Metric Perspectives of the Ricci Flow Appliedto Disjoint Unions

Abstract

In this paper we consider compact, Riemannian manifolds M1, M2 each equipped with a oneparameterfamily of metrics g1(t), g2(t) satisfying the Ricci flow equation. Adopting the characterization ofsuper-solutions to the Ricci flow developed by McCann-Topping, we define a super Ricci flow for a family ofdistance metrics defined on the disjoint union M1 ⊔ M2. In particular, we show such a super Ricci flow propertyholds provided the distance function between points in M1 and M2 is itself a super solution of the heatequation on M1 × M2. We also discuss possible applications and examples.

Keywords

• super ricci flow
• disjoint union
• heat kernel