03.02.26 · differential-geometry / manifolds

Harmonic maps: energy, tension field, and the harmonic-map equation

shipped3 tiersLean: none

Anchor (Master): Jost Riemannian Geometry and Geometric Analysis Ch. 8-9; Eells-Sampson Amer. J. Math. 86 (1964) §1-2; Eells-Lemaire Bull. LMS 10 (1978) and 20 (1988) reports

Intuition Beginner

Picture a thin rubber sheet stretched over a curved surface — a globe, say, or a saddle. The sheet wants to sit as slack as possible. Every place where it is pulled tight stores stretching energy, and the sheet relaxes until the total stored energy is as small as it can be. A harmonic map is the mathematical name for a configuration of the sheet that has settled into such a relaxed shape.

The "map" part means we are recording, for each point of one space, where it lands on the other. A flat map of a rubber band onto a wire is one example; draping a film over a frame is another. The stretching energy measures how much the map distorts distances. A harmonic map is the draping that minimizes this stretching, or at least balances it so no small wiggle lowers the total.

Why bother? Many natural objects are harmonic maps in disguise: the shortest path between two towns, the steady temperature in a metal plate, and a soap film spanning a wire loop. One idea, the relaxed sheet, ties all of these together.

Visual Beginner

A rubber sheet is shown draped from a flat square domain onto a curved target surface. Arrows along the sheet indicate how much each little patch is stretched: long arrows where the sheet is pulled tight, short arrows where it lies slack. A second copy of the same sheet, after relaxing, shows the arrows shrunk and evened out — the stretching has been spread as smoothly as possible, with no leftover pull at any interior point.

The picture shows the two things at once. The total length of all the arrows is the energy. The relaxed sheet is the one whose leftover pull — the imbalance of arrows at each point — has been driven to zero everywhere inside. That leftover pull is what the formal theory calls the tension, and a harmonic map is exactly a map with no tension.

Worked example Beginner

Take the domain to be the line segment from $0$ to $1$ , and the target to be the flat plane. A map sends each point $t$ of the segment to a point $f (t)$ in the plane — this is just a curve. The stretching energy adds up the square of the speed along the curve.

Try the straight-line curve from the point $(0, 0)$ to the point $(2, 0)$ , namely $f (t) = (2 t, 0)$ . Its speed is constant: each step of size $0.1$ in $t$ moves $0.2$ in the plane, so the speed is $2$ everywhere. The energy is half the speed squared summed along the segment, which works out to half of $4$ times $1$ , giving $2$ .

Now bend the curve so it bows out and comes back — same endpoints, but a longer, faster path. Its speed is larger somewhere, so its energy is more than $2$ . The straight line wins. The leftover pull along the straight line is zero: the curve is not accelerating, it moves at steady speed in a fixed direction. What this tells us is that for a target with no curvature, the harmonic maps from a segment are exactly the constant-speed straight lines — the shortest, most relaxed curves.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M, g)$ and $(N, h)$ be Riemannian manifolds, with $M$ compact and oriented (or the integrals taken over compact pieces), and let $f : M \to N$ be a smooth map. The differential $df$ is a section of the bundle $T^{*} M \otimes f^{*} T N$ , where $f^{*} T N$ is the pullback of the tangent bundle of $N$ along $f$ . Using $g$ on $T^{*} M$ and $h$ on $f^{*} T N$ , the bundle $T^{*} M \otimes f^{*} T N$ carries a fibre metric; the squared norm $∣ df ∣^{2}$ is the energy density $e (f)$ , equal in local coordinates to $g^{ij} h_{α β} (f) \partial_{i} f^{α} \partial_{j} f^{β}$ . The Dirichlet energy of $f$ follows Jost ^{[Jost Ch. 8 §8.1]}: $$ E(f) = \frac{1}{2} \int_M |df|^2 , d\mathrm{vol}g = \frac{1}{2} \int_M g^{ij} h{\alpha\beta}(f) , \partial_i f^\alpha , \partial_j f^\beta , d\mathrm{vol}_g . $$

The pullback bundle $f^{*} T N$ inherits a connection $f^{*} \nabla^{N}$ from the Levi-Civita connection $\nabla^{N}$ of $(N, h)$ via 03.02.27; combined with the Levi-Civita connection of $(M, g)$ on $T^{*} M$ , it gives a connection $\nabla$ on $T^{*} M \otimes f^{*} T N$ . The covariant derivative $\nabla df$ of the section $df$ is then a section of $T^{*} M \otimes T^{*} M \otimes f^{*} T N$ , symmetric in its two $T^{*} M$ slots; it is the second fundamental form of the map $f$ , written $\nabla df$ or $β (f)$ .

