Eigenvalue Estimate for the basic Laplacian on manifolds with foliated boundary

TL;DR

This paper establishes a sharp lower bound for the first eigenvalue of the basic Laplacian on manifolds with foliated boundary, revealing geometric conditions for equality and deriving rigidity results for specific boundary geometries.

Contribution

It provides a new sharp lower bound for the basic Laplacian's first eigenvalue on manifolds with foliated boundary, linking eigenvalue estimates to geometric and boundary conditions.

Findings

Sharp lower bound for the first eigenvalue of the basic Laplacian.

Characterization of boundary geometry in the limiting case.

Rigidity results for manifolds with specific boundary structures.

Abstract

In this paper, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. The limiting case gives rise to a particular geometry of the flow and the boundary. Namely, the flow is a local product and the boundary is $\eta$-umbilical. This allows to characterize the quotient of $\mathbb R\times B'$ by some group $\Gamma$ as being the limiting manifold. Here $B'$ denotes the unit closed ball. Finally, we deduce several rigidity results describing the product $\mathbb{S}^1\times \mathbb{S}^n$ as the boundary of a manifold.