Hamilton's Maximum Principle for Systems

The Theorem

The scalar maximum principle is a cornerstone of parabolic PDE theory: if a scalar function satisfies $\partial_t u \leq \Delta u + f(u)$ and $u \leq C$ initially, then $u \leq C(t)$ where $C(t)$ solves the ODE $\dot{C} = f(C)$. Hamilton generalised this to sections of vector bundles.

Theorem (Hamilton, 1986)

Let $\mathcal{E} \to M$ be a vector bundle over a closed Riemannian manifold with a time-dependent metric-compatible connection. Let $u$ be a section satisfying the reaction-diffusion equation

$$\frac{\partial u}{\partial t} = \Delta u + F(u)$$

where $\Delta$ is the connection Laplacian. Let $K \subset \mathcal{E}$ be a closed subset that is:

fibrewise convex: $K_x = K \cap \mathcal{E}_x$ is convex for each $x \in M$
O(n)-invariant: $K$ is preserved by parallel transport
ODE-preserved: the ODE $\dot{u} = F(u)$ preserves $K$

Then $K$ is preserved by the PDE: if $u(x, 0) \in K_x$ for all $x \in M$, then $u(x, t) \in K_x$ for all $x$ and $t > 0$.

Proof Sketch

The proof reduces the vector-valued problem to a family of scalar problems using supporting linear functionals.

Since $K_x$ is closed and convex, at any boundary point $u_0 \in \partial K_x$there exists a supporting hyperplane: a linear functional $\ell: \mathcal{E}_x \to \mathbb{R}$such that

$$\ell(u_0) = \sup_{v \in K_x} \ell(v), \qquad \ell(v) \leq \ell(u_0) \;\;\forall\, v \in K_x.$$

Define the scalar function $\varphi(x, t) = \ell(u(x, t))$. Using the O(n)-invariance of $K$ (which ensures $\ell$ can be extended by parallel transport) and the fibrewise convexity, one shows

$$\frac{\partial \varphi}{\partial t} \leq \Delta \varphi + \ell(F(u)).$$

The ODE-preservation condition gives $\ell(F(u_0)) \leq 0$ at the boundary of $K$, and the scalar maximum principle then prevents $\varphi$ from exceeding its initial bound. Since this holds for every supporting functional, $u$ remains in $K$.

Application: Preservation of Non-Negative Curvature

The principal application in Ricci flow is to the curvature operator $\mathcal{R}$, viewed as a section of the bundle $\mathrm{Sym}^2(\Lambda^2 T^*M)$. The curvature operator evolves by

$$\frac{\partial}{\partial t}\mathcal{R} = \Delta \mathcal{R} + \mathcal{R}^2 + \mathcal{R}^{\#}$$

where $\mathcal{R}^{\#}$ is the Lie-algebra square. Take $K = \{\mathcal{R} \geq 0\}$, i.e., the cone of non-negative definite curvature operators. We must verify that the ODE $\dot{\mathcal{R}} = \mathcal{R}^2 + \mathcal{R}^{\#}$ preserves this cone.

In dimension $n = 3$, the eigenvalues $\lambda_a$ of $\mathcal{R}$evolve by $\dot{\lambda}_a = \lambda_a^2 + \lambda_b \lambda_c$. At the boundary of $K$ we have $\lambda_1 = 0$ with $\lambda_2, \lambda_3 \geq 0$. The key quadratic form identity is:

$$\langle (\mathcal{R}^2 + \mathcal{R}^{\#})\varphi, \varphi \rangle = \sum_{a < b} (\lambda_a + \lambda_b)^2 \lambda_a \lambda_b \geq 0$$

when $\mathcal{R} \geq 0$, since each factor is non-negative. By Hamilton's maximum principle, $\mathcal{R} \geq 0$ is preserved under Ricci flow in dimension $n \leq 3$. This is a crucial ingredient in Hamilton's original proof that Ricci flow on closed 3-manifolds with positive Ricci curvature converges to a space form.

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