arXiv research

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…

We will present a new proof of the Gromoll-Grove diameter rigidity theorem.

Estimates submanifold diameters in curved spaces.

Sharp estimates for mean curvature flow confirm bounded diameter conjecture.

For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of th…

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

In this paper, we study the complete bounded $λ$-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded $λ$-hypersurfaces with $|A|\leqα$ and get some applications of the volume comparison theorem. Secondly, we consider the relation amo…

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…

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

The paper estimates surface diameter in conformal spaces.

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…

The paper finds an upper limit for the length of geodesic chords on Riemannian manifolds.

The paper proves inequalities for submanifolds in Riemannian manifolds.

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 …

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…

In this paper we address the relationship between Gromov-Hausdorff limits and intrinsic flat limits of complete Riemannian manifolds. In \cite{SormaniWenger2010, SormaniWenger2011}, Sormani-Wenger show that for a sequence of Riemannian manifolds with nonnegative Ricci curvature, a uniform upper bound on diameter, and n…

The paper proves stability of manifolds with boundary under volume and distance constraints.

We endow each closed, orientable Alexandrov space $(X, d)$ with an integral current $T$ of weight equal to 1, $\partial T = 0 and \set(T) = X$, in other words, we prove that $(X, d, T)$ is an integral current space with no boundary. Combining this result with a result of Li and Perales, we show that non-collapsing sequ…

Study shows convergence of volumes on manifolds with boundary under area constraints.

We present the Tetrahedral Compactness Theorem which states that sequences of Riemannian manifolds with a uniform upper bound on volume and diameter that satisfy a uniform tetrahedral property have a subsequence which converges in the Gromov-Hausdorff sense to a countably $\mathcal{H}^m$ rectifiable metric space of the…

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

The paper examines sequences of metric spaces converging to compact limits with specific properties.

We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the integral current structure $T_j$ has weight $1$ and no boundary and, all $(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either the…

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.

Study on consensus formation in manifolds with curvature constraints.

We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this property. We extend Sormani's Tetrahedral Property to a less restrictive property and p…

Sharp bound on smallest diameter of hyperbolic surfaces.

Maximal diameter theorem for graphs with positive Ricci curvature.

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.

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.

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.

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

Paper bounds surface diameter and solves Plateau-Douglas problem.