Expanding Ricci solitons and Higgs bundles
Ramiro A. Lafuente, Adam Thompson
TL;DR
This work develops a framework to construct expanding Ricci solitons with nilpotent symmetry on twisted bundles by reducing the soliton equation to twisted harmonic-Einstein equations. On Riemann surfaces, the method connects to Higgs bundles via the non-abelian Hodge correspondence, enabling large families of non-locally-homogeneous solitons, and yielding a complete 4D classification in the abelian case. It further shows that such solitons admit Einstein one-dimensional extensions, tying the soliton geometry to higher-dimensional Einstein metrics. The results culminate in explicit correspondences between solitons and Higgs-bundle data, the appearance of Hitchin components in moduli spaces, and concrete low-dimensional classifications (dimensions 4 and 5) illustrating the richness of non-homogeneous examples.
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.