Archive for January 2009

Isoperimetric Regions in Cones

Cones provide the simplest singular spaces and models for general singularities. The isoperimetric problem is a good way to explore their geometry. My students, collaborators, and I have a number of related publications (see my webpage), most recently “Isoperimetric balls in cones over tori” (Ann. Glob. Anal. Geom. 2008). Here I want to mention two other interesting cases which turn out to be trivial.

1. The cone over R. Here balls are isoperimetric, because this is the universal cover of the punctured plane, and balls are isoperimetric in the plane, even if multiplicity is allowed.

2. The cone over the line with Gaussian density. Here there are no isoperimetric sets, because you can do better and better by going farther out in the cone with smaller and smaller neighborhoods of +∞ in each slice.

Networks in Manifolds with Density

Abstract: The version of the shortest “Steiner” network problem in which you minimize length plus number of Steiner points has an interesting analog in manifolds with density. Continue reading ‘Networks in Manifolds with Density’ »

Surfaces, currents, and varifolds

What is a surface? Different technical definitions serve different purposes. Here we’ll focus on two-dimensional surfaces S in R3 or R4. Continue reading ‘Surfaces, currents, and varifolds’ »