Triviality, Rotational Symmetry, and Classification of Complete Expanding Gradient Yamabe Solitons
Shun Maeta
TL;DR
The paper addresses complete expanding gradient Yamabe solitons by comparing the scalar curvature $R$ to the soliton constant $λ$ and deriving rigidity and symmetry results. It uses the warped-product structure from Tashiro's theorem to convert the soliton equation into an ODE in the radial profile $F'(r)$ and to obtain a complete classification framework. In the regime $R<λ$, nontrivial solitons are classified as warped products with base manifolds of flat or negative constant scalar curvature $R̄$, and with asymptotic behavior $F'(r)→√(R̄/λ)$ as $r→∞$. In the regime $R>λ$, nontrivial solitons are rotationally symmetric warped products with either a sphere factor or with a base of nonpositive curvature, and the results extend previous partial classifications without assuming local conformal flatness.
Abstract
In this paper, we rigorously analyze the scalar curvature of complete expanding gradient Yamabe solitons. We show that if the scalar curvature is less than the soliton constant by an arbitrarily small positive amount, then the soliton is trivial. Additionally, we show that if the scalar curvature is greater than the soliton constant by an arbitrarily small positive amount, then the soliton is rotationally symmetric. Furthermore, we completely classify nontrivial complete expanding gradient Yamabe solitons in both cases: when the scalar curvature is greater than the soliton constant and when it is less than the soliton constant.