Definition (tension field). The tension field of $f$ is the trace of its second fundamental form with respect to $g$ , $$ \tau(f) = \mathrm{tr}g \nabla df = g^{ij} (\nabla df)(\partial_i, \partial_j) \in \Gamma(f^* TN) . $$ In local coordinates $x^{i}$ on $M$ and $y^{α}$ on $N$ , with $\Gamma^\alpha{\beta\gamma} $t h e C h r i s t o f f e l sy mb o l so f$ h $an d$ \Delta_g = g^{ij}(\partial_i\partial_j - ,{}^M\Gamma^k_{ij}\partial_k) $t h e L a pl a ce - B e l t r ami o p er a t or o n$ M$, $$ \tau(f)^\alpha = \Delta_g f^\alpha + g^{ij}, \Gamma^\alpha_{\beta\gamma}(f), \partial_i f^\beta , \partial_j f^\gamma . $$

Definition (harmonic map). The map $f$ is harmonic if $τ (f) = 0$ , equivalently if it satisfies the harmonic-map equation $$ \Delta_g f^\alpha + g^{ij}, \Gamma^\alpha_{\beta\gamma}(f), \partial_i f^\beta , \partial_j f^\gamma = 0, \qquad \alpha = 1, \dots, \dim N . $$

This is a semilinear elliptic system: the leading term $Δ_{g} f^{α}$ is linear, while the curvature of the target enters through the quadratic first-order term $Γ_{β γ}^{α} (f) \partial_{i} f^{β} \partial_{j} f^{γ}$ . The system is linear only when the target Christoffel symbols vanish, that is, when $N$ is flat in the relevant chart.

Counterexamples to common slips

The tension field is not the componentwise Laplacian $Δ_{g} f^{α}$ alone. Dropping the target-curvature term $g^{ij} Γ_{β γ}^{α} (f) \partial_{i} f^{β} \partial_{j} f^{γ}$ gives a quantity that depends on the chart on $N$ and is not a section of $f^{*} T N$ . The curvature term is exactly what makes $τ (f)$ a genuine geometric object.
A map can fail to be harmonic even with zero energy density at a point: harmonicity is the vanishing of $τ (f)$ , a second-order condition, not the vanishing of $∣ df ∣^{2}$ , a zeroth-order one. A constant map has both, but these are independent conditions in general.
"Energy-minimizing" and "harmonic" are not interchangeable. Every smooth minimizer is harmonic (it is a critical point), but a harmonic map can be a saddle or maximum of $E$ , not a minimum. Harmonicity is criticality, not minimality.

Key theorem with proof Intermediate+

Theorem (first variation of energy; the tension field is the negative gradient). *Let $f : (M, g) \to (N, h)$ be smooth with $M$ compact, and let $f_{t}$ be a smooth variation with $f_{0} = f$ and variation field $V = \frac{\partial f _{t}}{\partial t}_{t = 0} \in Γ (f^{*} T N)$ . Then* $$ \frac{d}{dt}\bigg|_{t=0} E(f_t) = - \int_M h\big(\tau(f),, V\big) , d\mathrm{vol}_g . $$ Consequently $f$ is a critical point of $E$ (with respect to all compactly supported variations) if and only if $τ (f) = 0$ .

