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Gregory R. Chambers

Publications and source records attributed to Gregory R. Chambers.

24 records · Page 2Linked to original sources
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↗
arXiv2015

Splitting a contraction of a simple curve traversed $m$ times

Suppose that $M$ is a $2$-dimensional oriented Riemannian manifold, and let $γ$ be a simple closed curve on $M$. Let $m γ$ denote the curve formed by tracing $γ$ $m$ times. We prove that if $m γ$ is contractible through curves of length less than $L$, then $γ$ is contractible through curves of length less than $L$. In the last section we state several open questions about controlling length and the number of self-intersections in homotopies of curves on Riemannian surfaces.

math.DG↗
arXiv2015

Geometric stability of the Coulomb energy

The Coulomb energy of a charge that is uniformly distributed on some set is maximized (among sets of given volume) by balls. It is shown here that near-maximizers are close to balls.

math.FA↗
arXiv2015

Proof of the Log-Convex Density Conjecture

We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.

math.DG↗
arXiv2014

Converting homotopies to isotopies and dividing homotopies in half in an effective way

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by L + epsilon. If the manifold is orientable, then for any epsilon > 0 we show that, if we can contract a curve gamma traversed twice through curves of length bounded by L, then we can also contract gamma through curves bounded in length by L + epsilon. Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.

math.DG↗