on the distance between homotopy classes of maps between spheres

J. Fixed Point Theory Appl. 17 (2007) 1–32DOI 10.1007/s11784-014-0156-5© Springer Basel 2014

Journal of Fixed Point Theoryand Applications

On the distance between homotopy classesof maps between spheres

Shay Levi and Itai Shafrir

Dedicated with great respect to Haim Brezis on the occasion of his 70th birthday

Abstract. Certain Sobolev spaces of maps between manifolds can bewritten as a disjoint union of homotopy classes. Rubinstein and Shafrir[Israel J. Math. 160 (2007), 41–59] computed the distance between ho-motopy classes in the spaces W 1,p(S1, S1) for different values of p, andin the space W 1,2(Ω, S1) for certain multiply connected two-dimensionaldomains Ω. We generalize some of these results to higher dimensions.Somewhat surprisingly we find that in W 1,p(S2, S2), with p > 2, thedistance between any two distinct homotopy classes equals a universalpositive constant c(p). A similar result holds in W 1,p(Sn, Sn) for anyn ≥ 2 and p > n.

Mathematics Subject Classification. 58D15, 46E35.

Keywords. Homotopy classes, Sobolev maps.

1. Introduction

Sobolev spaces of maps from a domain or a manifold with values in spheres ap-pear naturally in Geometry and Analysis, especially in the study of harmonicmaps. Motivation to study such maps comes from several areas of Physicslike Liquid Crystals and Superconductivity. In general, decomposition of therelevant space W 1,p(D,Sn) (where D is either a domain or a manifold) intohomotopy classes is a very subtle issue (see [2, 4, 7, 8, 9, 10]). In some casesa decomposition of the form

W 1,p(D,Sn) =⋃d

Ed, (1.1)

of the space into a disjoint union of homotopy classes, holds, where d iseither an integer or a vector of integers. Such partitions for Sobolev spaces ofmappings between two Riemannian manifolds were developed by White [19].The existence of such a partition for maps from a two-dimensional disk toS2 (for p = 2) was proved by Brezis and Coron [3] who used it to establish

S. Levi and I. Shafrir JFPTA

existence of certain harmonic maps. Rubinstein and Sternberg [14] used sucha partition for maps from the solid torus to S1 to explain persistent currentsin Superconductivity and in a later work, with Kim [11], to predict newstructures in Liquid Crystals (here again p = 2). The partition (1.1), whereD is a multiply connected domain, m = 1 and p = 2, arises naturally in thestudy, by Bethuel, Brezis and Helein [1], of minimizers of Ginzburg–Landau-type energy (in this case d is a vector of length n, where n is the number ofholes).

Usually partitions like (1.1) are used to prove existence of nontrivialp-harmonic maps, as minimizers of the p-energy within specific homotopyclasses. The study of the distance between two distinct homotopy classesseems to be initiated by Rubinstein and Shafrir in [13]. They considered twoclasses of maps. The first is

H1(S1, S1) = W 1,2(S1, S1) =⋃d∈Z

Ed =⋃d∈Z

{u : deg u = d},

where for any two integers d1 �= d2 the distance δ(d1, d2) between the homo-topy classes Ed1 and Ed2 is defined by

δ2 (d1, d2) = inf

{∫S1

| (u1 − u2)′ |2 : u1 ∈ Ed1 , u2 ∈ Ed2

}. (1.2)

Rubinstein and Shafrir found an explicit formula for δ (d1, d2), namely

δ2 (d1, d2) =8(d2 − d1)

2

π. (1.3)

They also proved analogous formula for different values of p. Here we studythe distance between homotopy classes of self-maps of spheres in higher di-mension. Since the case of maps from Sn to Sn turns out to be essentially thesame for any n ≥ 2, we restrict ourselves below to the case n = 2 (see Sec-tion 3 for details on the n-dimensional case). A well-defined notion of degreefor maps in W 1,p

(S2, S2

)exists only for p ≥ 2 (see Section 2 for details). We

then have

W 1,p(S2, S2

)=

⋃d∈Z

Ed =⋃d∈Z

{u : deg u = d}.

