The harmonic-map heat flow and the Eells–Sampson theorem

Intuition Beginner

Imagine a stretchy rubber sheet that you have draped over a curved surface — a sphere, say, or a saddle-shaped hill. The way you first lay it down is probably bunched up and creased, storing a lot of stretching energy in the rubber. If you let go, the sheet relaxes: it slides and smooths itself until it settles into the tautest, lowest-energy shape it can reach without tearing or jumping off the surface.

The harmonic-map heat flow is the mathematical version of that relaxation. A map from one curved space to another carries an energy that measures how much it stretches things. The flow nudges the map, moment by moment, in whatever direction lowers that stretching energy fastest. You feed in any starting map and let the clock run.

The destination of this relaxation has a name: a harmonic map. It is the map the sheet settles into, the one that cannot lower its energy by any small wiggle.

The deep question is whether the relaxation always succeeds, or whether the sheet can bunch up worse and worse forever. Eells and Sampson found a clean answer when the target surface curves the right way.

Visual Beginner

A bunched, wrinkled sheet is shown draped over a gently saddle-shaped target surface. A row of three frames runs left to right: the first frame shows the sheet creased and folded; the second shows it part-way relaxed, the big folds gone but small ripples remaining; the third shows it pulled taut and smooth, hugging the surface evenly. Small downhill arrows along the bottom track the steadily dropping energy value beneath each frame.

The picture makes two points at once. First, the flow only ever moves downhill in energy, so the sheet never spontaneously crumples. Second, the final taut state in the third frame is not torn loose from the starting state: it is reached by a continuous relaxation, so it belongs to the same family of drapings you began with. That family is what a topologist calls a homotopy class.

Worked example Beginner

Take a rubber band wrapped once around a smooth circular post, and let the band relax by sliding freely along the post. Mark four beads on the band at uneven spacing, bunched on one side.

The stretching energy is largest where the beads crowd together, because the band is locally over-stretched to span the rest of the post. The relaxation slides each bead toward the average of its two neighbours, the way heat smooths a temperature spike along a wire. Start with bead positions, as fractions of the way around the post, at $0$, $0.1$, $0.2$, and $0.5$. After one smoothing step that replaces each bead by the midpoint of its neighbours, the crowded beads at $0.1$ and $0.2$ spread out toward $0.1$ and $0.3$, while the lonely gap near $0.5$ pulls its neighbours apart.

Repeat the smoothing again and again. The beads drift toward even spacing at $0$, $0.25$, $0.5$, $0.75$ around the post. That evenly spaced wrap is the lowest-energy way to go around once: the harmonic map. What this tells us is that the flow does not change how many times the band wraps the post — it stays wrapped once — it only redistributes the stretch until it is uniform.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds with $M$ compact, and let $f:M\to N$ be a smooth map. The differential $df$ is a section of $T^{\ast }M\otimes f^{\ast }TN$, where $f^{\ast }TN$ is the pullback bundle carrying the pulled-back Levi-Civita connection of $N$. The energy density of $f$ is $$ e(f) = \tfrac12 |df|^2 = \tfrac12, g^{ij}, h_{\alpha\beta}, \partial_i f^\alpha, \partial_j f^\beta , $$ and the Dirichlet energy is $E(f)={\int }_{M}e(f)d{V}_g$, the functional whose theory is developed in 03.02.26.

The second fundamental form of $f$ is $\nabla df\in \Gamma (T^{\ast }M\otimes T^{\ast }M\otimes f^{\ast }TN)$, the total covariant derivative of $df$ with respect to the connection on $T^{\ast }M\otimes f^{\ast }TN$ induced by the Levi-Civita connections of $g$ and $h$. Its metric trace is the tension field $$ \tau(f) = \operatorname{tr}g \nabla df = g^{ij}!\left( \partial_i \partial_j f^\gamma - {}^M\Gamma{ij}^k, \partial_k f^\gamma + {}^N\Gamma_{\alpha\beta}^\gamma, \partial_i f^\alpha, \partial_j f^\beta \right) \partial_\gamma , $$ a section of $f^{\ast }TN$. The first variation of $E$ reads $\frac{d}{dt}{|}_{0}E(f_t)=-{\int }_M⟨\tau (f),{\partial }_t{f}_t⟩d{V}_g$, so $\tau (f)$ is the negative $L^2$-gradient of the energy. A map with $\tau (f)\equiv 0$ is a harmonic map; these are the critical points of $E$ studied in 03.02.26.

