Critical domains for certain Dirichlet integrals in weighted manifolds

TL;DR

This paper investigates the critical domains for Dirichlet integrals in weighted manifolds, deriving variational formulas, establishing Morse index results, and proving global rigidity theorems including an Alexandrov-type soap bubble theorem.

Contribution

It extends variational and Morse index formulas to unbounded domains in weighted manifolds and proves new global rigidity results, including in Gaussian half-spaces.

Findings

Derived variational formulas for Dirichlet eigenvalues in weighted manifolds.

Established Morse index formulas for critical domains.

Proved global rigidity theorems, including an Alexandrov-type soap bubble theorem.

Abstract

We start by revisiting the derivation of the variational formulae for the functional assigning to a bounded regular domain in a Riemannian manifold its first Dirichlet eigenvalue and extend it to (not necessarily bounded) domains in certain weighted manifolds. This is further extended to other functionals defined by certain Dirichlet energy integrals, with a Morse index formula for the corresponding critical domains being established. We complement these infinitesimal results by proving a couple of global rigidity theorems for (possibly critical) domains in Gaussian half-space, including an Alexandrov-type soap bubble theorem. Although we provide direct proofs of these latter results, we find it worthwhile to point out that the main tools employed (specifically, certain Pohozhaev and Reilly identities) can be formally understood as limits (when the dimension goes to infinity) of tools previously established by Ciarolo-Vezzoni and Qiu-Xia to handle similar problems in round hemispheres, with the notion of "convergence" of weighted manifolds being loosely inspired by the celebrated Poincar\'e's limit theorem in the theory of Gaussian random vectors.