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, ‎it confidently investigates the $\kappa$-noncollapsing‎ ‎estimates along the RBF‎.