability class. 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.
Paper uses language models to predict MBTI personality types with high accuracy.
problem Predicting Myers-Briggs personality types from text.
method Fine-tuned BERT model for predicting MBTI types and generating personality-specific language.
result BERT model achieves high accuracy in predicting MBTI types and personality-specific language generation.
New surgeries found in 3D shapes without 2-spheres.
problem Finding non-hyperbolic surgeries in 3-manifolds.
method Combining work on hyperbolic knots and 3-manifolds with observations about surgeries.
result Every 3-manifold with certain properties contains a hyperbolic knot with a non-trivial surgery.
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.
We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and L2 harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compa…
We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension…
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.
Proves positive energy conjecture for a specific metric class.
problem Proving the positive energy conjecture for a class of AHM metrics.
method Analyzes asymptotically Horowitz-Myers (AHM) metrics on R2imesTn−2. result Generalizes previous results on positive energy conjecture.
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.
This is the second article of a series or two, proving a generalisation of the uniqueness theorem of the Schwarzschild solution. The theorem to be shown classifies all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without any further assumptio…
The paper refines Steinerberger curvature for block graphs and bridges.
problem Understanding curvature in graph theory.
method Formulas and relations for curvature in block graphs and graph bridges.
result Self-centered Bonnet-Myers sharp graphs are antipodal.
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.
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.
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 paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.
problem Analyzing diffusion processes on Riemannian manifolds with weighted metrics.
method Proving a Laplacian comparison theorem and applying it to various geometric and analytic properties of diffusion processes.
result Optimal conditions on m-Bakry-Émery Ricci tensor for various geometric and analytic properties to hold on weighted complete Riemannian manifolds. New proofs confirm travel time data determine simple metrics on a disc.
problem Determining a simple Riemannian metric from travel time data.
method Proofs based on Myers-Steenrod theorem, Lipschitz-type stability estimate.
result Travel time data determine a simple Riemannian metric on a disc up to natural gauge.
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.
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.