Cohomogeneity two Ricci solitons with sub-Euclidean volume

TL;DR

This paper constructs new four-dimensional Ricci solitons with cohomogeneity two and volume collapsing ends, using a reduction to a degenerate Monge-Ampère equation, and classifies their local solutions.

Contribution

It introduces explicit families of Ricci solitons with volume collapsing ends and provides classification and existence results for these solutions.

Findings

Explicit complete expanding Ricci solitons constructed.

Existence results for shrinking and steady solitons with boundary.

Rigidity results for local solutions of the Monge-Ampère equation.

Abstract

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Amp\`{e}re equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge-Amp\`{e}re equation to prove rigidity for these solitons.