Proof. Let $F : M \times (- ε, ε) \to N$ , $F (x, t) = f_{t} (x)$ , and write $\partial_{t} = \frac{\partial}{\partial t}$ . Extend the connection $\nabla$ to the pullback $F^{*} T N$ over $M \times (- ε, ε)$ . The energy density of $f_{t}$ is $e (f_{t}) = \frac{1}{2} g^{ij} h (d F (\partial_{i}), d F (\partial_{j}))$ , so

\frac{d}{d t} E (f_{t}) = \int_{M} g^{ij} h (\nabla_{\partial_{t}} d F (\partial_{i}), d F (\partial_{j})) d vol_{g},

using that $\nabla$ is compatible with $h$ to differentiate the fibre inner product. The connection $F^{*} \nabla^{N}$ is torsion-free along the commuting coordinate fields $\partial_{t}, \partial_{i}$ , so $\nabla_{\partial_{t}} d F (\partial_{i}) = \nabla_{\partial_{i}} d F (\partial_{t}) = \nabla_{\partial_{i}} V$ at $t = 0$ (the symmetry $\nabla d F (\partial_{t}, \partial_{i}) = \nabla d F (\partial_{i}, \partial_{t})$ ). Hence at $t = 0$

\frac{d}{d t}_{t = 0} E (f_{t}) = \int_{M} g^{ij} h (\nabla_{\partial_{i}} V, df (\partial_{j})) d vol_{g} .

The integrand is $⟨ \nabla V, df ⟩$ in the fibre metric on $T^{*} M \otimes f^{*} T N$ . Define the $1$ -form $ω$ on $M$ by $ω (X) = h (V, df (X))$ . A computation in the metric $g$ gives the divergence

div_{g} ω^{♯} = g^{ij} h (\nabla_{\partial_{i}} V, df (\partial_{j})) + g^{ij} h (V, \nabla_{\partial_{i}} df (\partial_{j})) = ⟨ \nabla V, df ⟩ + h (V, τ (f)),

since $g^{ij} \nabla_{\partial_{i}} df (\partial_{j}) = tr_{g} \nabla df = τ (f)$ (the metric-divergence terms from $T^{*} M$ assemble into $Δ_{g}$ and reproduce the trace). Integrating over the compact $M$ and applying the divergence theorem, $\int_{M} div_{g} ω^{♯} d vol_{g} = 0$ , so

\int_{M} ⟨ \nabla V, df ⟩ d vol_{g} = - \int_{M} h (V, τ (f)) d vol_{g} .

Substituting back gives the stated first-variation formula. If $τ (f) = 0$ the right side is zero for every $V$ , so $f$ is critical. Conversely, if $f$ is critical, taking $V = ϕ τ (f)$ for a bump function $ϕ \geq 0$ forces $\int_{M} ϕ ∣ τ (f) ∣^{2} = 0$ , whence $τ (f) \equiv 0$ . $□$

Bridge. This first-variation formula builds toward the existence theory of 03.02.29: once the tension field is identified as the negative $L^{2}$ -gradient of the energy, the natural way to produce a harmonic map is to run the gradient flow $\partial_{t} f = τ (f)$ , the harmonic-map heat flow, and ask it to converge. The foundational reason the formula takes this clean form is the torsion-freeness of the Levi-Civita connection of 03.02.27, which lets the $t$ - and $x$ -derivatives of $d F$ be interchanged; this is exactly the same symmetry $\nabla df (\partial_{i}, \partial_{j}) = \nabla df (\partial_{j}, \partial_{i})$ that makes the second fundamental form a symmetric tensor. The harmonic-map equation generalises three classical equations at once — the geodesic equation, the Laplace equation, and the minimal-surface system — by letting both the domain and the target be curved, and this same equation appears again in 03.02.29 as the equilibrium condition for the heat flow. Putting these together, the tension field is the single object that turns "critical point of a stretching energy" into a concrete elliptic system, and the bridge is that solving $τ (f) = 0$ is the same problem as finding the equilibria of the heat flow whose convergence 03.02.29 establishes under a curvature hypothesis on the target.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Let $ι : (M, g) ↪ (N, h)$ be an isometric immersion, so $g = ι^{*} h$ . Show that $τ (ι)$ equals the mean curvature vector of $M$ in $N$ , and conclude that $ι$ is harmonic if and only if $M$ is minimal.