The distance between Ed1 and Ed2 is defined by

δpp (d1, d2) = inf

{∫S2

|∇ (u1 − u2) |p : u1 ∈ Ed1 , u2 ∈ Ed2

}.

We compute explicitly δp (d1, d2), and somewhat surprisingly, the resultsare quite different from those of [13] for the case n = 1. First, in the casep = 2 we found in Theorem 2.7 that δ2 (d1, d2) = 0 for every d1, d2 ∈ Z.Second, when p > 2, for every pair d1 �= d2, the distance δp (d1, d2) equalsa fixed positive value (independently of the degrees) that is given explicitly

Distance between homotopy classes

by 2Cp

, where

Cp = (2π)−1/p

⎡⎣√πΓ(

p−22p−2

)Γ(

2p−32p−2

)⎤⎦1−1/p

(see Theorem 2.10).

The constant Cp arises as the best constant in a Sobolev-type inequality ontwo-dimensional spheres, which is due to Talenti [18].

A brief explanation for the value 2Cp

goes as follows. It is not difficult

to see that for any two maps, u1 ∈ Ed1and u2 ∈ Ed2

, the (scalar) functionv = |u2 − u1| must take both the values 0 and 2 somewhere (we assume herefor simplicity that d2 �= −d1). Then, Talenti’s inequality applied to v yields∫

S2

|∇ (u1 − u2) |p ≥∫S2

|∇v|p ≥ maxS2 v −minS2 v

Cp=

2

Cp.

This is essentially the proof of the lower bound of Theorem 2.10. The proofof the upper bound uses an explicit construction based on the profile of theoptimal function in Talenti’s inequality. The main difference from the casen = 1 is explained by the extra dimension, that allows us to construct mapspossessing k-axial symmetry, i.e., maps of the form

u(ϕ, θ) =(sinΦ(ϕ) cos(kθ), sinΦ(ϕ) sin(kθ), cosΦ(ϕ)

).

The degrees of these maps are the result of rotations around the z-axis thatdo not affect the distance between the maps; see the proof of Theorem 2.10for details. The rather straightforward generalization of the above results tomaps between higher-dimensional spheres is given in Section 3.

Remark 1.1. The M. Sc. thesis [12] also contains a generalization of the aboveresults for the distance between homotopy classes of maps in W 1,p

(S2,Σ

),

where Σ = ∂K is a surface which is the boundary of a convex body K ⊂ R2 of

class C2+ (see [15]). There it is proved that for p > 2 we have δp (d1, d2) =

WCp

,

where W is the width of the convex body K (i.e., the minimal distance be-tween two parallel planes bounding K, see [15] for details). In [12] one canfind also a generalization of the result of [13] for W 1,p(S1, S1) to the case ofthe space W 1,p(S1, C), where the closed curve C is the boundary of convexbody in R

2.

2. Maps from S2 to S2

Let S2 = {x ∈ R3 : x2

1 + x22 + x2

3 = 1} denote the unit sphere in R3.

Denote the north and south poles by N = (0, 0, 1) and S = (0, 0,−1). Witha slight abuse of notation each v : S2 → R can be also viewed as a map from[0, π]× [0, 2π] to R such that v(0, θ) and v(π, θ) are independent of θ and alsov(ϕ, 0) = v(ϕ, 2π) for all 0 < ϕ < π. Hence we can also write any u : S2 → S2

as u = (v1, v2, v3) where vi : [0, π] × [0, 2π] → R. Here θ (longitude) and ϕ


(colatitude) are geographical coordinates on S2. Thus

x1 = cos θ sinϕ,

x2 = sin θ sinϕ,

x3 = cosϕ,

where 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ π.

Note that H2(dx) = sinϕdϕdθ where x runs over S2 and H2 denotesthe Hausdorff two-dimensional measure on S2. We also have

|∇vi| =√(

∂vi∂ϕ

)2

+

(1

sinϕ

∂vi∂θ

)2

, i = 1, 2, 3,

|∇u| =√|∇v1|2 + |∇v2|2 + |∇v3|2.

