Monotonicity, ε-Regularity and Tangent Cones

6C.7 Harmonic Maps and Their Heat Flow ended with two features of the harmonic map heat flow that it promised would recur in the Ricci flow: singularities are analysed by rescaling around the point where energy concentrates, and an $\varepsilon$-regularity theorem says that where the energy is small there is no singularity. This chapter develops those two ideas properly, for the two model problems of geometric analysis, minimal submanifolds and harmonic maps, and adds the third ingredient that makes them work: a monotonicity formula, an elliptic cousin of the parabolic formulas of 6A.13 Self-Similar Variables and Monotonicity, which says that a scale-invariant energy ratio can only grow as the scale grows.

Together the three ideas form the regularity theory of geometric variational problems. Monotonicity gives a density at every point and bounds rescalings. $\varepsilon$-regularity says that points of density close to the regular value are regular. Rescaling at a singular point produces a tangent cone or tangent map, which by the equality case of monotonicity is homogeneous; and dimension reduction (Federer's method) turns the classification of homogeneous singularities into a bound on the size of the singular set. In the critical dimension, where the energy is scale invariant, singularities appear instead as bubbles, with an energy identity that says no energy is lost in the necks between them. Every one of these has a Ricci flow counterpart, and the dictionary at the end of the chapter, from pseudolocality to Bamler's tangent flows, is the reason this machinery belongs in a guide to Perelman.

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

• prove the monotonicity formula for minimal submanifolds of $\mathbb{R}^N$ and identify its equality case as cones;
• define the density, compute it for planes, crossing planes and junctions, and state Allard's regularity theorem;
• derive the stationarity identity for harmonic maps from domain variations, and prove the monotonicity of $r^{2-n}\int_{B_r}|\nabla u|^2$ with its defect;
• state the $\varepsilon$-regularity theorems of Schoen–Uhlenbeck and Bethuel, and the dimension bounds for singular sets;
• explain tangent maps and cones, Federer's dimension reduction, the hedgehog $x/|x|$ and the Simons cone;
• describe bubble trees, the energy identity and the neck analysis in the conformally invariant case;
• match each idea with its Ricci flow counterpart: pseudolocality, tangent flows, canonical neighbourhoods, Cheeger–Colding–Naber and Bamler.

8.1Monotonicity for minimal submanifolds

Let $\Sigma$ be a smooth $k$-dimensional submanifold of $\mathbb{R}^N$ that is minimal: its mean curvature vector vanishes, equivalently it is a critical point of $k$-dimensional area for compactly supported variations (6C.6 Quasilinear Elliptic Equations and Minimal Surfaces, 9A.8 Submanifolds and Minimal Surfaces). The first variation formula says that for every vector field $X$ on $\mathbb{R}^N$ with compact support,

where $\operatorname{div}_\Sigma X = \sum_{i=1}^k\langle\nabla_{e_i}X, e_i\rangle$ for an orthonormal frame $e_i$ of the tangent space. The identity is all that minimality will be used for, and it makes sense for much rougher objects (stationary varifolds), which is why the theory extends beyond smooth surfaces.

Fix $x_0$, which we take to be $0$, and let $f(r) = |\Sigma\cap B_r|$ be the area of $\Sigma$ inside the ball of radius $r$. The position vector $X = x$ has $\operatorname{div}_\Sigma x = k$, because $\nabla_{e_i}x = e_i$. Cutting $X$ off at radius $r$ (rigorously, using $X = \phi(|x|)x$ and letting $\phi$ tend to the indicator of $[0, r]$), the first variation becomes

with $\eta$ the outward unit conormal of $\Sigma\cap B_r$ in $\Sigma$, and $\langle x, \eta\rangle = |x^T|$, the length of the tangential part of $x$, for almost every $r$.

