Joint IAS/PU Analysis Seminar

We develop the spectral analysis of the Jacobi operator on the interfaces of soap-bubble clusters. By Plateau's laws, these always meet in threes at $120^{\circ}$-angles, and thus naturally interact via 3 linearly independent "conformal" boundary conditions (a mixture of Dirichlet and Robin). This gives rise to a self-adjoint operator, whose spectral properties determine the \emph{stability} of the soap-bubbles -- whether an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In essence, stability is an infinitesimal notion of local isoperimetry, amounting to verifying a Poincar\'e-type inequality on soap-bubble clusters.

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters in $n$-dimensional Euclidean, spherical and hyperbolic spaces, which are conjectured to be global isoperimetric minimizers of total perimeter, are indeed stable. In fact, stability holds for all M\"obius-flat \emph{partitions}, in which several cells are allowed to have infinite volume. In the Gaussian setting (when volume and area are weighted relative to the Gaussian density), any partition obeying Plateau's laws and whose interfaces are all \emph{flat}, is stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential.

Based on joint work with Botong Xu.