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Regina Rotman

Publications and source records attributed to Regina Rotman.

14 recordsLinked to original sources
arXiv2025

Lengths of Orthogonal Geodesic Chords on Riemannian Manifolds

Let $N$ be a closed submanifold of a complete manifold, $M$. Then under certain topological conditions, there exists an orthogonal geodesic chord beginning and ending in $N$. In this paper we establish an upper bound for the length of such a geodesic chord in terms of geometric bounds on $M$. For example, if $N$ is a $2$-dimensional sphere embedded in a closed Riemannian $n$-manifold, then there exists an orthogonal geodesic chord in $M$ with endpoints on $N$ that has length at most $$ (4d+96D +8232\sqrt{A})(2n+1) $$ where $d$ is the diameter of $M$, and $A$ and $D$ are the area and intrinsic diameter of $N$, respectively.

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arXiv2024

Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below

Let $M$ be a simply connected Riemannian manifold in $\mathscr{M}_{k,v}^D(n)$, the space of closed Riemannian manifolds of dimension $n$ with sectional curvature bounded below by $k$, volume bounded below by $v$, and diameter bounded above by $D$. Let $c$ be the smallest positive real number such that any closed curve of length at most $2d$ can be contracted to a point over curves of length at most $cd$, where $d$ is the diameter of $M$. In this paper, we show that under these hypotheses there exists a computable rational function, $G(n,k,v,D)$, such that any continuous map of $S^l$ to $\Omega_{p,q}M$, the space of piecewise differentiable curves on $M$ connecting $p$ and $q$, is homotopic to a map whose image consists of curves of length at most $\exp(c\exp(G(n,k,v,D))$. In particular, for any points $p,q \in M$ and any integer $m>0$ there exist at least $m$ geodesics connecting $p$ and $q$ of length at most $m\exp(c\exp(G(n,k,v,D))$.

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arXiv2022

Geodesic nets on non-compact Riemannian manifolds

A geodesic flower is a finite collection of geodesic loops based at the same point $p$ that satisfy the following balancing condition: The sum of all unit tangent vectors to all geodesic arcs meeting at $p$ is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that in every complete non-compact manifold with locally convex ends there exists a non-trivial geodesic flower.

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arXiv2022

Positive Ricci Curvature and the Length of a shortest periodic geodesic

Let $M^n$ be a closed Riemannian manifold of dimension $n\geq 2$, with Ricci curvature $Ric \geq n-1$. We will show that any sphere of dimension $m$ in the space of closed loops on $M^n$ is homotopic to the sphere in the space of closed loops of length at most $8 \pi m$. It follows that the length of a shortest periodic geodesic on $M^n$ is bounded from above by $8 \pi (n-1)$.

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arXiv2020

Sweepouts of closed Riemannian manifolds

We show that for every closed Riemannian manifold there exists a continuous family of $1$-cycles (defined as finite collections of disjoint closed curves) parametrized by a sphere and sweeping out the whole manifold so that the lengths of all connected closed curves are bounded in terms of the volume (or the diameter) and the dimension $n$ of the manifold, when $n \geq 3$. An alternative form of this result involves a modification of Gromov's definition of waist of sweepouts, where the space of parameters can be any finite polyhedron (and not necessarily a pseudomanifold). We demonstrate that the so-defined polyhedral $1$-dimensional waist of a closed Riemannian manifold is equal to its filling radius up to at most a constant factor. We also establish upper bounds for the polyhedral $1$-waist of some homology classes in terms of the volume or the diameter of the ambient manifold. In addition, we provide generalizations of these results for sweepouts by polyhedra of higher dimension using the homological filling functions. Finally, we demonstrate that the filling radius and the hypersphericity of a closed Riemannian manifold can be arbitrarily far apart.

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arXiv2019

Wide short geodesic loops on closed Riemannian manifolds

It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 \over n}$, or $\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $\epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $\pi-\epsilon$ and $\pi$ with an explicit constant that depends both on $n$ and $\epsilon$. That is, let $\epsilon > 0$, and let $a = \lceil{ {1 \over {\sin ({\epsilon \over 2})}}} \rceil+1 $. We will prove that there exists a "wide" (i.e. with an angle that is wider than $\pi-\epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a "wide" geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad \leq 2 \cdot n(n+1)!^2a^{(n+1)^3} vol^{1 \over n}$. Here $FillRad$ is the Filling Radius of $M^n$.

