Fields Geometric Analysis Colloquium

Optimal bubble cluster problems concern the study of partitions of $\mathbb{R}^n$ into a finite collection of chambers, some with finite volume and some with infinite volume. One looks for local minimizers of interfacial area subject to volume constraints on the finite-volume chambers. The case of one infinite-volume chamber is the classical multiple bubble problem and has received much attention in recent decades, with a well-known conjecture of Sullivan predicting the existence of a unique minimizing configuration when there are not too many chambers, which has been partially verified to be true in low dimensions.

We study a variant of the multiple bubble problem with more than one infinite-volume chamber, in particular the simplest case of 1 finite-volume chamber and 2 infinite-volume chambers. Here, Bronsard & Novack showed that uniqueness of local minimizers also holds in sufficiently low dimensions. In stark contrast, we show that uniqueness fails in a large number of dimensions $n \geq 8$, and we provide some particular surprising phenomena that local minimizers can exhibit in these higher dimensions. This is based on joint work with Lia Bronsard, Robin Neumayer and Michael Novack.

http://www.fields.utoronto.ca/activities/25-26/geometric-analysis-colloquium