Hint

For an isometric immersion the second fundamental form of the map $\nabla d ι$ takes values in the normal bundle and agrees with the classical second fundamental form $II$ ; its $g$ -trace is the mean curvature vector.

Answer

For an isometric immersion the tangential part of $\nabla d ι$ vanishes (the induced connection on $T M$ is the Levi-Civita connection of $g$ ), so $\nabla d ι$ is the normal-valued classical second fundamental form $II$ . Its trace is $τ (ι) = tr_{g} II = m H$ , where $H$ is the mean curvature vector and $m = dim M$ . Thus $τ (ι) = 0$ exactly when $H = 0$ , i.e. when $M$ is a minimal submanifold. Rubric: full credit for identifying $\nabla d ι$ with $II$ and the trace with the mean curvature vector.

Exercise 5 (medium, short-answer).

Show that a constant map $f \equiv p_{0}$ is harmonic, and that composing a harmonic map with an isometry of the target (on the left) or precomposing with an isometry of the domain (on the right) preserves harmonicity.

Hint

Constant maps have $df = 0$ . Isometries preserve the metrics, hence the Levi-Civita connections and the energy.

Answer

If $f \equiv p_{0}$ then $df = 0$ , so $\nabla df = 0$ and $τ (f) = 0$ : constant maps are harmonic. If $Φ : N \to N^{'}$ is an isometry, then $Φ$ pushes the Levi-Civita connection of $h$ to that of $h^{'}$ and $d (Φ \circ f) = d Φ \circ df$ , giving $τ (Φ \circ f) = d Φ (τ (f))$ ; hence $Φ \circ f$ is harmonic iff $f$ is. If $Ψ : M^{'} \to M$ is an isometry, the energy and tension transform by pullback and $τ (f \circ Ψ) = Ψ^{*} τ (f)$ , again preserving harmonicity. Rubric: full credit for the constant case and both isometry invariances.

Exercise 6 (hard, short-answer).

Let $(M, g)$ and $(N, h)$ be Kähler manifolds and $f : M \to N$ a holomorphic map. Show that $f$ is harmonic.

Hint

In complex coordinates the Kähler condition means the Christoffel symbols are pure (only $Γ_{β γ}^{α}$ with all holomorphic indices survive), and the energy density splits into a $\partial$ -part and a $\overset{ˉ}{\partial}$ -part. Holomorphy kills one part and forces a Cauchy-Riemann structure on the other.

Answer

On a Kähler manifold the only non-vanishing Christoffel symbols of $h$ are the purely holomorphic $Γ_{α β}^{γ}$ (and their conjugates), and the Laplace-Beltrami operator in complex coordinates is $Δ_{g} = 2 g^{i \overset{ˉ}{j}} \nabla_{i} \nabla_{\overset{ˉ}{j}}$ . For a holomorphic map, $\partial_{\overset{ˉ}{j}} f^{α} = 0$ for every holomorphic target index $α$ . The harmonic-map equation's $α$ -component is $g^{i \overset{ˉ}{j}} (\nabla_{i} \nabla_{\overset{ˉ}{j}} f^{α} + \nabla_{\overset{ˉ}{j}} \nabla_{i} f^{α}) / \dots$ , and every term carries at least one antiholomorphic derivative $\partial_{\overset{ˉ}{j}} f^{α}$ or pairs a holomorphic $\partial_{i} f^{β}$ with the boundary operator's conjugate that vanishes by holomorphy; the Kähler purity of $Γ$ removes any surviving cross term. Hence $τ (f)^{α} = 0$ . So $\pm$ holomorphic maps between Kähler manifolds are automatically harmonic, with no minimality assumption. Rubric: full credit for using Kähler purity of the connection and the vanishing of antiholomorphic derivatives to kill $τ (f)$ .