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arXiv2019

Filling metric spaces

We prove a new version of isoperimetric inequality: Given a positive real $m$, a Banach space $B$, a closed subset $Y$ of metric space $X$ and a continuous map $f:Y \rightarrow B$ with $f(Y)$ compact $$\inf_FHC_{m+1}(F(X))\leq c(m)HC_m(f(Y))^{\frac{m+1}{m}},$$ where $HC_m$ denotes the $m$-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps $F:X\longrightarrow B$ such that $F(y)=f(y)$ for all $y\in Y$, and $c(m)$ depends only on $m$. Moreover, one can find $F$ with a nearly minimal $HC_{m+1}$ such that its image lies in the $C(m)HC_m(f(Y))^{1\over m}$-neighbourhood of $f(Y)$ with the exception of a subset with zero $(m+1)$-dimensional Hausdorff measure. The paper also contains a very general coarea inequality for Hausdorff content and its modifications. As an application we demonstrate an inequality conjectured by Larry Guth that relates the $m$-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov's systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply connected manifolds.

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arXiv2017

Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.

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arXiv2014

Contracting the boundary of a Riemannian 2-disc

Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\leq L + 200d\max\{1,\ln {\sqrt{A}\over d} \}$. This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked earlier by M.Gromov. We also prove that a Riemannian $2$-sphere $M$ of diameter $d$ and area $A$ can be swept out by loops based at any prescribed point $p\in M$ of length $\leq 200 d\max\{1,\ln{\sqrt{A}\over d} \}$. This estimate is optimal up to a constant factor. In addition, we provide much better (and nearly optimal) estimates for these problems in the case, when $A<<d^2$. Finally, we describe the applications of our estimates for study of lengths of various geodesics between a fixed pair of points on "thin" Riemannian $2$-spheres.

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arXiv2014

Lengths of three simple periodic geodesics on a Riemannian $2$-sphere

Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for "thin" spheres, when the area $A$ is much less than $d^2$, where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when ${A\over d^2}\longrightarrow 0$.

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arXiv2013

Monotone homotopies and contracting discs on Riemannian surfaces

We prove a "gluing" theorem for monotone homotopies; a monotone homotopy is a homotopy through simple contractible closed curves which themselves are pairwise disjoint. We show that two monotone homotopies which have appropriate overlap can be replaced by a single monotone homotopy. The ideas used to prove this theorem are used in [CL2] to prove an analogous result for cycles, which forms a critical step in their proof of the existence of minimal surfaces in complete non-compact manifolds of finite volume. We also show that, if monotone homotopies exist, then fixed point contractions through short curves exist. In particular, suppose that $\gamma$ is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which $\gamma$ bounds consisting of curves of length $\leq L$. If $\epsilon > 0$ and $q \in \gamma$, then there exists a homotopy that contracts $\gamma$ to $q$ over loops that are based at $q$ and have length bounded by $3L + 2d + \epsilon$, where $d$ is the diameter of the surface. If the surface is a disc, and if $\gamma$ is the boundary of this disc, then this bound can be improved to $L + 2d + \epsilon$.

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arXiv2006

Lengths of geodesics between two points on a Riemannian manifold

Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M of dimension n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of M. But what can be said about the length of the other geodesics? We conjecture that for every k there are k geodesics between x and y of length not exceeding kd, where d denotes the diameter of M.This conjecture is obviously true for round spheres and it is not difficult to prove it for all closed Riemannian manifolds with non-trivial torsion-free fundamental groups. In this paper we announce two further results in the direction of this conjecture. Our first result is that the length of the second shortest geodesic between x and y does not exceed 2nd. Our second result is that if n=2 and M is diffeomorphic to the two-dimensional sphere, then for every k every two points on M can be connected by k geodesics of length not exceeding $(k^2/2 + 3k/2 +2)d$.

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arXiv2002

Volume, diameter and the minimal mass of a stationary 1-cycle

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, where d is the diameter of a manifold M^n. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold M^n is bounded from above by 2(n+2)!Fill Rad(M^n) and, as a corollary, by 2(n+2)!(n+1)n^n(n!)^{1/2}(vol(M^n))^{1/n}, where Fill Rad(M^n) is the filling radius of the manifold, and vol(M^n) is its volume.

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