The Strong Maximum Principle and Splitting

The maximum principles of 11A.7 Maximum Principles under Ricci Flow say that nonnegative curvature is preserved. The strong maximum principle says much more: if the curvature is nonnegative and vanishes somewhere, in some direction, at a positive time, then it vanishes in that direction everywhere, and the manifold splits off a flat factor. Positivity spreads instantly through a connected manifold under a heat equation; so does the structure of the zeros.

This is one of the main tools in Perelman's analysis of $\kappa$-solutions (12B.1 κ-Solutions, 12B.3 The Structure of κ-Solutions), and Morgan and Tian use it in the form proved here at least a dozen times. A three-dimensional $\kappa$-solution has nonnegative curvature. If its curvature operator has a zero eigenvalue anywhere, the strong maximum principle makes it split, locally a surface times a line, and globally (after passing to a cover) a product $N^2\times\mathbb{R}$; and then the two-dimensional theory shows that $N$ is a round sphere and the solution is a neck. If not, its curvature is positive, and the soul theorem makes it a ball or compact. The other application is negative: a nonflat cone cannot occur as a time slice of a flow with nonnegative curvature, which is how blow-down limits of $\kappa$-solutions are shown to be lower-dimensional.

By the end of this chapter you will be able to:

• state the strong maximum principle for the heat equation with Dirichlet boundary values, and the comparison trick that transfers it to curvature;
• prove that a three-dimensional Ricci flow with nonnegative sectional curvature and $R = 0$ at one point of a later time is flat;
• prove that if, in addition, the Ricci curvature has a zero eigenvalue somewhere, the flow splits locally as a surface times a line, by showing that the null direction is parallel and constant in time;
• describe the global versions on covers, and the four possible covers;
• prove that a time slice of a flow with nonnegative curvature cannot contain an open piece of a nonflat cone;
• state Hamilton's null-space theorem in all dimensions and the classification of compact 3-manifolds with nonnegative Ricci curvature.

8.1The strong maximum principle for scalars

The first statement is the weak maximum principle. The second is the parabolic strong maximum principle of 6A.5 Maximum Principles (Nirenberg's), proved there with Hopf's barrier argument; equivalently, the Dirichlet heat kernel of $U$ is strictly positive in the interior for positive times. Heat started at one point is felt everywhere immediately.

The trick for transferring this to curvature is comparison with a Dirichlet solution. Suppose a quantity $s \geq 0$ built from the curvature is a supersolution of the heat equation, $(\partial_t - \Delta)s \geq 0$, at least in a weak sense. Put underneath it a solution $h$ of the heat equation with Dirichlet boundary values and with $0 \leq h(\cdot, 0) \leq s(\cdot, 0)$, positive somewhere. The weak maximum principle applied to $s - h$ gives $s \geq h$, and the strong maximum principle for $h$ gives $h > 0$ in the interior at positive times. So $s > 0$ there.

For a scalar such as $R$ this is immediate. For quantities like "the smallest eigenvalue of $\operatorname{Rm}$", which are not smooth and satisfy no exact equation, Morgan and Tian use the maximum principle for systems (11A.7 Maximum Principles under Ricci Flow) on the bundle $V\oplus\mathbb{R}$ with the convex set $\{(T, h) : s(T) \geq h \geq 0\}$: the conclusion is again $s \geq h$ (their Proposition 4.17). The conditions on $s$ are those that make this set convex and invariant: $s$ is concave on each fibre (so $\{s \geq h\}$ is convex), invariant under parallel transport, and the reaction term $\mathcal M^2 + \mathcal M^\#$ does not decrease it at points where it is nonnegative.

8.2Flatness from a single zero

Note what the theorem does not need: completeness, or any control at infinity. It is a local statement in space, which is why it applies to limits of flows that are only known on regions, as in Morgan and Tian's chapters 10 and 11.

8.3Splitting from a zero eigenvalue of Ricci

Step 4 is the heart of the matter. The null space persists in time (Step 2), so the quantity $\langle\mathcal M\sigma, \sigma\rangle$ is identically zero; differentiating it once in time and twice in space, and comparing through the evolution equation, leaves a sum of two nonnegative terms that must vanish, $2\langle\mathcal M\nabla\sigma, \nabla\sigma\rangle$ and $\langle\mathcal M^\#\sigma, \sigma\rangle$. The first says the null space cannot turn. Algebra alone (the curvature is nonnegative and vanishes on some planes) gives only first-order information and does not suffice; the parabolic equation supplies the second-order information.

