An isoperimetric inequality in the universal cover of the punctured plane

AbstractWe find the largest ϵ (approximately 1.71579) for which any simple closed path α in the universal cover R2∖Z2˜ of R2∖Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥ϵA(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2∖Z2

Bounds on exceptional Dehn filling

We show that for a hyperbolic knot complement, all but at most 12 Dehn fillings are irreducible with infinite word-hyperbolic fundamental group.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper15.abs.htm

Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference

Lectures on Minimal Surface Theory

An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.Comment: 46 pages, 6 figures. Some references added/corrected on August 2, 2014. A few minor corrections on October 16, 2015. Additional typos corrected on January 17, 201

Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.Comment: 102 pages, 17 figure

Cheeger constants of surfaces and isoperimetric inequalities

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most $g$, then the filling function in dimension 3 is `almost' linear.Comment: 28 page

Constant mean curvature surfaces in 3-dimensional Thurston geometries

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 × R, S2 × R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented.Ministerio de Educación y Ciencia MTM2007-65249Junta de Andalucía FQM325Junta de Andalucía P06-FQM-0164

Constant mean curvature surfaces in 3-dimensional Thurston geometries

This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented