arXiv research

Paper bounds surface diameter and solves Plateau-Douglas problem.

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

The paper estimates surface diameter in conformal spaces.

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman's $ν$ invariant. Here $n…

We introduce interactive structure discovery, a generic framework that encompasses many interactive learning settings, including active learning, top-k item identification, interactive drug discovery, and others. We adapt a recently developed active learning algorithm of Tosh and Dasgupta (2017) for interactive structu…

Estimates submanifold diameters in curved spaces.

Given an $m$-dimensional closed connected Riemannian manifold $M$ smoothly isometrically immersed in an $n$-dimensional Riemannian manifold $N$, we estimate the diameter of $M$ in terms of its mean curvature field integral under some geometric restrictions, and therefore generalize a recent work of Topping in the Eucli…

We first partially extend a theorem of Topping, on the relation between mean curvature and intrinsic diameter, from immersed submanifolds of $\mathbb{R} ^{n}$ to almost everywhere immersed, closed submanifolds of a compact Riemannian manifold. We use this to prove quantization of energy for pseudo-holomorphic closed c…

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

The paper proves inequalities for closed surfaces involving mean curvature.

Sharp diameter bounds for Calabi-Yau degenerations proved.

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

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

Sharp upper bound found for stable minimal surfaces.

Sharp bound on smallest diameter of hyperbolic surfaces.

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

Estimates Kähler metric diameters with entropy bound alone.

Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.

Study bounds Kähler current diameters on manifolds.

The paper proves a linear diameter bound for hyperbolic knot complexes.

Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.

Uniform diameter bound for reflection group disk patterns.

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (J. Differential Geom…

The paper shows mean curvature flow keeps diameter bounded under certain conditions.

The study finds a diameter bound for graphs with positive entropic Ricci curvature, with optimal bounds for arithmetic mean.

Sharp upper diameter limit found for Ricci solitons.

New bounds on diameters and generators for specific lattices and graphs.

In the celebrated book entitled Metric Structures for Riemannian and Non-Riemannian Spaces, so-called Green Book, Gromov presented a problem regarding a metric measure space. Gromov posed the question Bound the expansion coefficient from below in terms of the observable diameter. The overall aim of the current study is…

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

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…

The study finds a lower bound for the diameter of gradient ρ-Einstein solitons.

We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.

This note shows surfaces in stable 3D data are bounded by area and diameter.

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

Upper diameter bound for manifolds with positive scalar curvature.

Sharp estimates for mean curvature flow confirm bounded diameter conjecture.

Let G, a subset of O(4), act isometrically on the 3-sphere. In this article we calculate a lower bound for the diameter of the quotient spaces $S^3/G$. We find it to be ${1/2}\arccos(\frac{\tan(\frac{3 π}{10})}{\sqrt3})$, which is exactly the value of the lower bound for diameters of the spherical space forms. In the p…

The paper calculates the size of origami orbit graphs in complex surfaces.

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

We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.

We prove that the cardinality of the torsion subgroups in homology of a closed hyperbolic manifold of any dimension can be bounded by a doubly exponential function of its diameter. It would follow from a conjecture by Bergeron and Venkatesh that the order of growth in our bound is sharp. We also determine how the numbe…

Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance

Sharp bounds on diameter and eigenvalues for amply regular graphs.

Investigates properties of volume, entropy, and diameter in higher Teichmüller spaces.

The study bounds the effective diameter of graphs with positive Ollivier curvature.

Paper bounds Kähler manifolds' diameter using Orlicz spaces and complex Monge-Ampère equations.

It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.