A more careful version keeps the defect: $\Theta(r_2) - \Theta(r_1) = \frac{1}{\omega_k}\int_{\Sigma\cap(B_{r_2}\setminus B_{r_1})}\frac{|x^\perp|^2}{|x|^{k+2}}\,d\mu$, with $x^\perp$ the normal part of $x$, which measures exactly how far $\Sigma$ is from being a cone. Compare Huisken's formula (6A.10 Curve Shortening and the First Geometric Flows, 6A.13 Self-Similar Variables and Monotonicity): there the Gaussian weight made the monotone quantity insensitive to the cutoff, and the defect measured the failure to be a self-shrinker; here the balls of radius $r$ play the role of the Gaussians of scale $\sqrt{T - t}$.

The density. Since $\Theta(r)$ is nondecreasing, the density $\Theta(\Sigma, x_0) = \lim_{r\to0}\Theta(r)$ exists. At a point where $\Sigma$ is smooth, $\Sigma$ looks like its tangent plane at small scales, and $\Theta = 1$. Where two $k$-planes cross transversally, $\Theta = 2$; at the junction of three half-planes, $\Theta = \frac32$ (Exercise 8.5). Monotonicity gives two structural facts at once: the density is at least $1$ at every point of $\Sigma$ (indeed of its closure), and it is upper semicontinuous, $\Theta(x_0) \geq \limsup\Theta(x_j)$ when $x_j \to x_0$. And for a complete minimal submanifold with $\Theta(\infty) = \lim_{r\to\infty}\Theta(r) = 1$, monotonicity forces $\Theta \equiv 1$ and equality everywhere: the submanifold is a plane. This is a Bernstein-type theorem (6C.6 Quasilinear Elliptic Equations and Minimal Surfaces) proved by monotonicity alone.

William Allard proved this in 1972. It is the prototype of an $\varepsilon$-regularity theorem: a scale-invariant quantity, here the density ratio, that is close to its value at regular points controls everything else on a smaller ball. Combined with upper semicontinuity, it shows that the regular points are open and dense, and that singular points have density at least $1 + \varepsilon$.

8.2Monotonicity for harmonic maps

Let $u : B \to N \subset \mathbb{R}^L$ be a map from a ball $B \subset \mathbb{R}^n$ into a compact target, with energy $\int|\nabla u|^2$. A harmonic map is a critical point for variations of the values of $u$ (6C.7 Harmonic Maps and Their Heat Flow). A stationary harmonic map is, in addition, critical for variations of the domain: $u_s(x) = u(x + sX(x))$ for vector fields $X$ with compact support. Smooth harmonic maps are stationary, but the distinction matters for weak solutions, which can be harmonic without being stationary. Stationarity is equivalent to the vanishing divergence of the energy–momentum tensor of 6C.3 Symmetry, Noether’s Theorem and Moving Planes,

for all $X \in C^\infty_c(B; \mathbb{R}^n)$ (Exercise 8.7).

The exponent $2 - n$ is the scaling of the energy: if $u_\lambda(x) = u(\lambda x)$, then $\int_{B_r}|\nabla u_\lambda|^2 = \lambda^{2-n}\int_{B_{\lambda r}}|\nabla u|^2$. So $r^{2-n}\int_{B_r}|\nabla u|^2$ is the energy at scale $r$, invariant under rescaling, and monotonicity says the energy at small scales is controlled by the energy at large scales. Dimension two is critical: $r^0\int_{B_r}|\nabla u|^2$ is the energy itself, which is conformally invariant, and monotonicity is empty; dimensions $n \geq 3$ are supercritical, and monotonicity is strong.

$\varepsilon$-regularity. Richard Schoen and Karen Uhlenbeck (1982), for energy-minimising maps, and Fabrice Bethuel (1993), for stationary harmonic maps (after Lawrence Evans's 1991 proof for spherical targets), proved: there is $\varepsilon_0(n, N) > 0$ such that if

then $u$ is smooth on $B_{r/2}(x_0)$, with $\sup_{B_{r/2}}|\nabla u| \leq \frac Cr$ (statement). Combined with monotonicity, the singular set $\operatorname{sing}(u)$ consists of the points where the energy density $\lim_{r\to0}r^{2-n}\int_{B_r}|\nabla u|^2$ is at least $\varepsilon_0$, a closed set, and a covering argument (3A.6 Modes of Convergence and Differentiation) gives $\mathcal H^{n-2}(\operatorname{sing}u) = 0$ for stationary maps. For minimisers much more is true.