The harmonic-map heat flow is the initial-value problem $$ \partial_t f = \tau(f), \qquad f(,\cdot,, 0) = f_0 , $$ a section-valued semilinear parabolic system for a one-parameter family $f(\cdot ,t):M\to N$. In a single coordinate patch it is the inhomogeneous heat equation ${\partial }_t{f}^{\gamma }={\Delta }_g{f}^{\gamma }+g^{ij}{}^N{\Gamma }_{\alpha \beta }^{\gamma }{\partial }_i{f}^{\alpha }{\partial }_j{f}^{\beta }$, with the curved target entering only through the lower-order Christoffel quadratic. Because it is the gradient flow of $E$, energy is monotone: $$ \frac{d}{dt} E(f(\cdot, t)) = \int_M \langle \tau(f), \partial_t f\rangle, dV_g \cdot (-1) = -\int_M |\tau(f)|^2, dV_g \le 0 . $$

Counterexamples to common slips

• The tension field is not the coordinate Laplacian of the component functions. The term ${}^N{\Gamma }_{\alpha \beta }^{\gamma }{\partial }_i{f}^{\alpha }{\partial }_j{f}^{\beta }$ is essential; dropping it makes the equation depend on the chart and breaks the geometric meaning of $\tau (f)$ as a section of $f^{\ast }TN$.
• Non-positive sectional curvature of $N$ is the hypothesis, not non-positive Ricci or scalar curvature of $N$. The curvature of $N$ enters the energy-density evolution through the full Riemann tensor $R_N(df,df)df$, and only the sectional-curvature sign controls that term.
• The Ricci curvature in the Bochner formula is that of the domain $M$, while the sectional-curvature hypothesis is on the target $N$. Conflating the two reverses which manifold must be compact and which must be non-positively curved.

Key theorem with proof Intermediate+

The dynamics of the flow are governed by a parabolic Bochner identity for the energy density, in the spirit of the Bochner technique of 03.02.15 but with a time derivative adjoined.

Theorem (Bochner–Eells–Sampson formula). Along the harmonic-map heat flow, the energy density $e(f)=\frac{1}{2}|df{|}^2$ satisfies $$ (\partial_t - \Delta_g), e(f) = -|\nabla df|^2 - \langle \mathrm{Ric}_M, df, df\rangle + \langle R_N(df, df)df, df\rangle , $$ where ${\mathrm{Ric}}_M$ is the Ricci curvature of the domain and $R_N$ is the Riemann curvature of the target. If $N$ has non-positive sectional curvature ($K_N\le 0$), the final term is non-positive.

Proof. Work at a point and choose normal coordinates on $M$ so that ${}^M\Gamma$ vanishes there. Write $\nabla$ for the connection on $f^{\ast }TN$ and ${\nabla }_i$ for covariant differentiation along ${\partial }_i$. Then $e(f)=\frac{1}{2}{g}^{ij}h({\nabla }_{i}f,{\nabla }_{j}f)$ with ${\nabla }_{i}f:=df({\partial }_i)$. Differentiating the metric trace,

$${\Delta }_{g}e(f)=g^{ij}h({\nabla }_k{\nabla }_{i}f,{\nabla }_k{\nabla }_{j}f)\cdot g^{kk}+g^{ij}h({\nabla }_k{\nabla }_k{\nabla }_{i}f,{\nabla }_{j}f),$$

where the first group assembles to $|\nabla df{|}^2$. To convert ${\nabla }_k{\nabla }_k{\nabla }_{i}f$ into a time derivative, commute covariant derivatives. The Ricci identity on $f^{\ast }TN$ gives

$${\nabla }_k{\nabla }_i{\nabla }_{k}f={\nabla }_i{\nabla }_k{\nabla }_{k}f+{}^M{R}_{ki}{}^l{}_k{\nabla }_{l}f+R_N({\nabla }_{k}f,{\nabla }_{i}f){\nabla }_{k}f,$$

the two curvature corrections recording, respectively, that ${\partial }_k$ does not commute past the domain connection and that the pullback connection on $f^{\ast }TN$ has curvature $f^{\ast }{R}_N$. Tracing over $k$ turns ${}^M{R}_{ki}{}^l{}_k$ into the Ricci curvature ${\mathrm{Ric}}_M{}_i{}^l$ and turns ${\nabla }_k{\nabla }_{k}f$ into the tension field $\tau (f)={\mathrm{tr}}_g\nabla df$. Therefore

$${\Delta }_{g}e(f)=|\nabla df{|}^2+h({\nabla }_i\tau (f),{\nabla }_{i}f)+⟨{\mathrm{Ric}}_{M}df,df⟩-⟨R_N(df,df)df,df⟩.$$

