JOVANA
Explore Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Harmonic Maps & the Dirichlet Energy

A harmonic map is the right way to ask 'what is the smoothest map between two curved spaces?' We take the Dirichlet energy — the cost of stretching — and find its critical points, meeting the tension field, the geodesic equation in disguise, and the deep theorem that turns a map into a heat flow and lets it relax into a harmonic one.

The energy of a map: a stretching cost between curved spaces

Guide 1 measured the energy of a curve — a map from an interval into a manifold — and found geodesics as its critical points. Guide 2 saw the same trick rescue the Plateau problem: minimize the Dirichlet energy of a map from a disk rather than area directly, because energy is convex where area is not. This guide promotes that trick from a tool to the subject itself. Take ANY smooth map f: M -> N between two Riemannian manifolds, and ask the most natural question in the world: how much does it stretch? The answer is a single number, its Dirichlet energy, and the maps that make it critical are the harmonic maps — the right notion of 'the smoothest possible map' between curved spaces.

Build the energy honestly, because every later formula reads off this one. At a point p in M the differential df sends the tangent space T_p M into T_{f(p)} N — it is a linear map between two inner-product spaces, so it has a well-defined size. The pointwise energy density e(f) is half the squared Hilbert-Schmidt norm of df, in coordinates (1/2) g^ij h_ab (df^a/dx^i)(df^b/dx^j), where g is the metric on M (raised to g^ij) and h the metric on N. Then E(f) = integral over M of e(f) dV_g, the total energy, using the Riemannian volume of M. Notice both metrics appear: g says how to add up over the domain, h says how to measure stretch in the target. A harmonic map is a balance struck between two curved geometries at once.

The first variation: the tension field

Now do what this rung always does — compute the first variation. Push f through a family f_s of maps with variation field V = d/ds at s = 0, a section of the pulled-back tangent bundle f*TN (at each p, V(p) lives in T_{f(p)} N, telling you where the image point is nudged). Differentiate E(f_s), integrate by parts using the Levi-Civita connection on N pulled back along f, and the boundary term drops if V vanishes on the boundary or M is closed. What survives is dE/ds at 0 = - integral over M of h(V, tau(f)) dV_g. The object paired against the variation is the tension field tau(f) — the harmonic-map analogue of acceleration for a curve, or of the mean curvature vector H for a surface.

energy density:   e(f) = (1/2) g^ij h_ab (df^a/dx^i)(df^b/dx^j)
total energy:     E(f) = integral over M of e(f) dV_g

first variation:  dE/ds |_0 = - integral over M of h(V, tau(f)) dV_g

tension field (coordinates, Gamma~ = Christoffel symbols of N):
   tau(f)^a = Delta_M f^a  +  g^ij Gamma~^a_bc (df^b/dx^i)(df^c/dx^j)

HARMONIC MAP   <=>   tau(f) = 0

special cases:
   N = R          tau(f) = Delta f          (harmonic function)
   M = interval   tau(f) = nabla_{c'} c'    (geodesic equation)
The energy, its first variation, and the tension field whose vanishing defines a harmonic map — with the two limiting cases that justify the name.

Read the tension field's formula and you see the whole subject in one line: tau(f)^a = Delta_M f^a + g^ij Gamma~^a_bc (df^b/dx^i)(df^c/dx^j). The first piece is the Laplacian of the domain acting on each component — the LINEAR part. The second piece carries the Christoffel symbols of the TARGET, quadratic in the first derivatives of f — the NONLINEAR part, and it is exactly the curvature of N forcing its way in. So a harmonic map is one whose tension field vanishes, tau(f) = 0, a quasilinear elliptic system. When the target is flat it collapses to a decoupled set of Laplace equations; the target's curvature is the entire source of difficulty, precisely paralleling how the ambient curvature complicated the geodesic and minimal-surface equations.

