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APPEARED IN BULLETIN OF THE

arXiv:math/9201270v1 [math.DG] 1 Jan 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 1, Jan 1992, Pages 125-130NONUNIQUE TANGENT MAPS AT ISOLATEDSINGULARITIES OF HARMONIC MAPSBrian WhiteAbstract. Shoen and Uhlenbeck showed that “tangent maps” can be defined atsingular points of energy minimizing maps.Unfortunately these are not unique,even for generic boundary conditions.

Examples are discussed which have isolatedsingularities with a continuum of distinct tangent maps.Let Ωbe a bounded domain in Rm (or more generally a compact riemannianmanifold with boundary) and let N be a compact riemannian manifold. By theNash embedding theorem, N can be regarded as a submanifold of some euclideanspace.

The energy of a map f : Ω→N is defined to beE(f ) =ZΩ|Df|2. (Here f is allowed to be any measurable map from Ωto Rd such that f(x) ∈Nfor almost every x and such that the distributional first derivative of f is squareintegrable.) The map f is said to be energy minimizing if its energy is less thanor equal to the energy of each other map having the same boundary values.

It isfairly easy to prove that if g : Ω→N has finite energy, then there is an energyminimizing map f : Ω→N with the same boundary values as g. In [SU], Schoenand Uhlenbeck proved that if f is energy minimizing, then f is smooth except ona set K ⊂Ωof Hausdorffdimension at most m −3.Suppose f is energy minimizing and that x ∈Ωis a singularity of f. Schoen andUhlenbeck also proved that for every sequence ri of positive numbers converging tozero, a subsequence of the maps(1)y 7→f(x + riy)converges weakly to a map f∞: Rm →Ωthat is constant on rays through theorigin. Such a map is called a tangent map to f at x.

Intuitively, f∞is the result oflooking at f near x through a microscope with infinite magnification. The map f∞is simpler than f because it is constant on rays, but one would like to think thatit provides a good picture of f near x.

Note that f∞would not give a very goodpicture of f if there were more than one tangent map at x; that is, if a different1991 Mathematics Subject Classification. Primary 49F22, 35J60.The author was partially funded by NSF grants DMS85-53231 (PYI), DMS87-03537, andDMS9012-718Received by the editors April 3, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2BRIAN WHITEsubsequence of the maps (1) could converge to another limit map. Whether ornot such pathological behavior is possible has been perhaps the most basic openquestion about singularities of energy minimizing maps.There have been some positive results (ruling out pathological behavior).

First,Leon Simon [S] showed that if N is analytic and if f∞has an isolated discontinuity,then f∞is unique (i.e., it is the only tangent map at x). Second, Gulliver andWhite [GW] showed that if m = 3 and dim(N) = 2 (the lowest dimensions inwhich singularities are possible), then f∞is unique whether or not N is analytic.This paper is an announcement of the first example of nonuniqueness:Theorem 1.

There exists a C∞5-manifold N and a nonempty open set U ofsmooth maps φ : ∂B4 →N such that(1) each φ ∈U bounds one or more energy minimizing maps from B4 to N,and(2) if f : B4 →N is an energy minimizing map with f | ∂B4 ∈U, then f hasan isolated singularity x and a continuum of tangent maps at x. Each ofthe tangent maps is regular except at 0.The proof is too long to give here; see [W4].

However, we can prove a simpler butnonetheless interesting result that has the same flavor. Let f : Ω→N be a finiteenergy map that is smooth except at a finite set of discontinuities {pi : i = 1, .

. .

, k}.We say that f is harmonic if it satisfies the Euler-Lagrange partial differentialequations for the energy functional. Such an f is a critical point for energy, butit need not be a minimum.

Nonetheless, the existence of tangent maps and theuniqueness results of Simon and Gulliver-White in fact hold for such harmonicmaps. Thus it is interesting to note:Theorem 2.

There is a C∞4-manifold N and a harmonic map f : B3 →N suchthat f has an isolated singularity at 0 and a continuum of distinct tangent maps at0.Proof. Let N be the product S1 × R × S2 with the metricdx2 + dy2 + (2 −V (x, y))dz2(V will be specified later).

Note that this defines a complete metric on N providedV is everywhere less than 2. Of course N is not compact, but the image of theharmonic map we construct will be contained in a compact subset of N, so we couldeasily modify N to make it compact.Note that the orthogonal map O(3) acts on B3 and on N (on N by ρ: (x, y, z) 7→(x, y, ρz)).

We simplify the harmonic map equations by looking for solutions thatare O(3)-equivariant. It is not hard to see that every equivariant map is of theform:(2)p 7→(v1(|p|), v2(|p|), ±p/|p|) ∈S1 × R × S2.Here v: (0, 1] →S1 × R. It is convenient to introduce a change of variable.

