Expanding Ricci solitons and Higgs bundles

TL;DR

This paper investigates expanding Ricci solitons with nilpotent symmetry on vector bundles, linking them to twisted harmonic-Einstein equations and G-Higgs bundles, leading to new examples and classifications, especially in dimension four.

Contribution

It introduces a reduction of Ricci soliton equations to twisted harmonic-Einstein equations and establishes a correspondence with G-Higgs bundles on surfaces, producing new non-homogeneous examples.

Findings

Dimension reduction to twisted harmonic-Einstein equations

Correspondence with G-Higgs bundles on surfaces

Complete classification in dimension four

Abstract

Motivated by the long-time behavior of Ricci flows that collapse with bounded curvature, we study expanding Ricci solitons with nilpotent symmetry on vector bundles over a closed manifold. We prove that, under mild assumptions that are satisfied by Ricci flow limits, the equations dimension-reduce to the so-called twisted harmonic-Einstein equations. When the base is a surface, we establish a correspondence between solutions of the latter and a class of G-Higgs bundles. This allows us to produce infinite families of new examples that are not locally homogeneous, and in particular to obtain a complete description in dimension 4. We also show that all our examples admit Einstein one-dimensional extensions.