8.3Tangent maps and dimension reduction

At a singular point $x_0$ of an energy-minimising map, rescale: $u_\lambda(x) = u(x_0 + \lambda x)$, $\lambda \to 0$. By monotonicity the scale-invariant energies of $u_\lambda$ on $B_1$ are bounded by that of $u$ on $B_\lambda$, so a subsequence converges weakly in $H^1$; for minimisers, Schoen and Uhlenbeck proved that the convergence is strong (a compactness theorem that uses the minimising property, through a comparison argument of the kind later systematised by Luckhaus). The limit $\phi$, a tangent map, is again minimising, and its scale-invariant energy on every ball $B_r(0)$ equals $\lim_{\lambda\to0}\lambda^{2-n}\int_{B_\lambda(x_0)}|\nabla u|^2$, the density at $x_0$, independent of $r$. By the equality case of Theorem 8.3, $\phi$ is homogeneous of degree zero: $\phi(x) = \omega(\frac{x}{|x|})$ for a map $\omega$ from the sphere $S^{n-1}$ to $N$, which is itself harmonic. The singularity of $u$ at $x_0$ is modelled by a cone.

The hedgehog. The map $\frac{x}{|x|} : B^3 \to S^2$ is homogeneous of degree zero, harmonic away from the origin, with energy $\int_{B_r}|\nabla\frac{x}{|x|}|^2 = 8\pi r$ (Exercise 8.8), so its scale-invariant energy is constant: it is its own tangent map. Haïm Brezis, Jean-Michel Coron and Elliott Lieb (1986) proved that it is energy minimising among maps with its boundary values. It is the mathematical model of a point defect in a nematic liquid crystal, where the director field points radially away from a centre, and such "hedgehog" defects are observed in droplets of liquid crystal under polarised light.

Federer's dimension reduction. Herbert Federer introduced (1970, for minimal surfaces) a method for bounding the dimension of singular sets that Schoen and Uhlenbeck adapted to harmonic maps. Suppose the singular set of some minimiser had Hausdorff dimension greater than $n - 3$. Blowing up at a point of density of the singular set produces a tangent map whose singular set is at least as large and which is homogeneous; blowing up a homogeneous map at a singular point away from the origin produces a map that is invariant under translations in that direction, hence a minimiser on one dimension fewer. Iterating, one reaches a minimising map, homogeneous of degree zero, from $\mathbb{R}^2$ with an isolated singularity at the origin, that is, a harmonic map from a circle, extended homogeneously; but minimising maps from two-dimensional domains are smooth (Morrey, 1948), and a nonconstant homogeneous map of degree zero on $\mathbb{R}^2$ has infinite energy near the origin. The contradiction gives:

For minimal hypersurfaces the same method, with Simons's classification of stable minimal cones, gives singular sets of dimension at most $k - 7$ for $k$-dimensional area-minimising hypersurfaces (Federer, 1970). The bound is sharp: James Simons (1968) found the cone $\{x \in \mathbb{R}^8 : x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2\}$, a seven-dimensional minimal cone with an isolated singularity, and Enrico Bombieri, Ennio De Giorgi and Enrico Giusti (1969) proved that it is area minimising and used it to show that Bernstein's theorem fails for graphs over $\mathbb{R}^8$ (6C.6 Quasilinear Elliptic Equations and Minimal Surfaces).