The global version. On a complete manifold the local splitting globalises after passing to a cover.

The cover is needed because the line field may not be orientable, or its orthogonal surfaces may close up. Morgan and Tian list the possibilities (their Remark 4.21): the cover can be trivial, an infinite cyclic cover, a double cover, or a cover with group the infinite dihedral group $\mathbb{Z}/2 * \mathbb{Z}/2$. In the first two cases there is a global parallel unit vector field spanning the null direction; in the last two only a nonorientable line field. Examples: $S^2\times\mathbb{R}$ itself (trivial cover); $S^2\times S^1$ (covered by $S^2\times\mathbb{R}$ with group $\mathbb{Z}$); the cross-cap quotient $(S^2\times\mathbb{R})/\iota$ of 10A.6 Necks, Caps and Their Topology (double cover); and $\mathbb{RP}^3\#\mathbb{RP}^3$ with a product-like metric (group $\mathbb{Z}/2 * \mathbb{Z}/2$).

8.4Cones are excluded

The open cone over a Riemannian surface $(N, h)$ is $N\times(0, \infty)$ with $ds^2 + s^2h$. Its curvature is computed from the warped product formulas of 9A.5 Computing Curvature with warping function $\varphi(s) = s$: planes containing $\partial_s$ have curvature $-\frac{\varphi''}{\varphi} = 0$, and the plane tangent to $N\times\{s\}$ has curvature $\frac{K_h - \varphi'^2}{\varphi^2} = \frac{K_h - 1}{s^2}$. So the curvature operator of a cone has a 2-dimensional null space at every point, and its third eigenvalue $\frac{2(K_h - 1)}{s^2}$ scales like $s^{-2}$ along the rays. The cone is flat exactly when $K_h \equiv 1$, that is when $(N, h)$ is locally the unit sphere and the cone is locally $\mathbb{R}^3$.

This proposition is what makes asymptotic cones of $\kappa$-solutions useful. Rescaling a nonnegatively curved $\kappa$-solution down (a blow-down) produces, in the limit, a cone (9B.5 Splitting and Soul Theorems); if the limit were a smooth flow with nonnegative curvature through a nonflat cone, the proposition would force flatness. So the blow-down is either flat or singular, and in three dimensions Perelman uses this to show that the asymptotic volume of a $\kappa$-solution vanishes (12B.3 The Structure of κ-Solutions; Morgan and Tian, chapter 9, §6).

8.5All dimensions, and compact manifolds

Hamilton proved the strong maximum principle for the full curvature operator in every dimension (1986; statement here).

The proof is the argument of Step 4 above in general dimension: on the null space, $2\langle\mathcal M\nabla_i\sigma, \nabla_i\sigma\rangle + \langle\mathcal M^\#\sigma, \sigma\rangle = 0$, both terms are nonnegative, and the vanishing of the second, which is built from the Lie bracket, is what makes the image closed under brackets. In dimension three the possible null spaces are $0$ (positive curvature), a 2-dimensional subspace $\{V\wedge W\}$ (the splitting above), or all of $\Lambda^2$ (flat).

Combining with Hamilton's 1982 theorem gives the classification of compact 3-manifolds with nonnegative Ricci curvature (Hamilton 1986; statement): if $M$ is closed and has a metric with $\operatorname{Ric} \geq 0$, then under the Ricci flow either $\operatorname{Ric}$ becomes positive at once, and $M$ is a spherical space form (11A.11 Hamilton’s 1982 Theorem); or the flow splits locally, and $M$ is a quotient of $S^2\times\mathbb{R}$; or $\operatorname{Ric} \equiv 0$, which in three dimensions means flat, and $M$ is a quotient of $\mathbb{R}^3$. The middle case uses the analogue of Theorem 8.3 for Ricci curvature, whose proof is the same with $s = \mu + \nu$.

8.6History

Hamilton proved the strong maximum principle for the curvature operator, and the classification of compact 3-manifolds with nonnegative Ricci curvature and 4-manifolds with nonnegative curvature operator, in "Four-manifolds with positive curvature operator" (1986). Louis Nirenberg proved the strong maximum principle for parabolic equations in 1953. Perelman used the splitting consequences throughout his study of $\kappa$-solutions (2002), and Morgan and Tian's chapter 4, §4 gives the local versions used here.

