Harmonic Maps and Their Heat Flow

A harmonic function minimises the Dirichlet energy among functions with the same boundary values. A harmonic map does the same for maps between Riemannian manifolds: it is a critical point of

The class contains many familiar objects. If $M$ is an interval or a circle, harmonic maps are geodesics, parametrised proportionally to arc length. If $N = \mathbb{R}$, they are harmonic functions. If $M$ is a surface, conformal harmonic maps are minimal surfaces. Holomorphic maps between Kähler manifolds are harmonic. And the identity map of $M$, viewed as a map from $(M, g)$ to $(M, \tilde g)$, is harmonic exactly when a certain vector field built from the two connections vanishes: that vector field is the one in DeTurck's trick for the Ricci flow.

The natural way to find harmonic maps is to deform a given map down the energy gradient, the harmonic map heat flow $\partial_tu = \tau(u)$, introduced by James Eells and Joseph Sampson in 1964. It was the first geometric heat flow, two decades before Hamilton's Ricci flow, and Hamilton, whose 1975 monograph was on harmonic maps, modelled the Ricci flow on it: the 1982 paper uses the same strategy of maximum principles for curvature quantities combined with integral estimates. Eells and Sampson proved that if the target has non-positive curvature, the flow exists for all time and converges to a harmonic map. The proof rests on a Bochner formula for the energy density, in which the curvature of the target appears with a good sign; when the sign is wrong, the flow can develop singularities, and the energy concentrates in "bubbles".

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

• derive the harmonic map equation $\tau(u) = 0$ in local coordinates and for maps into a submanifold of Euclidean space;
• recognise geodesics, harmonic functions, minimal surfaces and the hedgehog of liquid crystals as harmonic maps;
• prove the energy identity for the heat flow, and the Bochner formula for $|du|^2$;
• prove the Eells–Sampson theorem: for non-positively curved targets, the flow exists for all time and subconverges to a harmonic map in the same homotopy class;
• explain why energy concentrates in two dimensions, and compute the tension field of the identity map, the DeTurck vector field.

7.1Liquid crystals

7.2The harmonic map equation

Let $(M^m, g)$ and $(N^n, h)$ be Riemannian manifolds, $M$ compact. In local coordinates $x^\alpha$ on $M$ and $y^i$ on $N$, a map $u$ has components $u^i(x)$, and its energy density is

The harmonic map equation is a semilinear elliptic system: the top-order part is the Laplacian, applied to each component, and the nonlinearity is quadratic in the first derivatives. That quadratic growth in $\nabla u$ is exactly critical in two dimensions, which is the source of both the beauty and the difficulty of the theory.

Examples. (1) If $M = S^1$ with arc length $t$, $\tau(u) = \ddot u^k + \Gamma^k_{ij}\dot u^i\dot u^j$, the geodesic equation: harmonic maps from a circle are closed geodesics, traversed at constant speed (Exercise 7.10). (2) If $N = \mathbb{R}$, $\tau(u) = \Delta_gu$. (3) If $M = N$ and $u$ is the identity, $\tau^k = g^{\alpha\beta}(\tilde\Gamma^k_{\alpha\beta} - \Gamma^k_{\alpha\beta})$ when the target carries a second metric $\tilde g$ (Exercise 7.16).

Embedded targets. By Nash's embedding theorem, $N$ can be isometrically embedded in some $\mathbb{R}^K$, and a map into $N$ is a map $u : M \to \mathbb{R}^K$ with $u(x) \in N$. Then $|du|^2$ is the ordinary $\sum_a|\nabla u^a|^2$, and the first variation in a tangential direction $v$ is $-\int\langle\Delta_gu, v\rangle$. So $u$ is harmonic exactly when

The normal part of $\Delta_gu$ is determined by the geometry of $N$: it equals $-A(u)(du, du)$, where $A$ is the second fundamental form of $N$ in $\mathbb{R}^K$ (9A.8 Submanifolds and Minimal Surfaces), so the harmonic map equation reads $\Delta_gu + A(u)(du, du) = 0$ in this convention. For the round sphere $S^n \subset \mathbb{R}^{n+1}$, differentiating $|u|^2 = 1$ twice gives $\langle\Delta u, u\rangle = -|du|^2$, so the normal part of $\Delta u$ is $-|du|^2u$, and

the equation already derived with a Lagrange multiplier in 6C.1 Regularity of Minimisers and Constraints.