8.4Bubbles in the critical dimension

In dimension two the energy is conformally invariant (6C.7 Harmonic Maps and Their Heat Flow) and monotonicity is empty, so a sequence of harmonic maps with bounded energy cannot be controlled by monotonicity. What survives is the $\varepsilon$-regularity theorem of Jonathan Sacks and Karen Uhlenbeck (1981): if $\int_{B_r}|\nabla u|^2 < \varepsilon_0$ for a harmonic map on a surface, then $|\nabla u| \leq \frac Cr$ on $B_{r/2}$, with all higher derivatives controlled. For a sequence $u_k$ with $\int|\nabla u_k|^2 \leq E$, at most $\frac{E}{\varepsilon_0}$ points can carry energy $\varepsilon_0$ in every small ball around them, so away from finitely many points $x_1, \dots, x_m$ the sequence is smoothly controlled and converges, after a subsequence, to a harmonic map $u_\infty$. At each $x_j$, rescale at the scale at which the energy concentrates: the rescaled maps converge to a non-constant harmonic map from $\mathbb{R}^2$, which by removability of point singularities extends to a harmonic map from $S^2$, a bubble. Further concentration inside a bubble produces bubbles on bubbles: a bubble tree.

Two questions remain: is any energy lost, and is any image lost, in the regions between the body and the bubbles? These regions are necks: annuli $\{\delta_k \leq |x - x_j| \leq \rho\}$ with $\delta_k \to 0$, conformally equivalent to long cylinders $S^1\times[0, L_k]$ with $L_k = \log\frac{\rho}{\delta_k} \to \infty$, on which the energy in each unit piece is small. Jürgen Jost (1991) and Thomas Parker (1996) proved the energy identity

and Parker showed that the images of the necks shrink to points (no neck), so the limiting image is connected: body and bubbles, joined at points. The neck analysis is a computation on a long cylinder with small energy: on such a cylinder the harmonic map equation is a perturbation of the Laplace equation, whose solutions with small energy decay exponentially from both ends toward the middle, like the Hermite modes of 6A.13 Self-Similar Variables and Monotonicity (on the cylinder, Fourier modes $e^{\pm ks}e^{ik\theta}$). For the harmonic map heat flow, the analogous energy identity at a singular time was proved by Jie Qing and Gang Tian (1997).

The same picture, with the same three ingredients (small-energy regularity, concentration at finitely many points, energy identity through necks), governs pseudo-holomorphic curves in symplectic geometry (Gromov compactness, 1985), Yang–Mills connections in dimension four (Uhlenbeck compactness, 1982, which underlies Donaldson theory), and the Yamabe problem of 6C.4 The Yamabe Problem and Critical Exponents, where bubbles are the concentrating profiles and Struwe's decomposition is the energy identity.

8.5The Ricci flow counterpart

The Ricci flow has every piece of this structure, with parabolic monotone quantities and with metrics in place of maps.

elliptic or harmonic-map idea	Ricci flow counterpart	where
monotone density ratio $r^{2-n}\int_{B_r}\lvert\nabla u\rvert^2$	reduced volume $\tilde V(\tau)$, entropy $\mu(g, \tau)$	12A.3 The 𝓦-Entropy, 12A.5 Reduced Distance and Reduced Volume
density $\Theta \geq 1$, $\Theta = 1$ at regular points	$\tilde V \leq 1$, $\tilde V \to 1$ at regular points	12A.5 Reduced Distance and Reduced Volume
$\varepsilon$-regularity (Allard, Schoen–Uhlenbeck)	pseudolocality: almost-Euclidean balls stay regular	12A.9 Pseudolocality
tangent cone, homogeneous of degree $0$	tangent flow: gradient shrinking soliton	12B.1 κ-Solutions
classification of minimising cones	classification of $\kappa$-solutions and canonical neighbourhoods	12B.3 The Structure of κ-Solutions, 12B.5 The Canonical Neighbourhood Theorem
dimension reduction	splitting at infinity, reduction to lower dimension	11B.4 Blow-Ups, Blow-Downs and Splitting at Infinity
bubbles and necks	caps and $\varepsilon$-necks	10A.6 Necks, Caps and Their Topology, 12B.5 The Canonical Neighbourhood Theorem

