arXiv research

The isospectral problem for p-widths is solved using Zoll metrics on S^2.

Study confirms a 2-sphere metric with three geodesics of minimal length.

Study finds geodesic networks for surfaces with convex boundary.

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

Derives generalizations of the long neck principle and spectral width inequality.

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

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…

Develops a new theory of width for embedded circles in Riemannian manifolds.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

Find simple geodesics in hyperbolic surfaces with bounded diameter.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.

Study on geodesics proving index and intersection bounds, with examples of multiplicity.

We compute the $k$-width of a round $2$-sphere for $k=1,\ldots,8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold.

Study calculates first $p$-widths of unit disk.

Proves principles and estimates for initial data sets in Einstein equations.

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout …

The paper proves rigidity theorems for area widths of Riemannian manifolds.

The study proves a tube theorem for complex hyperbolic manifolds.

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lo…

In this paper, we show a local energy convexity of $W^{1,2}$ maps into $CAT(K)$ spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of t…

We generalize the results of [AS], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each the lift of a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic …

Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

Geodesic envelopes stay uniformly bounded in specific Teichmüller spaces.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

The volume spectrum of fiber bundles is bounded by the product of the base's volume spectrum and the fiber's volume.

The paper extends the collar theorem to non-compact surfaces using new comparison theorems.

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

We give a simple procedure to estimate the smallest Lipshitz constant of a degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of 10. Using this procedure, we are able to prove several inequalities involving this Lipshitz constant. For instance, if the smallest Lipshitz constant is at least 1, …

For a Riemannian manifold $M^{n+1}$ and a compact domain $Ω\subset M^{n+1}$ bounded by a hypersurface $\partial Ω$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial Ω$ for the angle between the geodesic line joining a fixed interior point $O$ in $Ω$ to a point on…

Let $X$ be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group $Γ$ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $Γ$-action. Although the construction of Bowen-Margulis m…

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.

Width trees link link invariants and bridge number.

Lectures on deep learning properties in infinite and large-width networks.

Computed p-widths for hemisphere, first for manifolds with boundary.

Polygon $p$-widths are found via billiard trajectories.

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…

Computed p-widths for real projective plane.

Study bounds Urysohn width of manifolds under surgeries.

Residual networks with block width max(d_x, d_y) approximate all functions.

Proves conjecture about sphere widths under rotational symmetry.

New link invariants from diagram colorings match link widths.

Study infinite-depth limits of neural networks with fixed width.

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…

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

We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.