7.3The heat flow and the energy identity

The harmonic map heat flow is

a semilinear parabolic system. With an embedding, it reads $\partial_tu - \Delta_gu = A(u)(du, du)$.

The proof is the contraction argument of 6A.8 Nonlinear Parabolic Equations, carried out in parabolic Hölder spaces for the embedded system (Eells and Sampson, 1964; Lin and Wang, chapter 5), together with the fact that the solution stays in $N$ (Exercise 7.14). The continuation criterion says that the only way the flow can stop is through gradient blow-up: once $|du|$ is bounded, parabolic regularity bounds every higher derivative.

The heat flow is the gradient flow of $E$, exactly as the heat equation is the gradient flow of the Dirichlet energy (6A.11 Calculus of Variations and Gradient Flows). The energy identity gives two things: a bound $E(u(t)) \leq E(u_0)$, and a time integral bound on $\|\tau(u)\|_{L^2}^2$, which forces $\tau(u(t_k)) \to 0$ in $L^2$ along some sequence $t_k \to \infty$ if the flow lives forever. What it does not give is pointwise control of $|du|$. That is the job of the Bochner formula.

7.4The Bochner formula for maps

We use the conventions of 9A.4 Curvature and What It Means: $R(X, Y) = \nabla_X\nabla_Y - \nabla_Y\nabla_X - \nabla_{[X, Y]}$, $\operatorname{Ric}(X) = \sum_\beta R(X, e_\beta)e_\beta$, so that $\langle R(X, Y)Y, X\rangle$ is the sectional curvature times $|X\wedge Y|^2$. For a map $u$, $du$ is a section of the bundle $T^*M\otimes u^*TN$, which has a connection built from the Levi-Civita connections of $M$ and $N$. Its covariant derivative $\nabla du$ is the second fundamental form of the map, and it is symmetric: $(\nabla_Xdu)(Y) = (\nabla_Ydu)(X)$ (this is the torsion-freeness of both connections). Its trace is the tension, $\tau(u) = \operatorname{tr}\nabla du$.

For $N = \mathbb{R}$ and $u$ a function, this is the Bochner formula of 9A.6 The Laplacian and the Bochner Formula: $\frac12\Delta|\nabla u|^2 = |\nabla^2u|^2 + \langle\nabla\Delta u, \nabla u\rangle + \operatorname{Ric}(\nabla u, \nabla u)$. The new term is the last one, and its sign is what matters:

Non-positive target curvature helps. Geometrically: in negative curvature geodesics spread apart, so a map cannot wrap itself up in complicated ways at low energy, and energy cannot pile up.

7.5The Eells–Sampson theorem

Philip Hartman (1967) proved more for non-positively curved targets: the whole flow converges (not just a subsequence), and harmonic maps in a homotopy class are unique up to "sliding" along flat directions, and unique outright if $N$ has negative curvature and the map is not degenerate (for example, not constant and not mapping onto a closed geodesic).

7.6When the target is positively curved: bubbling

Without the curvature assumption, the theorem fails. James Eells and John Wood proved in 1976 that there is no harmonic map of degree one from the torus $T^2$ to the sphere $S^2$. Run the heat flow from a degree-one map: the degree is preserved as long as the flow is smooth, the energy decreases, and yet there is no harmonic limit in the homotopy class. Something must go wrong.

In two dimensions, Michael Struwe described what goes wrong.