Pseudolocality. Perelman's §10 proves that if a ball of radius $r_0$ at time $0$ has scalar curvature at least $-r_0^{-2}$ and an almost-Euclidean isoperimetric inequality for its subdomains, then for a definite time $(\varepsilon r_0)^2$ the curvature on a smaller ball is bounded by $\frac{\alpha}{t} + (\varepsilon r_0)^{-2}$. The hypothesis is a scale-invariant smallness condition, like $r^{2-n}\int_{B_r}|\nabla u|^2 < \varepsilon_0$ or $\Theta(1) \leq 1 + \varepsilon$; the conclusion is regularity on a smaller region, with estimates scaled to it. The proof is by contradiction and blow-up, with the entropy $\mu$ and the conjugate heat kernel playing the role of the monotone quantity: a sequence of counterexamples, rescaled at points of large curvature, would produce a limit with Euclidean entropy that is not Euclidean, contradicting the equality case of the log-Sobolev inequality.

Tangent cones of limit spaces. Jeff Cheeger and Tobias Colding (1996–97) proved that noncollapsed limits of manifolds with Ricci curvature bounded below have tangent cones that are metric cones, by a "volume cone implies metric cone" theorem: the equality case of the Bishop–Gromov monotonicity (9B.2 Volume Comparison), exactly like the equality case of Theorem 8.1. Cheeger and Aaron Naber (2015) proved that for noncollapsed limits with bounded Ricci curvature the singular set has codimension four. Richard Bamler (2020–23) built the corresponding theory for Ricci flows themselves: noncollapsed limits of Ricci flows are singular spaces whose tangent flows are singular gradient shrinking solitons, and whose singular set has codimension four, with Perelman's monotone quantities playing the role of the density. In dimension three, where the possible tangent flows are classified (the round sphere and cylinder and their quotients, 12B.1 κ-Solutions), the structure is simple enough for surgery; Bamler's theory and Kleiner–Lott's singular Ricci flows (12C.9 After Perelman) describe what happens without surgery.

8.6History

Plateau published his laws of soap films in 1873. Morrey proved regularity of two-dimensional minimisers in 1948. The monotonicity formula for minimal surfaces goes back to the 1960s (Fleming, Federer and others); Allard's regularity theorem appeared in 1972. Simons found his cone in 1968, and Bombieri, De Giorgi and Giusti proved it minimising in 1969; Federer's dimension reduction dates from 1970. Taylor proved Plateau's laws in 1976. Sacks and Uhlenbeck's paper on minimal spheres appeared in 1981, Schoen and Uhlenbeck's regularity theory in 1982, Uhlenbeck's compactness for Yang–Mills in 1982, and Price's monotonicity for stationary maps in 1983. Gromov's compactness for pseudo-holomorphic curves appeared in 1985, and Brezis, Coron and Lieb's work on the hedgehog in 1986. Evans (1991) and Bethuel (1993) proved $\varepsilon$-regularity for stationary maps; Jost (1991) and Parker (1996) proved the energy identity, and Qing and Tian (1997) its heat flow version. Cheeger and Colding's structure theory dates from 1996–2000, Perelman's pseudolocality from 2002, Cheeger and Naber's codimension-four theorem from 2015, and Bamler's theory from 2020–23.

8.7Exercises

Solution

(a) $|P\cap B_r| = \omega_kr^k$, so $\Theta = 1$. (b) The intersection of two transverse $k$-planes has measure zero in each, so $\Theta = 2$. (c) Each half-line contributes length $r$, and $\omega_1r = 2r$, so $\Theta = \frac{3r}{2r} = \frac32$. (d) Each half-plane contributes half of a disc's area, $\frac12\pi r^2$, so $\Theta = \frac{3}{2}$. A union of half-lines from $0$ is stationary when the unit vectors along them sum to zero (the first variation of length at the junction is minus that sum), which holds at $120°$; a union of two lines through $0$ is stationary too, but it is not minimising: near the crossing, replacing the X by two disjoint arcs shortens it, and stationary-but-not-minimising singularities are exactly what Taylor's theorem excludes for soap films.