Note that for p > 2, W 1,p(S2, S2) ⊂ C1−2/p(S2, S2), so that eachu ∈ W 1,p(S2, S2) has a well-defined degree. For p = 2, the degree is still welldefined, thanks to the density of C1(S2, S2) in the Sobolev space H1(S2, S2)(see [16]). This is a special case of the VMO (vanishing mean oscillation) de-gree that was developed by Brezis and Nirenberg in [5]. Thus, for p ≥ 2 wemay write

W 1,p(S2, S2

)=

⋃d∈Z

Ed =⋃d∈Z

{u : deg u = d}.

The distance between Ed1 and Ed2 is defined by

δpp (d1, d2) = inf

{∫S2

|∇ (u1 − u2) |p : u1 ∈ Ed1 , u2 ∈ Ed2

}. (2.1)

In this section we will compute δp(d1, d2) for any p ≥ 2 and d1, d2 ∈ Z.Interestingly, in contrast with the case of maps from S1 to S1 studied in [13],we will show that in dimension two (and higher) δp(d1, d2) is the same forany d1 �= d2.

It turns out that the computation of the distance between the homotopyclasses is related to the best constant in a certain Sobolev-type inequality onthe sphere. We recall below the relevant result which is due to Talenti [18].

Theorem 2.1. Let p > 2. If v ∈ W 1,p(S2,R), then

maxS2

v −minS2

v ≤ Cp ‖∇v‖Lp(S2) , (2.2)

where

Cp = (2π)−1/p

⎡⎣√πΓ(

p−22p−2

)Γ(

2p−32p−2

)⎤⎦1−1/p

.

Inequality (2.2) is sharp.


Let v be a smooth map from S2 to R. Without loss of generality, weassume v ≥ 0 . Let V (t) be the level sets of v, V (t) := {x ∈ S2 : v(x) > t}.The distribution function of v is given by μ(t) = H2(V (t)). Let

v∗ (s) =∫ ∞

0

χ[0,μ(t)](s) dt

denote the decreasing rearrangement of v in the sense of Hardy and Little-wood, i.e., the decreasing right-continuous map from [0, 4π] into [0,∞) whichis equidistributed with v. It can be shown that v∗ is locally Lipschitz contin-uous.

The spherical symmetric rearrangement v� of v is a function from S2

to [0,∞) which is equidistributed with v and whose level sets are concentricspherical caps. Hence, if ϕ is the colatitude of x and B(ϕ) is the area of thecap which is intercepted on S2 by a circular cone having its vertex in thecenter of S2 and aperture 2ϕ, then

v� (x) = v∗ (B (ϕ)) = v∗(4π sin2

ϕ

2

). (2.3)

An important property of v� is that it does not increase the Lp-norm of thegradient. In fact, the following lemma is a special case of a symmetrizationtheorem from [17].

Lemma 2.2. If p ≥ 1, then

‖∇v‖Lp(S2) ≥ ‖∇v�‖Lp(S2) =

[∫ 4π

0

[s (4π − s)]p/2

[dv∗

ds(s)

]pds

]1/p.

We give below a sketch of the proof for Theorem 2.1. For convenience,we assume v ≥ 0. By the definition of v∗,

maxS2

v = v∗ (0) ,

minS2

v = v∗ (4π−) ,

maxS2

v −minS2

v =

∫ 4π

0

−[dv∗

ds(s)

]ds.

Hence, Holder’s inequality gives

maxS2

v −minS2

v ≤[∫ 4π

0

[s (4π − s)]−p/2(p−1)

ds

]1−1/p

·[∫ 4π

0

[s (4π − s)]p/2

[dv∗

ds(s)

]pds

]1/p.

(2.4)

Inequality (2.2) follows from Lemma 2.2 since the first term on the right-handside of (2.4) equals Cp.