8.7Exercises

Solution

$h(x, t) = \sum_{k \geq 1}b_ke^{-k^2t}\sin kx$ with $b_k = \frac2\pi\int_0^\pi h_0\sin kx\,dx$. Positivity for all $t > 0$ follows from the representation $h(x, t) = \int_0^\pi p_t(x, y)h_0(y)\,dy$ with the Dirichlet heat kernel $p_t(x, y) > 0$ for $x, y \in (0, \pi)$ and $t > 0$ (a consequence of the strong maximum principle, or of the method of images, which writes $p_t$ as an alternating sum of Gaussians that is positive inside the interval). Since $h_0 \geq 0$ and is positive on a set of positive measure, $h(x, t) > 0$ everywhere in the interior. For large $t$, $h \approx b_1e^{-t}\sin x$ with $b_1 > 0$: positive everywhere, and with the shape of the first eigenfunction.

Solution

If the flow stayed flat near the second point $p$ up to some time $T > 0$, then $R(p, T) = 0$. The flow has nonnegative sectional curvature for all time (preserved, 11A.7 Maximum Principles under Ricci Flow), so Theorem 8.2 makes it flat everywhere on $[0, T]$, including at $t = 0$ near the first point, where it is round: a contradiction. The positive curvature near the first point immediately makes $R > 0$ everywhere.

Solution

For $dr^2 + \varphi(r)^2h$, 9A.5 Computing Curvature gives $K(\partial_r, X) = -\frac{\varphi''}{\varphi}$ and $K(X, Y) = \frac{K_h - \varphi'^2}{\varphi^2}$ for $X, Y$ tangent to $N$. With $\varphi = s$: $\varphi'' = 0$ and $\varphi' = 1$, giving $0$ and $\frac{K_h - 1}{s^2}$. A round sphere of radius $a$ has $K_h = \frac{1}{a^2} > 1$ when $a < 1$, so the tangential curvature $\frac{a^{-2} - 1}{s^2}$ is positive and blows up at the vertex $s = 0$, where the cone is singular: it is a cone of total angle less than that of $\mathbb{R}^3$, like a sharpened pencil tip in one dimension higher. By Proposition 8.5 no smooth flow with nonnegative curvature can pass through a time slice containing such a piece, away from the vertex.

Solution

$f(\tau) = \langle A(\tau)v, v\rangle \geq 0$ with $f(0) = 0$, so $f'(0) = \langle A'(0)v, v\rangle = 0$. For the last part, let $w \in \ker A(0)$ and apply the first part to $u = v + cw$, which is in $\ker A(0)$ for every real $c$: $0 = \langle A'(0)u, u\rangle = \langle A'(0)v, v\rangle + 2c\langle A'(0)v, w\rangle + c^2\langle A'(0)w, w\rangle$ for all $c$, so all three coefficients vanish, in particular $\langle A'(0)v, w\rangle = 0$. So $A'(0)v$ is orthogonal to $\ker A(0)$, which for a symmetric matrix means $A'(0)v \in \operatorname{im}A(0)$. This kind of first-order information is what a minimum provides; in the proof of Theorem 8.3 the second-order information from the evolution equation is what forces the null space to be parallel.

Solution

By the global splitting, a cover is $(N, h(t))\times\mathbb{R}$ with $(N, h(t))$ an ancient solution of the Ricci flow on a surface, with positive curvature (and complete, since $M$ is). If $N$ is compact it is $S^2$ (or $\mathbb{RP}^2$, covered by $S^2$), and compact ancient solutions on $S^2$ are the round shrinking spheres or the King–Rosenau sausages. The sausage is collapsed in the far past (it looks like two cigars), so if one knows that the factor is $\kappa$-noncollapsed it is round, and the cover is the round shrinking cylinder. If $N$ were noncompact, it would be a complete ancient surface solution with positive curvature, and the cigar is the model; it is collapsed at large scales. Perelman's $\kappa$-noncollapsing (and his classification of two-dimensional $\kappa$-solutions as round spheres, 12B.1 κ-Solutions) is the extra input: it excludes the sausage and the cigar, leaving the round cylinder and its quotients.