Lett = log r and u(t) = v(et), so u: (−∞, 0] →S1 × R. Then the energy of the map(2) isZ 0t=−∞(| ˙u|2 + (2 −V (u)))et dt.

NONUNIQUE TANGENT MAPS3(If the domain were k-dimensional, then et would be e(k−2)t.) The associated Euler-Lagrange equation is(3)¨u + ˙u + ∇V (u) = 0.Thus equivariant harmonic maps are equivalent to solutions of the ordinary dif-ferential equation (3). This equation has a physical interpretation: it is the equationof motion of a unit mass moving in S1 × R subject to a potential V and a viscousforce.

Thus the physical energy E(u, t) = 12| ˙u|2 +V (u) is monotonically decreasing.To see this mathematically, multiply (3) by ˙u:(4)ddt12 ˙u2 + V (u)= −˙u2.Now we choose V to beV (x, y) = −exp−1y2sinx + 1y.Lemma. (1) Let u be a solution of (3) with E(u, ·) constant.

Then u(t) ≡p forsome p ∈S1 × [0]. (2) Let u be a solution of (3) such that the physical energy E(u, 0) at time 0is negative.

Then the solution exists for all t ∈[0, ∞) and becomes unbounded ast →∞.Proof. If E(u, ·) is constant, then ˙u ≡0, since otherwise the particle would bedissipating physical energy to viscosity (see (4)).

Thus u is a constant p, so (3)implies that ∇V (p) = 0. But ∇V (x, y) = 0 if and only if y = 0.

This proves (1).Now suppose that u is a solution of (3) with E(u, 0) < 0. By elementary ODEtheory, the solution exists for all positive times unless the particle moves infinitelyfar in a finite time.

But 12| ˙u(t)|2 +V (u(t)) ≤E(u, 0) < 0, so 12| ˙u(t)|2 < −V (u(t)) ≤sup(−V ) = 1. Thus the solution exists for t ∈[0, ∞).Suppose u(t) remains in a bounded region of S1 × R. Then the set of pairs(u(n), ˙u(n)) is bounded, so a subsequence (u(ni), ˙u(ni)) converges.

It follows (fromthe smooth dependence of ODE solutions on initial conditions) that the solutionsui(t) = u(ni + t) converge smoothly to a solution v(t) of (3). Now(5)E(v, t) = limi→∞E(u, ni + t) = limt→∞E(u, t) ≤E(u, 0) < 0(where limt→∞E(u, t) exists because E(u, ·) is monotonic).

Thus E(v, ·) is con-stant, so by (1) of the lemma v(t) ≡p ∈S1 × [0]. But then E(v, t) = V (p) = 0,contradicting (5).

This proves (2).□Now let un : [0, ∞) →S1 × R be the solution to (3) with initial position un(0) =( 12π,12πn) and initial velocity ˙un(0) = 0. Note that the initial physical energy isnegative:12| ˙un(0)|2 + V (un(0)) = 0 + V (12π,12πn) < 0.Thus by the lemma, there is a first time tn > 0 at which un(tn) ∈S1 × {−1, 1}.If un(t) were ever in S1 × [0], then E(un, t) ≥V (un(t)) = 0, which is impossible.Thus un(tn) ∈S1 × [1] and un(t) ∈S1 × (0, 1) for t ∈(0, tn).

4BRIAN WHITENote that un(0) →( 12π, 0) and ˙un(0) ≡0, so the un converge to a solution w withw(0) = ( 12π, 0) and ˙w(0) = 0. By uniqueness of solutions to ODEs, w(t) ≡( 12π, 0).Thus limn→∞un(t) = ( 12π, 0), so tn →∞since un(tn) = (xn, 1).Now as in the proof of the lemma, there is a sequence n(i) such that the solutionsvi(t) = un(i)(tn(i) + t) converge smoothly to a solution v on (−∞, ∞).

Of coursev(0) ∈S1 × [1],v(t) ∈S1 × [0, 1]fort < 0,andE(v, t) ≤0for all t.In fact E(v, t) must be strictly negative for every t. For since E(v, t) is a nonpositiveand nonincreasing function of t, if it were 0 for some t = a, then it would be 0 foreach t ≤a. But then by the lemma, v(t) ≡p ∈S1 × [0] for all t ≤a.

By uniquecontinuation for ODE, v(t) ≡p ∈S1 ×[0] for all t. But v(0) ∈S1 ×[1]. This provesthat E(v, t) is strictly negative.Now I claim that v defines a harmonic map with a continuum of tangent mapsat the origin.