Remark 2.3. An inspection shows that equality holds in (2.2) if and only if vsatisfies

v∗ (s) = c1

∫ 1

s/4π

[t (1− t)]−p/2(p−1)

dt+ c2 (2.5)

for some constants c1 and c2. Using (2.2) we deduce the following corollary.

Corollary 2.4. Equality holds in (2.2) for a radially symmetric function v :S2 → R if and only if v satisfies

v (x) = c1

∫ π

ϕ

(sin t)−1/(p−1)

dt+ c2, (2.6)

where ϕ is the colatitude of x and c1, c2 are constants.

For p > 2 we define the function f0 = f(p)0 : [0, π] → [0, 1] by

f0 (ϕ) =

(∫ π

0

(sin t)−1/(p−1)

dt

)−1

·∫ ϕ

0

(sin t)−1/(p−1)

dt. (2.7)

Let v0(x) := f0(ϕ), where x ∈ S2 and ϕ is the colatitude of x. The functionf0 will be useful later in the proof of the upper bound for δp. We have

v0 (N) = f0 (0) = 0 and v0 (S) = f0 (π) = 1.

From Corollary 2.4 equality holds in (2.2) for the function v0. Thus,

‖∇v0‖Lp(S2) = 2π

∫ π

0

(f ′0 (ϕ)

)psinϕdϕ = C−1

p . (2.8)

Lemma 2.5. For d1, d2 ∈ Z, d1 �= −d2, p ≥ 2, let u1 and u2 be two continuousmaps in W 1,p(S2, S2) with deg ui = di, i = 1, 2. Then, there is a point x ∈ S2

such that u2(x) = u1(x).

Proof. We claim that there exist x ∈ S2 and t0 ∈ (0, 1) such that

t0u1(x) + (1− t0)(−u2(x)) = 0. (2.9)

Indeed, otherwise, the map I : [0, 1]× S2 → S2, given by

I (t, x) =tu1 (x) + (1− t) (−u2 (x))

‖tu1 (x) + (1− t) (−u2 (x))‖ ,

would be a homotopy between u1 and−u2. Since dim(S2) is even, deg(−u2) =−d2 . Hence d1 = −d2, contradicting our initial assumption. From (2.9) weget ‖t0u1(x)‖ = ‖(1− t0)(−u2(x))‖. Therefore, t0 = 1

2 and the result followsfrom (2.9). Note that the continuity assumption is needed only for p = 2since for p > 2 every u ∈ W 1,p(S2, S2) has a continuous representative. �

Lemma 2.6. If d1 �= −d2, p ≥ 2, then

δp (d1 + k, d2 + k) ≤ δp (d1, d2) for all k ∈ Z.


Proof. Take any u1 ∈ Ed1 , u2 ∈ Ed2 . We may assume, without loss of gen-erality, that u1, u2 are smooth maps. Since d1 �= −d2, by Lemma 2.5 thereis a point x ∈ S2 such that u1(x) = u2(x). We may choose the coordinatesaxes in the domain and in the range of the maps ui such that x = S andu1(x) = u2(x) = S. Thus, u1(S) = u2(S) = S.

For a small ε > 0 define the maps ui = u(ε)i , i = 1, 2, on S2 by

ui (ϕ, θ)

=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ui

(π

π − εϕ, θ

), ϕ ∈ [0, π − ε],(

sin[πε(π − ϕ)

]cos (−kθ) ,

sin[πε (π − ϕ)

]sin (−kθ) , cos

[πε (π − ϕ)

] ), ϕ ∈ (π − ε, π].

(2.10)

The maps ui belong to Edi+k, i = 1, 2, and satisfy∫S2

∣∣∇ (u2 − u1)∣∣p

=

∫ 2π

0

dθ

∫ π−ε

0

∣∣∣∣∇(u2

(π

π − εϕ, θ

)− u1(

π

π − εϕ, θ)

)∣∣∣∣ p sinϕdϕ

=π − ε

π

∫ 2π

0

dθ

∫ π

0

∣∣∇ (u2 (ϕ, θ)− u1 (ϕ, θ))∣∣p sin(π − ε

πϕ

)dϕ.

