Entropy rigidity for foliations by strictly convex projective manifolds

TL;DR

This paper establishes an entropy rigidity result for foliations with leaves modeled on strictly convex projective manifolds and hyperbolic manifolds, showing that entropy equality implies homothety of leaves.

Contribution

It introduces a new entropy rigidity theorem for foliations with convex projective and hyperbolic leaves, extending classical rigidity results to foliated settings.

Findings

Foliated volume entropies are well-defined for the considered foliations.

An inequality relating the entropies of the two foliations is proven: $h(M,\mathscr{F}_M) \leq h(N,\mathscr{F}_N)$.

Equality of entropies implies the leaves are homothetic.

Abstract

Let $N$ be a compact manifold with a foliation $\mathscr{F}_N$ whose leaves are compact strictly convex projective manifolds. Let $M$ be a compact manifold with a foliation $\mathscr{F}_M$ whose leaves are compact hyperbolic manifolds of dimension bigger than or equal to $3$. Suppose to have a foliation-preserving homeomorphism $f:(N,\mathscr{F}_N) \rightarrow (M,\mathscr{F}_M)$ which is $C^1$-regular when restricted to leaves. In the previous situation there exists a well-defined notion of foliated volume entropies $h(N,\mathscr{F}_N)$ and $h(M,\mathscr{F}_M)$ and it holds $h(M,\mathscr{F}_M) \leq h(N,\mathscr{F}_N)$. Additionally, if equality holds, then the leaves must be homothetic.