arXiv research

Sharp bound on smallest diameter of hyperbolic surfaces.

Sharp upper bound found for stable minimal surfaces.

Paper bounds surface diameter and solves Plateau-Douglas problem.

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

We prove that the minimal diameter of a hyperbolic compact orientable surface of genus $g$ is asymptotic to $\log g$ as $g \to \infty$. The proof relies on a random construction, which we analyse using lattice point counting theory and the exploration of random trivalent graphs.

The Kähler-Ricci flow yields bounded diameter and Ricci curvature for minimal models.

The paper estimates surface diameter in conformal spaces.

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.

Connectedness of small clusters in Riemannian and Finsler manifolds proven.

It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.

New bound for group action length without diameter restriction.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

The study optimizes cell membranes' shapes based on curvature and proves existence of minimizers.

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold M^n is bounded from above by (n+2)!d/3,…

We show that a group presented by a labelled oriented tree presentation in which the tree has diameter at most three is an HNN extension of a finitely presented group. From results of Silver, it then follows that the corresponding higher dimensional ribbon knots admit minimal Seifert manifolds.

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…

Study on Riemannian properties of SU_n using bi-invariant metric.

The paper proves inequalities for submanifolds in Riemannian manifolds.

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

New geometric invariant from min-max width of spheres on Riemannian 2-spheres.

We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers…

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.

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negativ…

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.

We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean $3$-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Mo…

Maximal diameter theorem for graphs with positive Ricci curvature.

Mathematical analysis of SNE and t-SNE for dimension reduction.

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.

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

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.

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.

We give a purely geometrical smooth characterization of closed infrasolv manifolds and orbifolds by showing that, up to diffeomorphism, these are precisely the spaces which admit a collapse with bounded curvature and diameter to compact flat orbifolds. Moreover, we distinguish irreducible smooth fake tori geometrically…

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

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 …

Upper diameter bound for manifolds with positive scalar curvature.

Let $Ω$ be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded $f$-minimal hypersurfaces contained in $Ω$. Using this estimat…

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

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

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