Since the maps u1, u2 are smooth and sin(π−επ ϕ

) ≤ sin (ϕ)+ε for 0 ≤ ϕ ≤ π,we get ∫

S2

|∇ (u2 − u1) |p ≤ 2π2Mpε+

∫S2

|∇ (u2 − u1) |p,

where M := ‖∇(u2 − u1)‖L∞(S2) < ∞. Hence

limε→0

∫S2

|∇ (u2 − u1) |p ≤∫S2

|∇ (u2 − u1) |p.

The result follows since ui can be chosen arbitrarily in Edi . �

Now we treat the case p = 2.

Theorem 2.7. For every d1, d2 ∈ Z we have δ2(d1, d2) = 0.

Proof. We begin with a brief description of our strategy. It would be enoughto deal with the case where one of the degrees is zero, and then use Lemma 2.6to deduce the general case. We shall construct two maps, both with m-axialsymmetry, one of degree zero and the second one of degree m = d2 − d1 �= 0.On a small sphere of order ε on S2, centered at the north pole, the two mapsare identical, each covering the upper hemisphere. On the remaining (muchlarger) part of S2 one of the maps “goes back” from the equator to the northpole, so its degree is zero. On the other hand, the values taken by the secondmap on that part of S2 are just the reflection with respect to the xy-plane


of the values taken by the first map. The degree of the second map thereforeequals m and the difference between the two maps has a nonzero componentonly in the z-direction. Using the fact that a point has zero 2-capacity indimension two, we can arrange to have arbitrarily small energy contributionfrom that component. The detailed construction is given below.

For any small ε > 0 define the maps Φ(ε)i : [0, π] → [0, π] by

Φi(ϕ) = Φ(ε)i (ϕ) =

⎧⎪⎪⎨⎪⎪⎩π

2εϕ, ϕ ∈ [0, ε],

π

2

(1− (−1)

i logϕ− log ε

log π − log ε

), ϕ ∈ (ε, π],

(2.11)

where i = 1, 2. A direct computation yields

limε→0

∫ π

ε

(Φ′

i(ϕ))2

sinϕdϕ = 0. (2.12)

We define the maps ui = u(ε)i from S2 to S2 by

ui(ϕ, θ) =(sinΦi(ϕ) cos(mθ), sinΦi(ϕ) sin(mθ), cosΦi(ϕ)

), i = 1, 2,

where 0 ≤ ϕ ≤ π, 0 ≤ θ ≤ 2π and m = d2 − d1 �= 0. Since |∇ui| < cε , i = 1, 2

(c is a positive constant, independent of ε) the maps belong to W 1,∞(S2, S2)and satisfy u1 ∈ Ed2−d1 and u2 ∈ E0. We have

u2 − u1 =

{(0, 0, 0), ϕ ∈ [0, ε],

(0, 0, 2 cosΦ2(ϕ)), ϕ ∈ (ε, π].

Thus, ∫S2

|∇ (u2 − u1) |2 ≤ 23π

∫ π

ε

(Φ′

2(ϕ))2

sinϕdϕ. (2.13)

From (2.12), (2.13) and Lemma 2.6 (applied to 0 and d2 − d1) we get

δ2 (d1, d2) ≤ δ2 (d2 − d1, 0) = 0. �

Remark 2.8. Theorem 2.7 is analogous to a result from [5] (see Lemma 6 andRemark 6 there) which states that the distance between homotopy classes inH1/2(S1, S1) is always zero.

Next we turn to the case p > 2. The lower bound is given by the followinglemma.

Lemma 2.9. If d1 �= d2, then for p > 2 we have

δp (d1, d2) ≥ 2

Cp.

