A maximal element of a moduli space of Riemannian metrics

TL;DR

This paper explores the concept of maximal metrics within the moduli space of Riemannian metrics on manifolds, providing new examples on Euclidean spaces and highlighting their significance in self-similar solutions to geometric flows.

Contribution

It introduces the notion of maximal metrics in the moduli space and constructs numerous examples on Euclidean spaces, advancing understanding of self-similar solutions in geometric analysis.

Findings

Constructed many examples of maximal metrics on Euclidean spaces

Maximal metrics serve as self-similar solutions for metric evolution equations

Enhanced understanding of the structure of the moduli space of Riemannian metrics

Abstract

For a given smooth manifold, we consider the moduli space of Riemannian metrics up to isometry and scaling. One can define a preorder on the moduli space by the size of isometry groups. We call a Riemannian metric that attains a maximal element with respect to the preorder a maximal metric. Maximal metrics give nice examples of self-similar solutions for various metric evolution equations such as the Ricci flow. In this paper, we construct many examples of maximal metrics on Euclidean spaces.