Now use the flow. Since ${\partial }_{t}f=\tau (f)$ and the time and space covariant derivatives commute up to no curvature (time is a parameter, not a manifold direction),

$${\partial }_{t}e(f)=h({\nabla }_t{\nabla }_{i}f,{\nabla }_{i}f)=h({\nabla }_i{\partial }_{t}f,{\nabla }_{i}f)=h({\nabla }_i\tau (f),{\nabla }_{i}f).$$

Substituting $h({\nabla }_i\tau (f),{\nabla }_{i}f)={\partial }_{t}e(f)$ into the displayed expression for ${\Delta }_{g}e(f)$ and rearranging,

$$({\partial }_t-{\Delta }_g)e(f)=-|\nabla df{|}^2-⟨{\mathrm{Ric}}_{M}df,df⟩+⟨R_N(df,df)df,df⟩.$$

For the sign of the last term, $⟨R_N(df,df)df,df⟩={\sum }_{i,j}h(R_N({\nabla }_{i}f,{\nabla }_{j}f){\nabla }_{j}f,{\nabla }_{i}f)$, a sum of sectional-curvature numerators of the planes spanned by the pairs ${\nabla }_{i}f,{\nabla }_{j}f$ in $TN$; when $K_N\le 0$ each such numerator is $\le 0$, so the whole term is $\le 0$. $\square$

Bridge. This identity builds toward the global-existence theorem and is the foundational reason the non-positive-curvature hypothesis matters: when $K_N\le 0$ the only term in $({\partial }_t-{\Delta }_g)e(f)$ that could be positive is gone, leaving $({\partial }_t-{\Delta }_g)e(f)\le -⟨{\mathrm{Ric}}_{M}df,df⟩$, a differential inequality the maximum principle can close. This is exactly the parabolic upgrade of the elliptic Bochner technique of 03.02.15, where the same Ricci-and-curvature bookkeeping forced harmonic forms to vanish; here it forces the energy density to stay bounded, and the curvature of the pullback bundle $f^{\ast }TN$ is what generalises the single curvature term of the classical Weitzenböck identity to a target-curvature contribution. The central insight is that the flow trades an algebraic obstruction — does a harmonic map exist in this class? — for an analytic one the heat equation can resolve, and putting these together with the energy monotonicity of the formal-definition section gives both the a-priori bound and the convergence. The bridge is that ${\nabla }_i\tau (f)={\nabla }_t{\nabla }_{i}f$ identifies the spatial divergence of the gradient with a time derivative, which is what couples the static Bochner geometry to the dynamic flow and reappears again in 03.02.26 as the first-variation formula read backward.

Exercises Intermediate+

Advanced results Master

The existence theorem is the payoff of the parabolic Bochner identity.

Theorem (Eells–Sampson 1964). Let $M$ be a compact Riemannian manifold and $N$ a complete Riemannian manifold of non-positive sectional curvature. For any smooth $f_0:M\to N$ the harmonic-map heat flow with initial data $f_0$ exists for all $t\in [0,\infty )$, the energy density stays uniformly bounded, and there is a sequence $t_k\to \infty$ along which $f(\cdot ,t_k)$ converges in $C^{\infty }$ to a harmonic map $f_{\infty }$ homotopic to $f_0$. Consequently every homotopy class of maps $M\to N$ contains a harmonic representative, which minimises energy in its class.

The architecture is three moves welded together. Short-time existence is pure parabolic theory and needs no curvature: the flow is a strictly parabolic semilinear system, solvable on a maximal interval $[0,T)$. Global existence is where $K_N\le 0$ enters: the parabolic maximum principle applied to $({\partial }_t-{\Delta }_g)e(f)\le -⟨{\mathrm{Ric}}_{M}df,df⟩$ bounds ${\sup }_{M}e(f)$ for all finite time by a Gronwall estimate against the fixed lower bound of ${\mathrm{Ric}}_M$ on the compact $M$. A finite-time singularity of a quasilinear parabolic system is forced by blow-up of the top-order norm, which the energy-density bound — bootstrapped through parabolic Schauder estimates to bounds on all derivatives — precludes, so $T=\infty$. Subconvergence then follows from the integrated energy identity ${\int }_0^{\infty }{\int }_M|\tau (f){|}^2=E(f_0)-E_{\infty }<\infty$, yielding times $t_k$ with $\Vert \tau (f(\cdot ,t_k)){\Vert }_{L^2}\to 0$ and, after Arzelà–Ascoli on the uniformly bounded derivatives, a smooth harmonic limit.

