Geometric and analytical results for $ρ$-Einstein solitons
Caio Coimbra
TL;DR
The paper investigates complete noncompact gradient ρ-Einstein solitons by deriving a Lichnerowicz–Obata-type spectral gap for the drifted Laplacian Δ_f and establishing volume and weighted volume growth estimates for geodesic balls. The authors leverage the gradient soliton structure Ric + Hess f = (ρR + λ)g to control the geometry and analysis, proving that the first nonzero eigenvalue satisfies λ1(Δ_f) ≥ λ with rigidity characterizing the Gaussian shrinking soliton in the equality case. They further obtain upper bounds for Vol(B_p(r)) and Vol_f(B_p(r)) that depend explicitly on the potential f and the soliton data, and provide corollaries comparing these volumes to those of Gaussian solitons when Ric_f remains positively bounded. These results extend classical volume-growth and spectral-gap phenomena to the setting of ρ-Einstein solitons, highlighting rigidity when equalities are attained and offering tools for further geometric analysis of smooth metric measure spaces.
Abstract
In this article, we study geometric and analytical features of complete noncompact $ρ$-Einstein solitons, which are self-similar solutions of the Ricci-Bourguignon flow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking $ρ$-Einstein solitons. Moreover, similar to classical results due to Calabi--Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact $ρ$-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds.