Continuity of the Alvarez class under deformations

TL;DR

This paper proves that the Alvarez class of a Riemannian foliation varies continuously under deformations, and under certain topological conditions, the property of minimizability remains invariant during such deformations.

Contribution

It demonstrates the continuity of the Alvarez class in smooth families and establishes invariance of minimizability under deformations under specific topological conditions.

Findings

Alvarez class varies continuously with the foliation parameter.

Minimizability is invariant under deformations given certain topological constraints.

The paper extends previous algebraic rigidity results to deformation invariance.

Abstract

A foliated manifold (M,F) is minimizable if there exists a Riemannian metric g on M such that every leaf of F is a minimal submanifold of (M,g). Alvarez Lopez defined a cohomology class of degree 1 called the Alvarez class of (M,F) whose triviality characterizes the minimizability of (M,F), when M is closed and F is Riemannian. In this paper, we show that the family of the Alvarez classes of a smooth family of Riemannian foliations is continuous with respect to the parameter. Since the Alvarez class has algebraic rigidity under certain topological conditions on (M,F) as the author showed in arXiv:0909.1125, we show that the minimizability of Riemannian foliations is invariant under deformation under the same topological conditions.