Jonathan Sacks and Karen Uhlenbeck (1981) had discovered the same phenomenon for minimising sequences, and it gave them their existence theorem for minimal two-spheres: either the minimiser exists, or a bubble, itself a harmonic sphere, is found. In the $T^2 \to S^2$ example, a degree-one bubble splits off and the map left behind has degree zero. Whether singularities can form in finite time was settled by Kung-Ching Chang, Wei-Yue Ding and Rugang Ye in 1992: for equivariant maps from the disc to $S^2$ with suitable boundary data, the flow blows up in finite time. In higher dimensions, the heat flow into spheres can blow up in finite time even from smooth data (Coron and Ghidaglia, 1989).

Two features of this story recur, transformed, in the Ricci flow. Singularities are analysed by rescaling around the point where the quantity concentrates, and the rescaled limits are special solutions (here harmonic spheres; for Ricci flow, solitons and $\kappa$-solutions). And an $\varepsilon$-regularity theorem says that if the energy in a small ball is below a threshold, there is no singularity there: this is the template for Perelman's pseudolocality theorem (12A.9 Pseudolocality).

7.7History

Bochner considered harmonic maps between Riemannian manifolds in 1940, and Fuller studied them in the 1950s, but the theory began in earnest with Eells and Sampson in 1964, whose paper introduced the heat flow and proved the theorem above; the Bochner formula for maps is theirs. Hartman's uniqueness theorem dates from 1967. Hamilton's monograph Harmonic Maps of Manifolds with Boundary appeared in 1975. Sacks and Uhlenbeck's bubbling paper is from 1981, Schoen and Uhlenbeck's regularity theory for minimisers from 1982, Struwe's heat flow theorem from 1985, and Chang, Ding and Ye's finite-time blow-up from 1992. Frédéric Hélein proved in 1991 that weakly harmonic maps from surfaces are smooth, using the conservation laws of 6C.3 Symmetry, Noether’s Theorem and Moving Planes.

7.8Exercises

Solution

(a) With $m = 1$ and $g = dt^2$, $\tau^k = \ddot\gamma^k + \Gamma^k_{ij}\dot\gamma^i\dot\gamma^j$, the coordinate expression of $\nabla_{\dot\gamma}\dot\gamma$. (b) Metric compatibility. (c) By Cauchy–Schwarz, $\ell = \int_0^L|\dot\gamma|\,dt \leq \sqrt L\big(\int_0^L|\dot\gamma|^2dt\big)^{1/2} = \sqrt{2LE}$, with equality iff $|\dot\gamma|$ is constant.

Solution

With $r = |x|$: $\Delta\frac{x_i}{r} = x_i\Delta\frac1r + 2\nabla x_i\cdot\nabla\frac1r = 0 - \frac{2x_i}{r^3}$. And $\partial_j\frac{x_i}{r} = \frac{\delta_{ij}}{r} - \frac{x_ix_j}{r^3}$, so $|\nabla u|^2 = \sum_{i,j}\big(\frac{\delta_{ij}}{r^2} - \frac{2\delta_{ij}x_ix_j}{r^4} + \frac{x_i^2x_j^2}{r^6}\big) = \frac{3}{r^2} - \frac{2}{r^2} + \frac{1}{r^2} = \frac{2}{r^2}$. Then $\Delta u + |\nabla u|^2u = -\frac{2x}{r^3} + \frac{2}{r^2}\frac xr = 0$. The energy is $\frac12\int_{B_1}\frac{2}{r^2}\,dx = \int_0^1\frac{1}{r^2}4\pi r^2\,dr = 4\pi$.