Advanced results Master

The energy and tension field organise into the second variation, which governs the stability of a harmonic map. For a harmonic $f$ and a variation field $V \in Γ (f^{*} T N)$ , the Hessian of $E$ at $f$ is $$ \mathrm{Hess},E_f(V,V) = \int_M \Big( |\nabla V|^2 - \sum_{i} h\big(R^N(V, df(e_i)), df(e_i),, V\big) \Big) d\mathrm{vol}_g , $$ where ${e_{i}}$ is a local orthonormal frame on $M$ and $R^{N}$ is the Riemann curvature of the target, the curvature tensor of 03.02.05. The sign of the curvature term is decisive: when $N$ has nonpositive sectional curvature the curvature contribution is nonnegative, so the Hessian is nonnegative and every harmonic map into a nonpositively curved target is stable. This is the variational shadow of the Eells-Sampson regime and reappears as the convexity that drives the heat-flow convergence of 03.02.29.

The Bochner technique converts harmonicity into a pointwise differential inequality for the energy density. For a harmonic map $f$ the energy density $e (f)$ satisfies the Bochner-Eells-Sampson formula $$ \Delta_g e(f) = |\nabla df|^2 + \sum_{i,j} h\big(df(\mathrm{Ric}^M e_i), df(e_i)\big) - \sum_{i,j} h\big(R^N(df,e_i, df,e_j),df,e_j, df,e_i\big) , $$ relating the Laplacian of the energy density to the full second fundamental form $∣\nabla df ∣^{2}$ , the Ricci curvature $Ric^{M}$ of the domain, and the sectional curvature of the target. When $Ric^{M} \geq 0$ and $N$ has nonpositive curvature, every term on the right except possibly the Ricci one is controlled, and the maximum principle on a compact domain forces $\nabla df \equiv 0$ : the harmonic map is totally geodesic. The proof of this formula and the existence theorem it powers are the content of the sequel; the present unit isolates the energy, the tension field, and the equation they produce.

Two further structural facts deserve recording. First, the harmonic-map equation is conformally invariant in domain dimension two: if $dim M = 2$ and $\tilde{g} = λ^{2} g$ is a conformal rescaling, then $τ$ scales by $λ^{- 2}$ and its zero set is unchanged, so harmonicity of $f : M^{2} \to N$ depends only on the conformal class of the domain metric. This is the foundational reason harmonic maps from surfaces are the natural objects in Teichmüller theory and string theory, where the domain carries a conformal rather than a metric structure. Second, the energy admits the Hopf-differential refinement: for $dim M = 2$ the $(2, 0)$ -part of the pullback $f^{*} h$ is a holomorphic quadratic differential precisely when $f$ is harmonic, an algebraic constraint that has no analogue in higher domain dimensions.

Synthesis. The tension field is the foundational reason a stretching energy becomes a geometric differential equation: $τ (f) = tr_{g} \nabla df$ is the negative $L^{2}$ -gradient of $E$ , and its vanishing is the harmonic-map system. This is exactly the unification the subject is built on — geodesics, harmonic functions, and minimal immersions are the three boundary cases obtained by flattening the domain to a line, flattening the target, or restricting to isometric immersions, and the second fundamental form $\nabla df$ is the single tensor whose trace, restriction, and normal part recover all three. Putting these together, the second variation and the Bochner formula show that the curvature of the target, drawn from 03.02.05, is the quantity that decides stability and rigidity: nonpositive target curvature makes the Hessian nonnegative and drives the energy density to a subharmonic function, which is the central insight behind the Eells-Sampson existence theorem of 03.02.29. The harmonic-map equation generalises the geodesic equation of 13.02.02 by allowing a curved, higher-dimensional domain, and it is dual to the heat flow $\partial_{t} f = τ (f)$ whose equilibria it describes — the bridge from the static variational problem to the parabolic existence proof. The conformal invariance in dimension two is what ties the whole apparatus to Riemann surfaces and to the worldsheet of string theory, where the same energy appears as the Polyakov action.