The curvature sign is not a technical convenience; it is the mechanism. On a target with positive curvature the term $⟨R_N(df,df)df,df⟩$ can be positive and large, the energy density can concentrate, and the flow can develop finite-time singularities — bubbling of harmonic spheres — exactly as in the two-dimensional domain case analysed later by Struwe and by Chang–Ding–Ye, where finite-time blow-up genuinely occurs. The non-positive-curvature hypothesis is the convexity that suppresses concentration: in the universal cover of $N$ the distance function is convex, geodesics between points are unique, and the energy is geodesically convex along homotopies, which is the global shadow of the pointwise sign in the Bochner formula.

The Hartman uniqueness addendum sharpens the conclusion. When $N$ has strictly negative curvature, the harmonic representative in a homotopy class is unique unless its image is contained in a closed geodesic, in which case the representatives form a one-parameter family translated along that geodesic; when $K_N\le 0$, any two homotopic harmonic maps are joined by a parallel family along which the energy is constant. The proof is the convexity of energy along the geodesic homotopy, the second-variation computation of Exercise 7. Hartman's theorem is the elliptic companion to Eells–Sampson's parabolic existence: existence comes from the flow, uniqueness from the convexity, and both rest on $K_N\le 0$.

The Schoen–Yau application turns this machinery into rigidity. Because a harmonic map into a non-positively curved target is unique and energy-minimising, its existence constrains the topology of the domain: Schoen and Yau used harmonic maps from surfaces to detect incompressible surfaces in three-manifolds and to prove rigidity of compact manifolds of non-positive curvature, and the same circle of ideas underlies the Eells–Sampson-to-Siu rigidity for Kähler targets and Mostow-type rigidity reproved through harmonic maps by Corlette and others. The flow manufactures the canonical map; the curvature hypothesis makes it unique; uniqueness against a symmetry of the situation forces the symmetry onto the map and thence onto the manifold.

Synthesis. The harmonic-map heat flow is the foundational reason existence of harmonic maps reduces to an a-priori energy-density estimate, and the Bochner–Eells–Sampson formula is the bridge that supplies that estimate. Putting these together, the three pillars — short-time parabolic existence, global existence by the maximum principle under $K_N\le 0$, and subconvergence by the integrated energy identity — assemble into the statement that every homotopy class into a non-positively curved target has an energy-minimising harmonic representative; this is exactly the parabolic realisation of the variational problem posed in 03.02.26. The curvature term $⟨R_N(df,df)df,df⟩$ generalises the single Weitzenböck curvature of the elliptic Bochner technique 03.02.15 to a target-curvature contribution, and its sign is dual to the geodesic convexity of energy that yields Hartman uniqueness: the central insight is that one inequality, $K_N\le 0$, simultaneously kills the bad term in the parabolic flow and convexifies the elliptic energy, so existence and uniqueness flow from a single hypothesis. This is the same relaxation-to-equilibrium pattern that recurs in geometric flows generally, where a curvature condition on the ambient data converts a gradient flow's monotonicity into long-time convergence.

Full proof set Master

The Bochner–Eells–Sampson formula and its non-positive-curvature sign are proved in full in the Key theorem section, and the global-existence architecture is laid out in Advanced results. The two structural propositions are recorded here with proofs.

Proposition (energy-density bound under $K_N\le 0$). Let $M$ be compact with ${\mathrm{Ric}}_M\ge -\Lambda g$ for a constant $\Lambda \ge 0$, and $K_N\le 0$. If the flow exists on $[0,T)$ with $T<\infty$, then ${\sup }_{M\times [0,T)}e(f)\le e^{2\Lambda T}{\sup }_{M}e(f_0)$.

