Theorem H — Flow Stability

Prerequisites: Compactness of $G^{F^{\circ }}$, real-analyticity of $\mathcal{E}$, Lojasiewicz inequality (Fact B.8).

Theorem H. If $h^{\ast }$ is a non-degenerate local minimum of ${\mathcal{E}}_{F^{\circ }}$, then the gradient flow ${\dot{h}}_s=-\mathrm{grad}\mathcal{E}(h_s)$ converges, and $[A_{\infty }]$ is a stable fixed point.

Convergence rate: If the Lojasiewicz exponent at $h^{\ast }$ is $\alpha =\frac{1}{2}$, convergence is exponential: $d(h_s,h^{\ast }\cdot {\mathcal{G}}_{\mathrm{const}})\le C_0{e}^{-\gamma s}$ for $\gamma >0$ depending on the Hessian of $\mathcal{E}$ at $h^{\ast }$.

Proof. (Complete; expanded from Chapter 7.)

Step 1 (Real-analyticity). The energy ${\mathcal{E}}_{F^{\circ }}:G^{F^{\circ }}\to \mathbb{R}$ is real-analytic. Each term $W_t(i,j)d_G(g_t^h(i,j),e{)}^2$ is analytic in $h$: the group multiplication $G\times G\to G$ is analytic, the bi-invariant distance squared $d_G(\cdot ,e{)}^2$ is analytic on $G$ (it equals $\Vert \log (\cdot ){\Vert }_{\mathfrak{g}}^2$ near $e$, and extends analytically by compactness and bi-invariance), and a finite sum of analytic functions is analytic.

Step 2 (Global existence). $G^{F^{\circ }}$ is compact, so the negative gradient flow exists for all $s\ge 0$.

Step 3 (Energy monotonicity). $\frac{d}{ds}\mathcal{E}(h_s)=-\Vert \mathrm{grad}\mathcal{E}(h_s){\Vert }^2\le 0$. Since $\mathcal{E}\ge 0$, $\mathcal{E}(h_s)\searrow {\mathcal{E}}^{\ast }$.

Step 4 (Lojasiewicz convergence). By Fact B.8, since $\mathcal{E}$ is real-analytic on the compact manifold $G^{F^{\circ }}$, for every critical value $c$: $\Vert \mathrm{grad}\mathcal{E}(h)\Vert \ge C|\mathcal{E}(h)-c{|}^{\alpha }$ for some $\alpha \in [\frac{1}{2},1)$ near ${\mathcal{E}}^{-1}(c)$.

Standard consequence (Simon, 1983): ${\int }_0^{\infty }\Vert {\dot{h}}_s\Vert ds<\infty$, so $h_s$ has a limit $h_{\infty }$.

Step 5 (Exponential rate for $\alpha =1/2$). At a non-degenerate minimum, the Hessian ${\nabla }^2\mathcal{E}{|}_{h^{\ast }}$ is positive-definite on $T_{h^{\ast }}(G^{F^{\circ }})/T_{h^{\ast }}(h^{\ast }\cdot {\mathcal{G}}_{\mathrm{const}})$ (the normal space to the gauge orbit). Denote its smallest eigenvalue by ${\lambda }_{\min }>0$. Then $\Vert \mathrm{grad}\mathcal{E}(h){\Vert }^2\ge {\lambda }_{\min }(\mathcal{E}(h)-{\mathcal{E}}^{\ast })$ near $h^{\ast }$, giving $\alpha =\frac{1}{2}$.

From $\frac{d}{ds}\mathcal{E}=-\Vert \mathrm{grad}\mathcal{E}{\Vert }^2\le -{\lambda }_{\min }(\mathcal{E}-{\mathcal{E}}^{\ast })$, Gronwall gives $\mathcal{E}(h_s)-{\mathcal{E}}^{\ast }\le (\mathcal{E}(h_0)-{\mathcal{E}}^{\ast })e^{-{\lambda }_{\min }s}$.

The distance to the critical orbit: $d(h_s,h^{\ast }\cdot {\mathcal{G}}_{\mathrm{const}}{)}^2\le C_0(\mathcal{E}(h_s)-{\mathcal{E}}^{\ast })\le C_0(\mathcal{E}(h_0)-{\mathcal{E}}^{\ast })e^{-{\lambda }_{\min }s}$. Set $\gamma ={\lambda }_{\min }/2$. $\square$