arXiv research

Maximal diameter theorem for graphs with positive Ricci curvature.

This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.

Max diameter Kahler manifolds with positive bisectional curvature are complex projective spaces.

We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian $n$-manifold $M$ having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …

Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.

Concerning the set of exceptional surgery slopes for a hyperbolic knot, Lackenby and Meyerhoff proved that the maximal cardinality is 10 and the maximal diameter is 8. Their proof is computer-aided in part, and both bounds are achieved simultaneously. In this note, it is observed that the diameter bound 8 implies the m…

We prove that if a complete connected $n$-dimensional Riemannian manifold $M$ has radial sectional curvature at a base point $p\in M$ bounded from below by the radial curvature function of a two-sphere of revolution $\widetilde M$ belonging to a certain class, then the diameter of $M$ does not exceed that of $\widetild…

We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result i…

In this paper, two lower bounds on the diameters of the boundary slope sets are given for Montesinos knots. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.

The main result of this article states that the (K;N)-cone over some metric measure space satisfies the reduced Riemannian curvature-dimension condition RCD^*(KN;N+1) if and only if the underlying space satisfies RCD^*(N-1;N). The proof uses a characterization of reduced Riemannian curvature-dimension bounds by Bochner…

The maximal analytic Schwarzschild spacetime is manifestly inextendible as a Lorentzian manifold with a twice continuously differentiable metric. In this paper, we prove the stronger statement that it is even inextendible as a Lorentzian manifold with a continuous metric. To capture the obstruction to continuous extens…

Study quotients of curve complex actions by mapping class group.

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.

We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to $\mathbb{R}$, but does not topologically split. The second space satisfies…

Characterizes Forman curvature bounds and proves curvature equivalence.

For an orientable surface of finite type equipped with a flat metric with holonomy of finite order q, the set of maximal embedded cylinders can be empty, non-empty, finite, or infinite. The case when q < 3 is well-studied as such surfaces are (semi-)translation surfaces. Not only is the set always infinite, the core cu…

The paper proves Laplacian comparison theorems for modified m-Bakry-Emery Ricci tensors on Riemannian manifolds.

Study on positive scalar curvature and its impact on Ricci limit spaces.

New simplicial complex for infinite-type surfaces shows graph properties.

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our r…

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

Improved lower bound for first Dirichlet eigenvalue using variance refinement.

We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus $g$ with a single boundary curve and $n$ marked points on this curve; we consider triangulations up to homeomor…

The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my previous papers cited in the present paper as [K1] and [K2], and for the space SU(3)/SO(…

Torus covers have controlled volume and diameter under curvature and diameter bounds.

Short proof shows infinite diameter for surface diffeomorphisms.

The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.

Exact diameter found for some Riemann surfaces.

The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.

Sharp diameter bounds for Calabi-Yau degenerations proved.

Sharp bound on smallest diameter of hyperbolic surfaces.

Spheres can be stretched to have larger diameter than antipodal distance.

Study on compact Quasi-Einstein manifolds yields diameter estimates and Hitchin-Thorpe inequality conditions.

Makeev proved that among centrally symmetric four-dimensional polytopes, with more than twenty facets and circumscribed about the Euclidean ball of diameter one, there is no universal cover for the family of unit diameter sets. In this paper we examine the converse problem, and prove that each centrally symmetric polyt…

Small sub-Riemannian balls have diameter close to twice their radius.

Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.

Uniform estimates for Kaehler metrics' diameters and volumes.

Study on RCD(0,N) spaces with small linear diameter growth.

Estimates Kaehler metrics' diameter in big cohomology classes.

Study bounds Kähler current diameters on manifolds.

Study systoles and diameters on hyperbolic surfaces, finding an upper bound for their ratio.

New framework explains data augmentation's role in machine learning.

Upper diameter bound for manifolds with positive scalar curvature.

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

Paper bounds surface diameter and solves Plateau-Douglas problem.

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.

Estimates graph curvature and diameter using Laplacian eigenvalues.