Solution

(a) The sum is invariant under the symmetry group of the tetrahedron, which fixes no nonzero vector, so it is $0$. Then $0 = |\sum v_i|^2 = 4 + 12c$, where $c$ is the common inner product of distinct vectors (all equal by symmetry), so $c = -\frac13$. (b) The tetrahedral cone is the cone from the centre over the six edges of the tetrahedron projected to the sphere; its singular curves are the four rays through the vertices, so they meet at the angle between $v_i$ and $v_j$. Along each ray three of the six sheets meet, at $120°$ by the three-fold symmetry about that ray: the two Plateau laws are consistent, and Taylor's theorem says they are the only possibilities.

Solution

Write $\Phi_s(x) = x + sX(x)$, a diffeomorphism for small $s$, and $u_s = u\circ\Phi_s$. Then $\partial_iu_s = (\partial_ku)\circ\Phi_s\,\partial_i\Phi_s^k$, with $\partial_i\Phi_s^k = \delta_{ik} + s\partial_iX^k$. Change variables $y = \Phi_s(x)$, $dx = \det(D\Phi_s)^{-1}dy = (1 - s\operatorname{div}X + O(s^2))dy$, and $(D\Phi_s)\circ\Phi_s^{-1} = I + sDX + O(s^2)$ evaluated at $y$ up to $O(s^2)$. So $\int|\nabla u_s|^2dx = \int\sum_i\big|\partial_iu + s\,\partial_iX^k\partial_ku\big|^2(1 - s\operatorname{div}X)\,dy + O(s^2)$. Differentiating at $s = 0$: $\int\big(2\langle\partial_iu, \partial_ku\rangle\partial_iX^k - |\nabla u|^2\operatorname{div}X\big)$. Setting this to zero for all $X$ is the stationarity identity (the overall sign is immaterial).

Solution

(a) $\partial_iu^j = \frac{\delta_{ij}}{|x|} - \frac{x_ix_j}{|x|^3}$, so $|\nabla u|^2 = \sum_{ij}\big(\frac{\delta_{ij}}{|x|} - \frac{x_ix_j}{|x|^3}\big)^2 = \frac{n}{|x|^2} - \frac{2}{|x|^2} + \frac{1}{|x|^2} = \frac{n - 1}{|x|^2}$. (b) In polar coordinates, $\int_{B_r}\frac{n - 1}{|x|^2}dx = (n - 1)|S^{n-1}|\int_0^r\rho^{n-3}d\rho = \frac{(n - 1)|S^{n-1}|}{n - 2}r^{n-2}$; for $n = 3$ this is $8\pi r$. So $r^{2-n}\int_{B_r}|\nabla u|^2$ is constant, the equality case of monotonicity. (c) The radial integral $\int_0^r\rho^{n-3}d\rho$ converges at $0$ iff $n \geq 3$; for $n = 2$ it diverges logarithmically, consistent with the fact that harmonic maps from surfaces have no point singularities.

Solution

(a) A direct computation (or the standard fact that stereographic projection is conformal with factor $\frac{2}{1 + |x|^2}$) gives orthogonal partial derivatives of equal length $\frac{2}{1 + |x|^2}$. (b) $|\nabla\sigma|^2 = \frac{8}{(1 + |x|^2)^2}$, which is twice the area density $\frac{4}{(1 + |x|^2)^2}$ of the pulled-back round metric; integrating gives $2\cdot4\pi$. (c) For any map $u : S^2 \to S^2$, pointwise $|\partial_1u|^2 + |\partial_2u|^2 \geq 2|\partial_1u\times\partial_2u|$, and integrating, $\int|\nabla u|^2 \geq 2\int|u^*(dA)| \geq 2\cdot4\pi|\deg u|$. Harmonic maps $S^2 \to S^2$ are holomorphic or antiholomorphic (Lemaire; Wood), for which equality holds, so a non-constant harmonic sphere has $|\deg u| \geq 1$ and energy $8\pi|\deg u| \geq 8\pi$. Each bubble costs at least $8\pi$: the quantum of energy concentration.