Three faces of the same idea

Harmonic maps are worth your attention because so many objects you already know are secretly harmonic maps in disguise. Lead with the examples, as this rung insists. First: when the target is R, you recover harmonic functions, Delta f = 0 — the prototype. Second: when the domain is a circle or interval, harmonic maps ARE geodesics, the very objects Guide 1 studied; a closed geodesic is a harmonic map from S^1. These two extremes pin down the concept from both sides before you ever meet a hard case.

The third face closes the circle Guide 2 opened, and it is the prettiest. Suppose the domain M is a SURFACE and the map f is conformal — it preserves angles, stretching equally in all directions at each point (think conformal parametrization). Then a remarkable identity holds: f is harmonic if and only if its image is a (possibly branched) minimal surface. This is exactly why Douglas and Rado could solve Plateau by minimizing energy instead of area — a conformal energy-minimizer is automatically area-minimizing and minimal. The 'brilliant swap' of Guide 2 was no accident: harmonic maps from surfaces and minimal surfaces are two views of one critical-point problem, and conformality is the dictionary between them.

Does a harmonic map exist? The heat flow that relaxes a map

Existence is the real prize, and it is where the deepest idea of the rung enters. You are handed a homotopy class of maps f: M -> N and asked: is there a harmonic map in it? The naive plan — take maps with energy dropping toward the infimum and pass to a limit — runs into the same wall as Plateau in Guide 2: the limit can concentrate energy at points and 'bubble off' a sphere, escaping the class. The masterstroke is dynamic. Instead of minimizing in one leap, let the map FLOW downhill in energy, following the steepest descent of E. That flow is the harmonic map heat flow, df/dt = tau(f), a nonlinear geometric heat equation: the tension field is minus the gradient of energy, so the map continuously relaxes, shedding energy as a hot body sheds heat.

  1. Start the flow from your given map f_0 and evolve it by df/dt = tau(f), the negative gradient flow of the Dirichlet energy; energy E(f_t) is monotonically non-increasing in time t, so the map can only get smoother.
  2. If the flow exists for all time and converges as t -> infinity, the limit has tau = 0 — it is a harmonic map, homotopic to f_0, exactly what you wanted; the flow does the minimizing for you.
  3. The crux is whether the flow can run forever without forming singularities; this is decided by the curvature of the TARGET N, and is the precise content of the existence theorem.

The landmark result is the Eells-Sampson theorem (1964): if M is compact and the target N has nonpositive sectional curvature, then the harmonic map heat flow from any smooth f_0 exists for all time and converges to a harmonic map homotopic to f_0. State the hypotheses, never the slogan: COMPACT domain, NONPOSITIVELY CURVED target. The curvature condition is not decoration — it is what a Bochner-Weitzenbock computation needs to control the energy density along the flow and forbid blow-up, and it is exactly the hypothesis Guides 4 and 5 will isolate. Drop it and the conclusion can fail: maps into spheres (positively curved) really do bubble, and Eells-Sampson does not apply. This is a genuine theorem with sharp hypotheses, not a universal guarantee.

What you have built, and where it flows next

Step back and the rung's grand parallel is now three rows tall. From the Dirichlet energy of a map you extracted a first variation; its vanishing is tau(f) = 0, the definition of a harmonic map — generalizing harmonic functions (target R), geodesics (domain an interval), and, conformally, minimal surfaces. Existence is won not by a static limit but by a dynamic heat flow that relaxes any map downhill in energy, and the Eells-Sampson theorem secures convergence under an honest curvature hypothesis. The same three moves — energy, critical equation, flow — that organized geodesics and minimal surfaces now organize maps.

Two threads run straight into the rest of the rung. The Laplace-Beltrami operator that sat at the heart of the tension field is the next two guides' explicit subject: Guide 4 builds it and the Hodge Laplacian and proves the Hodge decomposition, and Guide 5 develops the Bochner-Weitzenbock formula — the very identity that powered Eells-Sampson — into eigenvalue estimates and a first serious look at geometric flows. The harmonic map heat flow you just met is the gentle cousin of mean curvature flow and Ricci flow waiting at the end of this rung; you have already seen, in miniature, how a flow and a curvature hypothesis cooperate to force existence. Carry that pattern forward — it is the engine of modern geometric analysis.