Proof. By the Bochner–Eells–Sampson formula and $K_N\le 0$, $$ (\partial_t - \Delta_g), e(f) \le -\langle \mathrm{Ric}_M, df, df\rangle \le \Lambda, |df|^2 = 2\Lambda, e(f), $$ using ${\mathrm{Ric}}_M\ge -\Lambda g$ to bound $-⟨{\mathrm{Ric}}_{M}df,df⟩\le \Lambda |df{|}^2$. Set $u(x,t)=e^{-2\Lambda t}e(f)(x,t)$. Then $({\partial }_t-{\Delta }_g)u=e^{-2\Lambda t}[({\partial }_t-{\Delta }_g)e(f)-2\Lambda e(f)]\le 0$, so $u$ is a subsolution of the heat operator on the compact $M\times [0,T)$. The parabolic maximum principle gives ${\sup }_{M\times [0,t]}u={\sup }_{M}u(\cdot ,0)={\sup }_{M}e(f_0)$, hence $e(f)(x,t)=e^{2\Lambda t}u(x,t)\le e^{2\Lambda T}{\sup }_{M}e(f_0)$ on the whole interval. $\square$

Proposition (no finite-time singularity). Under the hypotheses above, the maximal existence time is $T=\infty$.

Proof. Suppose $T<\infty$. The previous proposition bounds $\sup e(f)$, hence $\sup |df|$, uniformly on $[0,T)$. The flow ${\partial }_{t}f={\Delta }_{g}f+g^{ij}{}^N{\Gamma }_{\alpha \beta }^{\gamma }{\partial }_i{f}^{\alpha }{\partial }_j{f}^{\beta }$ then has its inhomogeneous term bounded in $C^0$, so interior parabolic Schauder estimates give a uniform $C^{2,\alpha }$ bound on $f$ over $M\times [\delta ,T)$ for each $\delta >0$; bootstrapping, differentiating the equation and reapplying Schauder, yields uniform $C^{k,\alpha }$ bounds for every $k$. Thus $f(\cdot ,t)$ and all its derivatives extend continuously to $t=T$, defining a smooth map $f(\cdot ,T)$; restarting the short-time existence theorem from $f(\cdot ,T)$ continues the flow past $T$, contradicting maximality. Hence $T=\infty$. $\square$

Proposition (subconvergence to a harmonic map). Under the hypotheses above, there is $t_k\to \infty$ with $f(\cdot ,t_k)\to f_{\infty }$ in $C^{\infty }(M,N)$, where $\tau (f_{\infty })=0$ and $f_{\infty }\simeq f_0$.

Proof. The energy identity $\frac{d}{dt}E=-{\int }_M|\tau (f){|}^2$ integrates to ${\int }_0^{\infty }{\int }_M|\tau (f){|}^{2}d{V}_{g}dt=E(f_0)-{\lim }_{t\to \infty }E(f)\le E(f_0)<\infty$, the limit existing because $E$ is non-increasing and non-negative. Hence ${\mathrm{liminf}}_{t\to \infty }{\int }_M|\tau (f(\cdot ,t)){|}^2=0$, so pick $t_k\to \infty$ with ${\int }_M|\tau (f(\cdot ,t_k)){|}^2\to 0$. The uniform $C^{k,\alpha }$ bounds of the previous proposition make $\{f(\cdot ,t_k)\}$ precompact in $C^{\infty }$ by Arzelà–Ascoli; pass to a convergent subsequence $f(\cdot ,t_k)\to f_{\infty }$. The limit satisfies ${\int }_M|\tau (f_{\infty }){|}^2={\lim }_k{\int }_M|\tau (f(\cdot ,t_k)){|}^2=0$, so $\tau (f_{\infty })\equiv 0$ and $f_{\infty }$ is harmonic. Each $f(\cdot ,t_k)$ is homotopic to $f_0$ through the flow path, and $C^0$-close maps are homotopic, so $f_{\infty }\simeq f_0$. $\square$

Connections Master

The Dirichlet energy of maps, the tension field, and harmonic maps 03.02.26 are the upstream input this unit dynamises. The functional $E(f)=\frac{1}{2}{\int }_M|df{|}^2$ and its critical points — the harmonic maps with $\tau (f)=0$ — are defined there as a static variational problem; here the negative gradient flow ${\partial }_{t}f=\tau (f)$ of that functional turns the search for critical points into a parabolic evolution, and the Eells–Sampson theorem is precisely the assertion that the gradient flow reaches a critical point whenever the target curves the right way. The first-variation formula of 03.02.26 is what makes the flow energy-monotone, and the energy-minimising property of the limit is what closes the variational circle.

The Bochner technique and curvature vanishing theorems 03.02.15 supply the algebraic identity at the heart of the proof. The elliptic Bochner formula there, relating the rough Laplacian to the Hodge Laplacian by a curvature term and forcing harmonic forms to vanish under positive Ricci curvature, is the time-independent ancestor of the parabolic identity $({\partial }_t-{\Delta }_g)e(f)=-|\nabla df{|}^2-⟨{\mathrm{Ric}}_{M}df,df⟩+⟨R_N(df,df)df,df⟩$. The Ricci-curvature term enters in the same Weitzenböck way; the target-curvature term $R_N(df,df)df$ is the new ingredient that the maps-between-manifolds setting forces and that the bundle $f^{\ast }TN$ carries.

