[Paper Review] Super Ricci flow for disjoint unions

Sajjad Lakzian, Michael Munn|arXiv (Cornell University)|Nov 12, 2012

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper introduces a metric-based characterization of super Ricci flow for disjoint unions of compact Riemannian manifolds by defining a family of distance metrics $D^t$ on $M_1 igsqcup M_2$ that evolve such that the Lipschitz constant of heat flow solutions is non-increasing in time. The key result shows that this super Ricci flow property holds if the distance between points in different components evolves by the heat equation, i.e., $\partial_t D^t \geq \Delta_{M_1^t \times M_2^t} D^t$. This provides a metric-level criterion for super Ricci flow in disconnected settings.

ABSTRACT

In this paper we consider compact, Riemannian manifolds $M_1, M_2$ each equipped with a one-parameter family of metrics $g_1(t), g_2(t)$ satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-Topping, we introduce the notion of a super Ricci flow for a family of distance metrics defined on the disjoint union $\MM$. In particular, we show such a super Ricci flow property holds provided the distance function between points in $M_1$ and $M_2$ evolves by the heat equation. We also discuss possible applications and examples.

Motivation & Objective

• To extend the concept of super Ricci flow from smooth Riemannian metrics to disconnected metric spaces formed by disjoint unions of manifolds.
• To define a family of distance metrics $D^t$ on $M_1 \sqcup M_2$ that are compatible with evolving Ricci flows on each component.
• To establish a sufficient condition—based on metric evolution via the heat equation—for the entire family $(M_1 \sqcup M_2, D^t)$ to satisfy the super Ricci flow property.
• To provide a metric-theoretic framework that persists through singularities, enabling analysis beyond finite-time Ricci flow breakdown.

Proposed method

• Define a family of distance metrics $D^t$ on the disjoint union $M_1 \sqcup M_2$ such that $D^t|_{M_i} = d_{g_i(t)}$, preserving the intrinsic geometry of each component.
• Adopt the McCann-Topping characterization of super Ricci flow: a family $D^t$ is a super Ricci flow if the Lipschitz constant of any heat flow solution is non-increasing in time.
• Use the heat equation on the product space $M_1^t \times M_2^t$ to define the Laplacian $\Delta_{M_1^t \times M_2^t}$ acting on the distance function $D^t(x,y)$ for $x \in M_1, y \in M_2$.
• Establish the key inequality $\partial_t D^t \geq \Delta_{M_1^t \times M_2^t} D^t$ as a sufficient condition for the super Ricci flow property.
• Apply the maximum principle to the ratio $\overline{u}/D^t$ for heat flow solutions $\overline{u}$, showing that $\partial_t \operatorname{Lip}(u,t) \leq 0$ under the stated condition.
• Generalize the result to $k$-component disjoint unions by requiring the same inequality $\partial_t D^t \geq \Delta_{M_i^t \times M_j^t} D^t$ for all $i \neq j$.

Experimental results

Research questions

• RQ1 Under what conditions does a family of distance metrics on a disjoint union of manifolds qualify as a super Ricci flow?
• RQ2 Can the super Ricci flow property be characterized purely in terms of the evolution of inter-component distances, independent of smooth metric evolution?
• RQ3 How does the heat equation on the product space $M_1^t \times M_2^t$ govern the time evolution of distances between distinct components in a super Ricci flow?
• RQ4 What is the role of the maximum principle in proving the non-increasing Lipschitz constant of heat flow solutions on disconnected metric spaces?
• RQ5 Can the super Ricci flow framework be extended to multiple disconnected components with inter-component distance evolution governed by heat-type inequalities?

Key findings

• The super Ricci flow property holds on $M_1 \sqcup M_2$ if $\partial_t D^t \geq \Delta_{M_1^t \times M_2^t} D^t$ for all $x \in M_1, y \in M_2$, ensuring the Lipschitz constant of any heat flow solution is non-increasing in time.
• The condition $\partial_t D^t \geq \Delta_{M_1^t \times M_2^t} D^t$ is sufficient for $D^t$ to define a super Ricci flow on the disjoint union, even when the individual metrics $g_i(t)$ are only super Ricci flows.
• The result extends to $k$-component disjoint unions: if the inter-component distance evolution satisfies the same inequality for all $i \neq j$, then $D^t$ is a super Ricci flow on $M_1 \sqcup \cdots \sqcup M_k$.
• If $D^0(x,y) \geq c > 0$ initially, then $D^t(x,y) \geq c$ for all $t > 0$, as guaranteed by the maximum principle applied to the distance evolution.
• The proof relies on analyzing the Laplacian of the ratio $\overline{u}/D^t$ and deriving the inequality $\partial_t \overline{u} \leq \frac{\overline{u}}{D^t} \partial_t D^t$, which implies $\partial_t \operatorname{Lip}(u,t) \leq 0$.
• The framework provides a metric-level alternative to classical Ricci flow, enabling analysis of geometric evolution beyond finite-time singularities through distance-based evolution.