That is, I claim that v(t) has a continuum of subsequential limits ast →−∞.As in the proof of the lemma, every sequence of t’s tending to −∞has a subse-quence τi such that the solutions wi(t) = v(t + τi) converge to a solution w(t). OfcourseE(w, t) = limi→∞E(v, t + τi) =limt→−∞E(v, t) ≤0(where limt→−∞E(v, t) exists because E(v, ·) is monotonic).

Thus E(w, ·) is con-stant, so by the lemma w(t) ≡p, where p ∈S1 × [0].What we have shown is limt→−∞v2(t) = 0, where v2(t) is the second componentof v(t) = (v1(t), v2(t)) ∈S1 × R.Now the set Z = {p ∈S1 × R : V (p) = 0} consists of S1 × [0] together witha collection of curves that wind around the cylinder infinitely many times as theyapproach S1 × [0]. Since V (v(t)) ≤E(v, t) < 0, v(t) is never in Z.

Thus v(t)must also wind around the cylinder infinitely many times as t →−∞. This provesTheorem 2.

(To make this last argument more formal, note from the definition of V thatfor each x ∈S1 and each ε > 0, the set Z ∪([x] × (−ε, ε)) divides S1 × R intoinfinitely many connected components, the closure of each of which is disjoint fromS1 × [0]. Since v(t) approaches S1 × [0] as t →−∞, the particle must cross theset Z ∪([x] × (−ε, ε)).

Since it never crosses Z, it must cross [x] × (−ε, ε). As thisholds for every x and ε, each (x, 0) ∈S1 × [0] is a subsequential limit of v(t).

)□RemarksExactly the same construction provides examples of harmonic maps from Bm toN = S1 × R × Sm−1 (metrized as above) with a continuum of tangent maps at anisolated singularity. The only difference is that the viscosity (i.e., the coefficient infront of ˙u in (3)) is m −2 instead of 1.

NONUNIQUE TANGENT MAPS5In all those examples, the dimension of the target manifold is one more than thedimension of the domain. But we can also prove that there is a harmonic map ffrom B4 to the 4-manifold N of Theorem 2 such that f has a continuum of tangentmaps at an isolated singularity.

The proof is the same as the proof of Theorem 2,except that we consider maps of the formf : p →(f 1(|p|), f 2(|p|), h(p/|p|)),where h : S3 →S2 is the Hopf fibration.Open questions1. Must tangent maps be unique if the target manifold N is 2-dimensional?

Theanswer is “yes” if the domain is 3-dimensional [GW].2. Must tangent maps be unique for generic metrics on the target manifold N?3.

If T is a minimal variety in a riemannian manifold N, then at each singular pointx ∈T there are one or more tangent cones (i.e., subsequential limits of images ofT under dilations about x). Can there be more than one?

See [AA; T1,2; W1–3],and Simon [S] for results in special cases. Simon [S] proved that if a tangent conehas multiplicity one and has an isolated singularity, then it is unique.

Unlike hisanalogous result for harmonic maps, this does not require that the metric on N beanalytic.The construction in this paper does not seem to have any analogue in the caseof minimal varieties.References[AA]W. K. Allard and F. J. Almgren, Jr., On the radial behavior of minimal surfaces and theuniqueness of their tangent cones, Ann. of Math.

(2) 113 (1981), 215–256. [GW] R. Gulliver and B.

White, The rate of convergence of a harmonic map at a singular point,Math. Ann.

283 (1989), 539–549.[SU]R. Schoen and K. Uhlenbeck, A regularity theory for harmonic mappings, J. DifferentialGeom.

17 (1982), 307–335.[S]L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications togeometric problems, Ann.

of Math. (2) 118 (1983), 525–571.[T1]J.

E. Taylor, Regularity of the singular sets of two-dimensional area-minimizing flat chainsmodulo 3 in R3, Invent. Math.

22 (1973), 119–159. [T2], The structure of singularities in solutions to ellipsoidal variational problems withconstraints in R3, Ann.

of Math. (2) 103 (1976), 541–546.[W1]B.

White, The structure of minimizing hypersurfaces mod 4, Invent. Math.

53 (1979),45–58. [W2], Tangent cones to 2-dimensional area-minimizing integral currents are unique,Duke Math.

J. 50 (1983), 143–160.

[W3], Regularity of the singular sets in immiscible fluid interfaces and in solutions toother plateau-type problems, Proc. Centre for Math.

Anal., Canberra, Australia, 1985, pp.244–249. [W4], Nonunique tangent maps at isolated singularities of energy-minimizing maps (inpreparation).Mathematics Department, Stanford University, Stanford, California 94305

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