The sectional, Ricci, and scalar curvatures 03.02.05 are the curvature invariants whose signs drive the entire dichotomy. The hypothesis $K_N\le 0$ on the target's sectional curvature is what gives the curvature term its good sign; the Ricci curvature of the domain ${\mathrm{Ric}}_M$ controls the maximum-principle estimate; and the contrast with positive curvature, where bubbling occurs, is a statement about exactly these invariants. The unit cannot be read without the curvature taxonomy fixed there.

The gradient-like vector field and gradient flow on a cobordism 03.02.27 is the finite-dimensional cousin of the construction here. There a gradient flow on a finite-dimensional manifold relaxes a Morse function to its critical points; here a gradient flow on the infinite-dimensional space of maps relaxes the energy functional to its harmonic critical points. Both are instances of the same principle — follow the negative gradient of a functional to a critical point — and the analytic difficulty here, global existence and convergence in infinite dimensions, is exactly what the compactness of the finite-dimensional setting hands one for free there.

The generalised Poincaré conjecture and Ricci flow remarks 03.02.24 connect laterally through the geometric-flow paradigm. The Eells–Sampson flow is the earliest successful geometric heat flow, predating Hamilton's Ricci flow used in Perelman's resolution of the three-dimensional Poincaré conjecture referenced there; both are gradient-like parabolic flows of a geometric functional whose long-time behaviour is governed by curvature, and the maximum-principle techniques and singularity analysis pioneered for the harmonic-map flow are direct ancestors of the corresponding Ricci-flow methods.

Historical & philosophical context Master

The harmonic-map heat flow and the non-positive-curvature existence theorem were introduced 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 defined harmonic maps as critical points of the Dirichlet energy, derived the tension field and the second-variation theory, introduced the parabolic flow ${\partial }_{t}f=\tau (f)$, and proved that for compact domain and non-positively curved target the flow converges to a harmonic representative of any homotopy class. The Bochner-type identity for the energy density, generalising Bochner's 1946 vanishing technique to maps between manifolds, is the analytic core of that paper. Philip Hartman supplied the uniqueness companion in On homotopic harmonic maps (Canad. J. Math. 19, 1967, 673–687) ^{[Hartman 1967]}, establishing convexity of the energy along geodesic homotopies and the resulting uniqueness of harmonic maps into non-positively curved targets up to translation along closed geodesics.

The method became a standard instrument of geometric analysis over the following two decades. Richard Schoen and Shing-Tung Yau, in Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature (Ann. of Math. 110, 1979, 127–142) ^{[Schoen-Yau 1979]} and related work, turned harmonic and minimal-map existence into topological and rigidity conclusions, and the same uniqueness-against-symmetry mechanism reappears in Siu's strong-rigidity theorem for Kähler manifolds and in Corlette's and Gromov–Schoen's extensions to non-Riemannian targets. The two-dimensional-domain case, outside the scope of the all-time existence theorem because the energy is conformally invariant and concentration is possible, was settled separately: Sacks and Uhlenbeck described the bubbling of harmonic spheres in 1981, and Michael Struwe constructed a global weak solution of the flow with finitely many singular times in 1985, completing the picture of when the Eells–Sampson relaxation succeeds and when it fails.

Bibliography Master

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  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{hartman1967,
  author  = {Hartman, Philip},
  title   = {On homotopic harmonic maps},
  journal = {Canadian Journal of Mathematics},
  volume  = {19},
  pages   = {673--687},
  year    = {1967}
}

@article{schoenyau1979,
  author  = {Schoen, Richard and Yau, Shing-Tung},
  title   = {Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature},
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  volume  = {110},
  number  = {1},
  pages   = {127--142},
  year    = {1979}
}

@article{sacksuhlenbeck1981,
  author  = {Sacks, J. and Uhlenbeck, K.},
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  number  = {1},
  pages   = {1--24},
  year    = {1981}
}

@article{struwe1985,
  author  = {Struwe, Michael},
  title   = {On the evolution of harmonic mappings of {R}iemannian surfaces},
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}

@book{linwang2008,
  author    = {Lin, Fanghua and Wang, Changyou},
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}