[Paper Review] The mixed scalar curvature flow on a fiber bundle

Vladimir Rovenski, Leonid Zelenko|arXiv (Cornell University)|Mar 28, 2012

Geometric Analysis and Curvature Flows2 references3 citations

TL;DR

This paper introduces a conformal flow of metrics on a fiber bundle to evolve the mixed scalar curvature toward positivity, showing that if the non-umbilicity of the orthogonal distribution is sufficiently small relative to its non-integrability, the flow converges to a metric with positive mixed scalar curvature. The mean curvature vector evolves according to a forced Burgers equation, which reduces to a linear Schrödinger equation with a potential tied to the distribution's geometric non-umbilicity.

ABSTRACT

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case of integrable orthogonal distribution the flow velocity is proportional to the mixed scalar curvature. We observe that the mean curvature vector $H$ of $D$ satisfies along the leaves the forced Burgers equation, this reduces to the linear Schrödinger equation, whose potential function is a certain "non-umbilicity" measure of $D$. On order to show convergence of the solution metrics $g_t$ as $t o\infty$, we normalize the flow, and instead of a foliation consider a fiber bundle $π: M o B$ of a Riemannian manifold $(M, g_0)$. In this case, if the "non-umbilicity" of $D$ is smaller in a sense then the "non-integrability", then the limit mixed scalar curvature function is positive. For integrable $D$, we give examples with foliated surfaces and twisted products.

Motivation & Objective

• To determine under what geometric conditions a foliation on a Riemannian manifold admits a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive.
• To analyze the evolution of the mixed scalar curvature via a conformal flow restricted to the orthogonal distribution of a foliation.
• To establish convergence criteria for the flow in the setting of a fiber bundle, particularly when the orthogonal distribution is not necessarily integrable.
• To link the dynamics of the mean curvature vector to the forced Burgers equation and its reduction to a linear Schrödinger operator with a geometric potential.

Proposed method

• The authors define a conformal flow of metrics on a fiber bundle $\pi: M \to B$, evolving the metric only on the orthogonal distribution $D$ to control the mixed scalar curvature.
• The flow velocity is proportional to the mixed scalar curvature when $D$ is integrable, and includes the integrability tensor $T$ of $D$ in the evolution operator for non-integrable cases.
• The mean curvature vector $H$ of $D$ satisfies a forced Burgers equation, which reduces to a linear Schrödinger equation with potential function measuring the non-umbilicity of $D$.
• The flow is normalized to ensure long-time existence and convergence; the limit metric is analyzed using spectral theory of the Schrödinger operator $\mathcal{H} = -\Delta - f(x)\,\text{id}$.
• Sobolev embedding and eigenfunction regularity are used to prove uniform convergence of series expansions of solutions, ensuring smoothness of the limit metric.
• The convergence to a metric with positive mixed scalar curvature is established under the condition that the non-umbilicity of $D$ is smaller than its non-integrability in a precise geometric sense.

Experimental results

Research questions

• RQ1 Under what geometric conditions does a foliation on a Riemannian manifold admit a metric making the leaves totally geodesic and the mixed scalar curvature positive?
• RQ2 How does the mean curvature vector of the orthogonal distribution evolve under the conformal flow, and what PDE governs its dynamics?
• RQ3 What is the relationship between the non-integrability of the orthogonal distribution and the non-umbilicity of the mean curvature, and how do they affect the sign of the mixed scalar curvature?
• RQ4 Can the conformal flow on a fiber bundle converge to a metric with positive mixed scalar curvature, and under what conditions?

Key findings

• The mean curvature vector $H$ evolves according to a forced Burgers equation, which reduces to a linear Schrödinger equation with potential $f(x)$ measuring the non-umbilicity of the distribution $D$.
• For a fiber bundle with normalized flow, if the non-umbilicity of $D$ is sufficiently small relative to its non-integrability, the limit mixed scalar curvature is positive.
• The eigenfunctions of the Schrödinger operator $\mathcal{H} = -\Delta - f(x)\,\text{id}$ are smooth and belong to $C^\infty(F)$, ensuring regularity of the limit metric.
• The series expansion of any smooth function $u$ on the fiber $F$ in terms of eigenfunctions of $\mathcal{H}$ converges absolutely and uniformly, with derivatives also converging uniformly.
• The spectral theory of $\mathcal{H}$, combined with Sobolev embeddings, ensures that $L^2$-solutions of the flow converge uniformly to smooth limits, guaranteeing the existence of a smooth limiting metric.
• The paper provides explicit examples with foliated surfaces and twisted products, illustrating the applicability of the flow in geometrically distinct settings.