Solution

$\nabla u_\lambda(x) = \lambda(\nabla u)(\lambda x)$, so $\int|\nabla u_\lambda|^2dx = \lambda^2\int|\nabla u(\lambda x)|^2dx = \lambda^{2-m}\int|\nabla u(y)|^2dy$ with $y = \lambda x$. For $m = 2$ the energy is unchanged: as $\lambda \to \infty$, $u_\lambda$ has the same energy, all of it within distance $R/\lambda$ of the origin if most of the energy of $u$ is in $B_R$. A bounded-energy sequence can therefore converge weakly to a map with less energy, the difference concentrating at a point.

Solution

$u_x$ and $u_y$ are tangent to the sphere at $u$, and $|u\cdot(u_x\times u_y)| = |u_x\times u_y| \leq |u_x||u_y| \leq \frac12(|u_x|^2 + |u_y|^2)$. Equality requires $u_x \perp u_y$ and $|u_x| = |u_y|$, i.e. $u_x = \pm u\times u_y$ (rotation by $\pm90°$ in the tangent plane): the Cauchy–Riemann equations for a holomorphic or antiholomorphic map. The energy density in isothermal coordinates is $\frac12\lambda^{-2}(|u_x|^2 + |u_y|^2)$ and the area element is $\lambda^2dx\,dy$, so $E(u) = \frac12\int(|u_x|^2 + |u_y|^2)\,dx\,dy \geq \big|\int u\cdot(u_x\times u_y)\,dx\,dy\big| = 4\pi|\deg u|$, since $u\cdot(u_x\times u_y)\,dx\,dy$ is the pullback of the area form of $S^2$, of total area $4\pi$.

Solution

$\partial_t|u|^2 = 2u\cdot\Delta u + 2|du|^2|u|^2$, and $\Delta|u|^2 = 2u\cdot\Delta u + 2|du|^2$. Subtracting, $\partial_t\rho - \Delta\rho = 2|du|^2(|u|^2 - 1) = 2|du|^2\rho$. This is a linear equation for $\rho$ with bounded coefficient $2|du|^2$ on $[0, T']$ for $T' < T$ and zero initial data, so $\rho \equiv 0$ by the maximum principle (applied to $e^{-Ct}\rho$ and $-e^{-Ct}\rho$, as in 6A.5 Maximum Principles).

Solution

By the maximum principle for bounded subsolutions on $\mathbb{R}^m$ (6A.3 The Heat Equation on ℝⁿ), $f(x, t) \leq \int\Phi(x - y, 1)f(y, t - 1)\,dy$, and the heat kernel at time $1$ is bounded by $(4\pi)^{-m/2}$. In Eells–Sampson the integral $\int f(y, t - 1)\,dy$ is the energy $E(u(t - 1))$ (times $e^{-2K(t-1)}$), bounded by $E(u_0)$ thanks to the energy identity: the integral bound comes from the gradient flow structure, the pointwise bound from the maximum principle.

Solution

(a) For $f = x^k$: $\partial_{\alpha\beta}x^k = 0$ and $\partial_\gamma x^k = \delta^k_\gamma$, so $\Delta_gx^k = -g^{\alpha\beta}\Gamma^k_{\alpha\beta}$. The identity map has components $u^k = x^k$ and $\partial_\alpha u^i = \delta^i_\alpha$, so $\tau^k = -g^{\alpha\beta}\Gamma^k_{\alpha\beta} + g^{\alpha\beta}\tilde\Gamma^k_{\alpha\beta}$. (b) The difference of two connections is a tensor (9A.2 Connections), so $\tilde\Gamma - \Gamma$ is a $(1, 2)$-tensor and its trace with $g^{\alpha\beta}$ is a vector field. (c) The identity is harmonic exactly when $W = 0$, which is a gauge condition ("$g$ is harmonic with respect to $\tilde g$"), the analogue of harmonic coordinates. Fixing the diffeomorphism freedom by asking the identity to be harmonic removes the degeneracy that diffeomorphism invariance causes, just as harmonic gauge makes Einstein's equations hyperbolic (6B.14 Hyperbolic Equations and Systems). And the diffeomorphisms that relate the two flows move by the heat flow of harmonic maps.