Hedlund-Metrics and the Stable Norm

TL;DR

This paper explores which norms on the first homology group of a compact Riemannian manifold can be realized as stable norms, providing an alternative smooth Riemannian metric construction for polyhedral norm balls.

Contribution

It offers a new geometric method to construct smooth Riemannian metrics whose stable norms match given polyhedral norms with rational vertices.

Findings

Any polyhedral norm with rational vertices can be realized as a stable norm of a smooth Riemannian metric.

The new construction stays within the smooth Riemannian framework, avoiding singular metrics.

The approach generalizes previous results to a broader class of norms.

Abstract

The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space $H_1(M,\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the dimension of $M$ is at least three, I. Babenko and F. Balacheff proved in \cite{baba} that every polyhedral norm ball in $H_1(M,\mathbb{R})$, whose vertices are rational with respect to the lattice of integer classes in $H_1(M,\mathbb{R})$, is the stable norm ball of a Riemannian metric on $M$. This metric can even be chosen to be conformally equivalent to any given metric. The proof in \cite{baba} uses singular Riemannian metrics on polyhedra which are finally smoothed. Here we present an alternative construction of such metrics which remains in the geometric framework of smooth Riemannian metrics.