Full proof set Master

The first-variation formula and its corollary (criticality $\Leftrightarrow$ $τ (f) = 0$ ) are proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (the harmonic-map equation in local coordinates). In local coordinates $x^{i}$ on $M$ and $y^{α}$ on $N$ , the tension field of $f$ has components $$ \tau(f)^\alpha = \Delta_g f^\alpha + g^{ij}, \Gamma^\alpha_{\beta\gamma}(f), \partial_i f^\beta, \partial_j f^\gamma , $$ where $Δ_{g} = \frac{1}{d e t g} \partial_{i} (det g g^{ij} \partial_{j})$ and $Γ_{β γ}^{α}$ are the Christoffel symbols of $h$ .

Proof. By definition $τ (f) = g^{ij} (\nabla df) (\partial_{i}, \partial_{j})$ , where $\nabla$ is the tensor-product connection on $T^{*} M \otimes f^{*} T N$ . Write $df = \partial_{i} f^{α} d x^{i} \otimes (\partial_{α} \circ f)$ . Applying $\nabla_{\partial_{i}}$ and using the Leibniz rule across the tensor product, $$ (\nabla df)(\partial_i,\partial_j) = \partial_i\partial_j f^\alpha,\partial_\alpha - {}^M\Gamma^k_{ij},\partial_k f^\alpha,\partial_\alpha + \partial_i f^\beta,\partial_j f^\gamma, \Gamma^\alpha_{\beta\gamma}(f),\partial_\alpha , $$ where $^{M} Γ_{ij}^{k}$ are the domain Christoffel symbols (from $\nabla$ on $T^{*} M$ ) and $Γ_{β γ}^{α} (f)$ the pulled-back target Christoffel symbols (from $f^{*} \nabla^{N}$ acting on $\partial_{α} \circ f$ , since $\nabla_{\partial_{i}} (\partial_{α} \circ f) = \partial_{i} f^{γ} Γ_{γ α}^{β} \partial_{β}$ ). Contracting with $g^{ij}$ , the first two terms combine into $g^{ij} (\partial_{i} \partial_{j} f^{α} -^{M} Γ_{ij}^{k} \partial_{k} f^{α}) = Δ_{g} f^{α}$ , and the third yields $g^{ij} Γ_{β γ}^{α} (f) \partial_{i} f^{β} \partial_{j} f^{γ}$ . $□$

Proposition (geodesics, harmonic functions, minimal immersions as special cases). (i) If $dim M = 1$ with $M = R$ or a circle, harmonic maps are geodesics of $N$ . (ii) If $N = R$ with the flat metric, harmonic maps are harmonic functions $Δ_{g} f = 0$ . (iii) If $f$ is an isometric immersion, $f$ is harmonic if and only if $f (M)$ is a minimal submanifold of $N$ .

Proof. (i) For $dim M = 1$ the single component of $g^{ij}$ contracts the equation to $\ddot{f}^{α} + Γ_{β γ}^{α} (f) \dot{f}^{β} \dot{f}^{γ} = 0$ , the geodesic equation. (ii) For $N = R$ all $Γ_{β γ}^{α}$ vanish, leaving $τ (f) = Δ_{g} f$ ; harmonicity is $Δ_{g} f = 0$ . (iii) For an isometric immersion the induced connection on $T M$ is the Levi-Civita connection of $g = ι^{*} h$ , so the tangential component of $\nabla d ι$ vanishes and $\nabla d ι = II$ , the normal-valued second fundamental form. Then $τ (ι) = tr_{g} II = m H$ with $H$ the mean curvature vector and $m = dim M$ ; $τ (ι) = 0$ iff $H = 0$ , the definition of minimality. $□$

Proposition (holomorphic maps between Kähler manifolds are harmonic). Let $(M, g, J_{M})$ and $(N, h, J_{N})$ be Kähler and $f : M \to N$ holomorphic. Then $f$ is harmonic.

