Fiberwise symmetrizations for variational problems on fibred manifolds

TL;DR

This paper develops a fiberwise symmetrization method to establish lower bounds for energy functionals on fibred manifolds and applies it to compare the first eigenvalues of Laplacians on warped products.

Contribution

It introduces a novel fiberwise symmetrization framework for variational problems on fibred manifolds, enabling new comparison theorems for eigenvalues.

Findings

Established a lower bound for Dirichlet-type energy functionals.

Proved a comparison theorem for the first Laplacian eigenvalue on warped products.

Provided a new analytical tool for variational problems on fibred manifolds.

Abstract

We establish a framework for fiberwise symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to prove a comparison theorem of the first eigenvalue of the Laplacian on a warped product manifold.