Semicontinuity of eigenvalues under intrinsic flat convergence

TL;DR

This paper establishes the semicontinuity of eigenvalues and min-max values of Dirichlet energy under intrinsic flat convergence of integral current spaces, extending spectral analysis to nonsmooth geometric limits.

Contribution

It introduces a Dirichlet energy for Lipschitz functions on integral currents and proves semicontinuity of eigenvalues under intrinsic flat convergence.

Findings

Min-max values of normalized energy are semicontinuous under convergence.

Eigenvalues below the essential spectrum are semicontinuous if volume converges.

A framework for spectral analysis on nonsmooth limit spaces is developed.

Abstract

We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of the Lipschitz functions. We define min-max values based on the normalized energy and show that when integral current spaces converge in the intrinsic flat sense without loss of volume, the min-max values of the limit space are larger than or equal to the upper limit of the min-max values of the currents in the sequence. In particular, the infimum of the normalized energy is semicontinuous. On spaces that are infinitesimally Hilbertian, we can define a linear Laplace operator. We can show that semicontinuity under intrinsic flat convergence holds for eigenvalues below the essential spectrum, if the total volume of the spaces converges as well.