On non-compact gradient solitons

TL;DR

This paper extends the theory of generalized solitons to non-compact cases, showing under certain conditions that such solitons are stationary and flat, with applications to various geometric flows.

Contribution

It generalizes existing results for $q$-solitons to non-compact manifolds, establishing conditions for stationarity and flatness.

Findings

Non-compact $q$-solitons are stationary and $q$-flat under certain conditions.

Regularity and curvature conditions are crucial for the results.

Applications to ambient obstruction, Cotton, and Bach solitons demonstrate the theorems' utility.

Abstract

In this paper we extend existing results for generalized solitons, called $q$-solitons, to the complete case by considering non-compact solitons. By placing regularity conditions on the vector field $X$ and curvature conditions on $M$, we are able to use the chosen properties of the tensor $q$ to see that such non-compact $q$-solitons are stationary and $q$-flat. We conclude by applying our results to the examples of ambient obstruction solitons, Cotton solitons, and Bach solitons to demonstrate the utility of these general theorems for various flows.