Proof. Choose holomorphic coordinates $z^{i}$ on $M$ and $w^{α}$ on $N$ . The Kähler condition makes the only non-vanishing Christoffel symbols of $h$ the purely holomorphic $Γ_{α β}^{γ}$ and conjugates, and gives $Δ_{g} = 2 g^{i \overset{ˉ}{j}} \nabla_{\overset{ˉ}{j}} \nabla_{i}$ acting on functions. Holomorphy of $f$ means $\partial_{\overset{ˉ}{j}} f^{α} = 0$ for holomorphic target indices $α$ . Decompose $τ (f)^{α} = 2 g^{i \overset{ˉ}{j}} (\partial_{i} \partial_{\overset{ˉ}{j}} f^{α} + Γ_{β γ}^{α} (f) \partial_{i} f^{β} \partial_{\overset{ˉ}{j}} f^{γ})$ plus the conjugate-index contributions. Each surviving term contains an antiholomorphic derivative $\partial_{\overset{ˉ}{j}} f^{γ}$ or $\partial_{\overset{ˉ}{j}} f^{α}$ , which vanishes; the mixed term $\partial_{i} \partial_{\overset{ˉ}{j}} f^{α} = \partial_{\overset{ˉ}{j}} (\partial_{i} f^{α})$ vanishes because $\partial_{i} f^{α}$ is itself holomorphic. The Kähler purity of $Γ$ excludes any mixed Christoffel term that would couple a holomorphic and an antiholomorphic derivative. Hence $τ (f)^{α} = 0$ for all $α$ . $□$

Connections Master

Sectional, Ricci, and scalar curvature 03.02.05 supply the target-curvature data that enters every refined statement about harmonic maps. The Riemann tensor $R^{N}$ appears in the second variation of energy with a sign that decides stability, and the Ricci curvature of the domain enters the Bochner-Eells-Sampson identity; nonpositive target curvature is the hypothesis that makes the energy density subharmonic, which is the analytic engine of the existence theory. This unit consumes that curvature language to write down the Hessian and the Bochner formula but defers their use to the sequel.

The Levi-Civita connection, exponential map, and gradient-like fields 03.02.27 are the differential-geometric apparatus on which the tension field is built. The pullback connection $f^{*} \nabla^{N}$ of 03.02.27's Levi-Civita connection is what makes $\nabla df$ a well-defined tensor, and its torsion-freeness is exactly what symmetrises the second fundamental form and lets the first-variation proof interchange the domain and variation derivatives. The exponential map of that unit also underlies the normal-coordinate computations behind the Bochner formula.

Variational calculus 03.04.08 provides the Euler-Lagrange machinery this unit specialises. The Dirichlet energy is the geometric instance of an integral functional, and the harmonic-map equation is its Euler-Lagrange system; the first-variation theorem here is the manifold-valued generalisation of the scalar variational principle, with the tension field playing the role of the Euler-Lagrange operator and the divergence theorem supplying the integration by parts.

Geodesics and parallel transport 13.02.02 are recovered as the one-dimensional-domain case of the harmonic-map equation: a harmonic map from $R$ is precisely a geodesic of the target, and the curvature term $Γ_{β γ}^{α} (f) \partial_{i} f^{β} \partial_{j} f^{γ}$ of the harmonic-map equation reduces to the geodesic acceleration term $Γ_{β γ}^{α} \dot{f}^{β} \dot{f}^{γ}$ . The harmonic-map equation is the higher-dimensional-domain generalisation of the geodesic equation studied there.

The harmonic-map heat flow and the Eells-Sampson existence theorem 03.02.29 is the immediate downstream consumer. The first-variation formula proved here identifies $τ (f)$ as the negative $L^{2}$ -gradient of $E$ , which makes $\partial_{t} f = τ (f)$ the gradient flow; the sequel proves that for nonpositively curved targets this flow exists for all time and converges to a harmonic map, using the Bochner formula recorded here. Everything in this unit is the static, equation-level input to that parabolic existence proof.

Historical & philosophical context Master

The harmonic-map equation in its modern form was written down by James Eells and Joseph Sampson in their 1964 paper Harmonic mappings of Riemannian manifolds (Amer. J. Math. 86, 109–160) ^{[Eells-Sampson 1964]}, which introduced the energy functional $E (f)$ , identified its critical points through the tension field $τ (f) = tr_{g} \nabla df$ , and proved the first general existence theorem by the heat-flow method for nonpositively curved targets. The special cases had long been known in isolation: geodesics go back to the calculus of variations of Euler and Lagrange, harmonic functions to potential theory, and minimal surfaces to Lagrange and Plateau; Eells and Sampson's contribution was to see all of them as instances of one variational problem for maps between curved spaces. The systematic state of the field is recorded in the two reports of Eells and Luc Lemaire (A report on harmonic maps, Bull. London Math. Soc. 10, 1–68, 1978; and the 1988 sequel) ^{[Eells-Lemaire 1978]}, which remain the standard surveys, and in Jost's textbook treatment ^{[Jost Ch. 8]}.

