The Sobolev inequalities along the Ricci-Bourguignon flow and applications

Abstract

Consider a closed Riemannian manifold $M$ where the metrics $g(t)$ evolve by the Ricci-Bourguignon flow (RBF).The paper first derives a Sobolev inequality along the RBF. Then, by applying this Sobolev inequality we obtain a linear parabolic estimate and an upper bound estimate for the heat kernel  along the RBF. Additionally, {this paper investigates} the $\kappa$-noncollapsing  estimates along the RBF.

Keywords

• Ricci-Bourguignon flow
• eigenvalue
• monotonicity formula
• Sobolev inequality

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