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Gregory R. Chambers

Publications and source records attributed to Gregory R. Chambers.

At least 19 recordsLinked to original sources
arXiv2025

Mountain Pass Critical Points of the Volume Constrained Area Functional

We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map.

math.AP↗
arXiv2024

Diameter, Area, and Mean curvature

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean curvature and the area of the manifold. We apply our diameter bounds to minimal, constant mean curvature, and prescribed mean curvature surfaces arising in min-max constructions.

math.DG↗
arXiv2023

A note on Kalai's $3^d$ Conjecture

Suppose that $C$ is a centrally symmetric $d$-dimensional convex polytope; in 1989 Kalai conjectured that $C$ has at least $3^d$ facets. We prove this result if there are $d$ hyperplanes with orthogonal normal vectors so that $C$ is symmetric about all of them.

math.CO↗
arXiv2023

On the relative isoperimetric problem for the cube

In this article, we solve the relative isoperimetric problem in $[0,1]^3$ for orthogonal polyhedra. Up to isometries of the cube or sets of measure $0$, the minimizers are of the form $[0,\epsilon]^3$, $[0,\epsilon]^2 \times [0,1]$, or $[0,\epsilon] \times [0,1]^2$ for some $\epsilon > 0$. This should be compared to the conjectured minimizers for the unconstrained relative isoperimetric problem in $[0,1]^3$, which are (up to isometries and sets of measure $0$) of the form $\left( B^3(\epsilon) \right) \cap [0,1]^3$, $\left( B^2(\epsilon) \times [0,1] \right) \cap [0,1]^3$, or $[0,\epsilon] \times [0,1]^2$ for some $\epsilon > 0$. Here, $B^k(\epsilon)$ is the closed ball in $\mathbb{R}^k$ of radius $\epsilon$ centered at the origin.

math.DG↗
arXiv2022

Uryson width and pants decompositions of hyperbolic surfaces

Suppose that $M$ is a hyperbolic surface of genus $g$ and with $n$ cusps. Then we can find a pants decomposition of $M$ composed of simple closed geodesics so that each curve is contained in a ball of diameter at most $C\sqrt{g + n}$, where $C$ is a universal constant.

math.GT↗
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.

math.DG↗
arXiv2022

On the square peg problem

We show that if $γ$ is a Jordan curve in $\mathbb{R}^2$ which is close to a $C^2$ Jordan curve $β$ in $\mathbb{R}^2$, then $γ$ contains an inscribed square. In particular, if $κ> 0$ is the maximum unsigned curvature of $β$ and there is a map $f$ from the image of $γ$ to the image of $β$ with $||f(x) - x|| < \frac{1}{10 κ}$ and $f \circ γ$ having winding number $1$, then $γ$ has an inscribed square of positive sidelength.

math.GT↗
arXiv2022

On minimal higher genus fillings

In this article, we prove that if $(M,g)$ is a genus $G$ orientable surface with a single boundary component $S^1$, and if $(D,g_0)$ is a disc such that interior points are connected by unique geodesics and $$d_{(D,g_0)}(x,y) \geq d_{(M,g)}(x,y)$$ for all $x,y \in \partial M = \partial D$, then $$(1 + \frac{2 G}π) \textrm{Area}(M,g) \geq \textrm{Area}(D,g_0).$$

math.DG↗
arXiv2021

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.

math.DG↗
arXiv2019

Existence of minimal hypersurfaces in complete manifolds of finite volume

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region $U$ can be swept out by a family of hypersurfaces of volume at most $V$, then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most $V + \varepsilon$.

math.DG↗
arXiv2018

Quantitative nullhomotopy and rational homotopy type

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemannian manifold $X$ and $Y$ a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected $Y$, asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.

math.GT↗
arXiv2017

Quantitative null-cobordism

For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces: for example, any simply connected Lie group. Then two homotopic L-Lipschitz maps $f, g : X \rightarrow Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.

math.GT↗
arXiv2017

Area of convex disks

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)) \leq 2(\fracπ{2 R})^2$ on the first nonzero Neumann eigenvalue $μ_2$ of the Laplacian in terms only of the radius. This has also been conjectured for $R$ up to half the injectivity radius.

math.DG↗
arXiv2016

Ergodic properties of folding maps on spheres

We consider the trajectories of points on $\mathbb{S}^{d - 1}$ under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is $d+1$.

math.DS↗
arXiv2016

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 $γ$ is a simple closed curve of a Riemannian surface, and that there exists a monotone contraction which covers a disc which $γ$ bounds consisting of curves of length $\leq L$. If $ε> 0$ and $q \in γ$, then there exists a homotopy that contracts $γ$ to $q$ over loops that are based at $q$ and have length bounded by $3L + 2d + ε$, where $d$ is the diameter of the surface. If the surface is a disc, and if $γ$ is the boundary of this disc, then this bound can be improved to $L + 2d + ε$.

math.DG↗
arXiv2016

Optimal sweepouts of a Riemannian 2-sphere

Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than L, we construct a second sweepout composed of curves of length less than L which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Freedman concerning the existence of min-max embedded geodesics, we partially answer a question due to N. Hingston and H.-B. Rademacher, and we also extend the results of [CL] concerning converting homotopies to isotopies in an effective way.

math.DG↗