The conceptual shift is the relocation of the unknown from a function valued in a vector space to a map valued in a manifold. A harmonic function takes values in $R$ , where addition and the Laplacian make immediate sense; a harmonic map takes values in a curved $N$ , where there is no addition and the "second derivative" must be the covariant one, with the target Christoffel symbols entering as a quadratic first-order term. This is why the harmonic-map system is semilinear rather than linear: the curvature of the target is not a perturbation but a structural feature of the equation. The same relocation later let Charles Morrey's and Schoen-Yau's regularity theory, Karen Uhlenbeck's bubbling analysis, and Andreas Floer's infinite-dimensional Morse theory treat geometric variational problems whose unknowns live on manifolds, with the tension field as the prototype of a geometric Euler-Lagrange operator.

Bibliography Master

@article{eellssampson1964,
  author  = {Eells, James and Sampson, J. H.},
  title   = {Harmonic mappings of {R}iemannian manifolds},
  journal = {American Journal of Mathematics},
  volume  = {86},
  number  = {1},
  pages   = {109--160},
  year    = {1964}
}

@article{eellslemaire1978,
  author  = {Eells, James and Lemaire, Luc},
  title   = {A report on harmonic maps},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {10},
  number  = {1},
  pages   = {1--68},
  year    = {1978}
}

@article{eellslemaire1988,
  author  = {Eells, James and Lemaire, Luc},
  title   = {Another report on harmonic maps},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {20},
  number  = {5},
  pages   = {385--524},
  year    = {1988}
}

@book{jost2017,
  author    = {Jost, J\"urgen},
  title     = {Riemannian Geometry and Geometric Analysis},
  edition   = {7th},
  publisher = {Springer},
  year      = {2017}
}

@book{xin1996,
  author    = {Xin, Yuanlong},
  title     = {Geometry of Harmonic Maps},
  series    = {Progress in Nonlinear Differential Equations and Their Applications},
  volume    = {23},
  publisher = {Birkh\"auser, Boston},
  year      = {1996}
}

Prerequisites

03.02.05
03.02.27
03.04.08

Tier anchors

beginner: Jost Riemannian Geometry and Geometric Analysis Ch. 8 (informal energy/soap-film picture); Eells-Lemaire Bull. LMS 10 (1978) §1 survey introduction
intermediate: Jost Riemannian Geometry and Geometric Analysis Ch. 8 §8.1-8.2 (energy, first variation, tension field); Xin Geometry of Harmonic Maps Ch. 1
master: Jost Riemannian Geometry and Geometric Analysis Ch. 8-9; Eells-Sampson Amer. J. Math. 86 (1964) §1-2; Eells-Lemaire Bull. LMS 10 (1978) and 20 (1988) reports

References

Jost — Riemannian Geometry and Geometric Analysis (7th ed., Springer 2017) · Ch. 8 §8.1-8.3 (energy, tension field, harmonic-map equation); Ch. 9 (existence, Eells-Sampson)
Eells, Sampson — Harmonic mappings of Riemannian manifolds (1964) · Amer. J. Math. 86, pp. 109--160; §1-2 (energy density, tension field, harmonic-map system), §3 (Bochner formula)
Eells, Lemaire — A report on harmonic maps (1978) · Bull. London Math. Soc. 10, pp. 1--68; §1-3 (definitions, examples, the special cases)
Xin — Geometry of Harmonic Maps (Birkhauser, 1996) · Ch. 1 §1.1-1.3 (second fundamental form of a map, tension field, holomorphic maps are harmonic)

Estimated time

beginner: 15m
intermediate: 35m
master: 70m