# Gregory R. Chambers

Since September 2014, I have been an L. E. Dickson Instructor at the University of Chicago. I completed my PhD at the University of Toronto in June 2014 under the supervision of Alexander Nabutovsky and Regina Rotman.

My research is in metric geometry and geometric analysis. I am particularly interested in quantitative topology, min-max theory, isoperimetric inequalities, stability estimates for geometric inequalities, and symmetrizations and their applications.

## Contact Information

Department of Mathematics
University of Chicago
5734 S University Ave
Chicago, IL
60637

Phone: (773) 702-9177

Email: chambers@math.uchicago.edu

Office: Eckhart 325

## Teaching

I am currently teaching MAT205 Section 45 (Accelerated). All course information is available on Chalk.

## Publications

### Submitted

1. Area of convex disks, with C. Croke, Y. Liokumovich, and H. Wen
arXiv:1701.06594

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 $$Area(B(p,R))\geq\frac{8}{\pi}R^2$$. This inequality has long been conjectured for $$R$$ less than half the injectivity radius. This result also yields the upper bound $$\mu_2(B(p,R))\leq 2(\frac{\pi}{2 R})^2$$ on the first nonzero Neumann eigenvalue $$\mu_2$$ of the Laplacian in terms only of the radius. This has also been conjectured for $$R$$ up to half the injectivity radius.

2. Quantitative nullhomotopy and rational homotopy type, with F. Manin, and S. Weinberger
arXiv:1611.03513

In [GrOrang], Gromov asks the following question: given a nullhomotopic map $$f:S^m \rightarrow S^n$$ 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 thickness (Lipschitz constant in the space variable) is $$C(m,n)(L+1)$$ and whose width (Lipschitz constant in the time variable) is $$C(m,n)(L+1)^2$$.
More generally, we prove a similar result for maps $$f : X \rightarrow Y$$ 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.

3. Quantitative null-cobordism, with D. Dotterrer, F. Manin, and S. Weinberger
arXiv:1610.04888

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 [GrQHT], 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 $$f,g:X \to Y$$ are homotopic via a $$CL$$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $$Y$$.

4. Existence of minimal hypersurfaces in complete manifolds of finite volume, with Y. Liokumovich
arXiv:1609.04058

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume.

5. A note on the affine-invariant plank problem
arXiv:1604.00456

Suppose that $$C$$ is a bounded convex subset of $$\mathbb{R}^n$$, and that $$P_1,\dots,P_k$$ are planks which cover $$C$$ in respective directions $$v_1,\dots,v_k$$ and with widths $$w_1,\dots,w_k$$. In 1951, Bang conjectured that $$\sum_{i=1}^k \frac{w_i}{w_{v_i}(C)} \geq 1,$$ generalizing a previous conjecture of Tarski. Here, $$w_{v_i}(C)$$ is the width of $$C$$ in the direction $$v_i$$. In this note we give a short proof of this conjecture under the assumption that, for every $$m$$ with $$1 \leq m \leq k$$, $$C \setminus \bigcup_{i=1}^m P_i$$ is a convex set.

### Published

1. Monotone homotopies and contracting discs on Riemannian surfaces, with R. Rotman
Journal of Topology and Analysis, to appear, arXiv:1311:2995

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 (10) 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 $$L + 2d + \epsilon$$.

2. Optimal sweepouts of a Riemannian 2-sphere, with Y. Liokumovich
Journal of the European Mathematics Society, to appear, arXiv:1411:6349

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 (2) concerning converting homotopies to isotopies in an effective way.

3. Proof of the Log-Convex Density Conjecture
Journal of the European Mathematics Society, to appear, arXiv:1311.4012

We completely characterize isoperimetric regions in $$\mathbb{R}^n$$ 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.

4. Ergodic properties of folding maps on spheres, with A. Burchard and A. Dranovski
Discrete and Continuous Dynamical Systems - Series A 37(3):1183-1200 (2017), DOI 10.3934/dcds.2017049, arXiv:1509.02454

We consider the trajectories of points on the $$(d-1)$$-dimensional sphere under certain folding maps associated with reflections. The main result gives a condition for a collection of such maps to produce dense trajectories. At least $$d+1$$ directions are required to satisfy the conditions.

5. Isoperimetric regions in $$\mathbb{R}^n$$ with density $$r^p$$, with W. Boyer, B. Brown, A. Loving, and S. Tammen
Analysis and Geometry in Metric Spaces 4(1):236-265 (2016), DOI 10.1515/agms-2016-0009, arXiv:1504:01720

We show that the unique isoperimetric hypersurfaces in $$\mathbb{R}^n$$ with density $$r^p$$ for $$n \geq 3$$ and $$p>0$$ are spheres that pass through the origin.

6. Splitting a contraction of a simple curve traversed $$m$$ times, with Y. Liokumovich
Journal of Topology and Analysis (2016), DOI 10.1142/S1793525317500157, arXiv:1510.03445

Suppose that $$M$$ is a 2-dimensional oriented Riemannian manifold, and let $$\gamma$$ be a simple closed curve on $$M$$. Let $$\gamma^m$$ denote the curve formed by tracing $$\gamma$$ $$m$$ times. We prove that if $$\gamma^m$$ is contractible through curves of length less than $$L$$, then $$\gamma$$ 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.

7. Geometric stability of the Coulomb energy, with A. Burchard
Calculus of Variations and PDE 54(3):3241-3250 (2015), DOI 10.1007/s00526-015-0900-8, arXiv:1407.1918

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.

8. Perimeter under multiple Steiner symmetrizations, with A. Burchard
Journal of Geometric Analysis (2015) 25:871, DOI 10.1007/s12220-013-9448-z, arXiv:1209.4521

Steiner symmetrization along $$n$$ linearly independent directions transforms every compact subset of $$\mathbb{R}^n$$ into a set of finite perimeter.

9. Converting homotopies to isotopies and dividing homotopies in half in an effective way, with Y. Liokumovich
Geometric and Functional Analysis (2014) 24:1080, DOI 10.1007/s00039-014-0283-6, arXiv:1311.0779

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.