[Paper Review] Stability of the Spinor Flow

Lothar Schiemanowski|arXiv (Cornell University)|Jun 28, 2017

Geometry and complex manifolds9 references3 citations

TL;DR

This paper establishes the stability of Ricci-flat metrics with parallel spinor fields under the spinor flow, proving that initial conditions near such critical points converge exponentially fast to a critical point. It further shows exponential convergence for volume-constrained minimizers under similar conditions, using a Łojasiewicz-Simon inequality and parabolic regularity estimates on the gauged spinor flow.

ABSTRACT

We show stability of pairs of Ricci flat metrics and parallel spinor fields with respect to the spinor flow, i.e. we show that the spinor flow with initial conditions near such pairs converges to a critical point with exponential speed. Moreover, we show stability of certain volume constrained critical points of the spinorial energy.

Motivation & Objective

• To establish the stability of critical points of the spinorial energy functional under the spinor flow on closed spin manifolds.
• To analyze the convergence behavior of the spinor flow near Ricci-flat metrics with parallel spinors, under the absence of Killing fields.
• To extend stability results to volume-constrained critical points of the spinorial energy, particularly volume-constrained minimizers.
• To prove exponential convergence in all $C^k$ norms for both standard and volume-normalized spinor flows.
• To generalize stability results from $G_2$-structures to general spinor flows using Łojasiewicz-Simon techniques.

Proposed method

• Derive a Łojasiewicz-Simon inequality for the spinorial energy functional, leveraging the smoothness of the critical set established in prior work.
• Apply parabolic estimates and regularity theory to the gauged spinor flow, which is a strongly parabolic system on the space of spinor fields and metrics.
• Use gradient estimates and decay bounds on the negative gradient $Q( ilde{ ho}_t)$ to control the $H^k$-norm of the flow velocity.
• Employ a modified flow via gauge-fixing to ensure convergence in Sobolev norms and to apply standard parabolic theory.
• Analyze the long-time behavior of the flow by integrating the velocity field and showing convergence to a critical point in $H^k$.
• Use the fact that the spinor flow is a generalization of the heat flow for $G_2$-structures, adapting techniques from Ricci flow stability.

Experimental results

Research questions

• RQ1 Does the spinor flow converge to a critical point when initialized near a pair of Ricci-flat metric and parallel spinor field?
• RQ2 What conditions ensure exponential convergence of the spinor flow to a critical point in $C^k$ norms?
• RQ3 How does the stability of volume-constrained critical points differ from standard critical points under the spinor flow?
• RQ4 Can the Łojasiewicz-Simon inequality be applied to the spinorial energy functional to ensure convergence?
• RQ5 What role do Killing fields play in the stability of critical points, and why is their absence required?

Key findings

• The spinor flow with initial data in a $C^inity$ neighborhood of a critical point $(\bar{g}, \bar{\varphi})$ converges smoothly to a critical point with exponential speed in all $C^k$ norms.
• Exponential convergence is established via a Łojasiewicz-Simon inequality and parabolic regularity estimates on the gauged spinor flow.
• The critical set near a volume-constrained minimizer must be a manifold, and under this condition, the volume-normalized spinor flow converges exponentially to a minimizer.
• The convergence rate is $O(e^{-\alpha t})$ for some $\alpha > 0$ in the standard case, and $O(t^{-(\gamma-1)})$ with $\gamma > 1$ in the case of weaker estimates.
• The absence of Killing fields (i.e., no torus factor) is essential for stability, as it ensures the critical set is isolated and the flow does not get trapped in non-trivial moduli.
• The results generalize stability results from $G_2$-structures to general spinor flows, using techniques from Ricci flow and geometric analysis.