Theorem H — Flow Stability

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

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

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 $\|\log(\cdot)\|^2_{\mathfrak{g}}$ 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 $s\ge0$.

Step 3 (Energy monotonicity). $\frac{d}{ds}\mathcal{E}(h_s)=-\|\mathrm{grad}\,\mathcal{E}(h_s)\|^2\le0$. Since $\mathcal{E}\ge0$, $\mathcal{E}(h_s)\searrow\mathcal{E}^*$.

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 $c$: $\|\mathrm{grad}\,\mathcal{E}(h)\|\ge C|\mathcal{E}(h)-c|^\alpha$ for some $\alpha\in[\frac{1}{2},1)$ near $\mathcal{E}^{-1}(c)$.

Standard consequence (Simon, 1983): $\int_0^\infty\|\dot h_s\|\,ds<\infty$, so $h_s$ has a limit $h_\infty$.

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

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

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