Proof. Take any u1 ∈ Ed1 , u2 ∈ Ed2 . Since d1 �= d2, Lemma 2.5 applied to u1

and −u2 implies that there is a point x1 ∈ S2 such that u2(x1) = −u1(x1).We may assume, without loss of generality, that d2 �= 0. We choose thecoordinates axes in the range so that u2(x1) = N . Since d2 �= 0, it followsthat there is x2 ∈ S2 such that u2(x2) = S. Let v3 : S2 → R be the third


component of u2 − u1. The function v3 belongs to W 1,p(S2,R) and satisfiesv3(x1) = 2, v3(x2) ≤ 0. From Theorem 2.1 we get

‖∇ (u2 − u1)‖Lp(S2) ≥ ‖∇v3‖Lp(S2)

≥ maxS2 v3 −minS2 v3Cp

≥ 2

Cp.

(2.14)

�

Next we prove the main result of this section.

Theorem 2.10. If d1 �= d2, then for p > 2 we have

δp (d1, d2) =2

Cp. (2.15)

Proof. Thanks to the lower bound of Lemma 2.9, it is enough to prove thatthe following upper bound holds:

δp (d1, d2) ≤ 2

Cp(2.16)

for all d1 �= d2. The construction of pairs of maps that realize (2.16) in thelimit shares some similarities with the construction used in the proof of Theo-rem 2.7. Indeed, once again it is enough to consider the case where one of thedegrees is zero and both maps are taken to be equal on a small sphere, whoseimage by each of the maps is the upper hemisphere. On the remaining partof S2 one map (the one of zero degree) covers again the upper hemispherewhile the second map is just the reflection with respect to the xy-plane ofthe first one. This time however we arrange so that the difference betweenthe two maps (which is in the direction of the z-axis) is equal approximatelyto f0 (see (2.7)–(2.8)) which is the profile of the minimizer in Theorem 2.1.

For any small ε > 0 consider the following approximation F = F (ε) :[ε, π] → [0, 1] of the function f0 (defined in (2.7)):

F (ε) (ϕ) = Jε(f0(ϕ)), (2.17)

where the map Jε : [f0 (ε) , 1] → [0, 1] is a Cm+1-map satisfying the followingproperties:

Jε (1) = 1, J ′ε (1) = · · · = J

(m)ε (1) = 0,

Jε (f0 (ε)) = 0, J ′ε (f0 (ε)) = · · · = J

(m)ε (f0 (ε)) = 0,

Jε (s) = s, 2f0 (ε) ≤ s ≤ 1− ε,

0 ≤ J ′ε (s) < c0, f0 (ε) ≤ s ≤ 1,∣∣∣J (m+1)

ε (s)∣∣∣ < c1

εm , f0 (ε) ≤ s ≤ 1,

c2εm <

∣∣∣J (m+1)ε (s)

∣∣∣ < c1εm , 1− ε

2 < s ≤ 1,

(2.18)

for some constants c0, c1, c2 (independent of ε) and m an integer that satisfiesm ≥ 1 + 2

p−2 .

In fact, we need to construct a function Jε on the two intervals

[f0 (ε) , 2f0 (ε)] and [1− ε, 1]


“connecting” the values Jε(f0(ε)) = 0 and Jε(1 − ε) = 1 − ε to the valuesJε(s) = s on the interval [2f0(ε), 1− ε] that satisfies the estimates in (2.18).This requires a change of order f0(ε) for Jε on the interval [f0(ε), 2f0(ε)]and of order ε on the interval [1 − ε, 1]. An appropriate Jε will then have aderivative of order O(1). But now the change of order 1 between J ′

ε(f0(ε))and J ′

ε(2f0(ε)) and between J ′ε(1− ε) = 1 and J ′

ε(1) = 0 requires J ′′ε of order

max

(1

f0(ε),1

ε

)=

1

ε

(since f0(ε) ∼ εp−2p−1 ε). Similarly, for higher-order derivatives we will get∣∣J (k)

ε (s)∣∣ ≤ c

εk−1.

Since what we are requiring is just interpolation between certain given valuesof the function and some of its derivatives at two pairs of points, it is clearthat we can even take a polynomial for Jε.

Using the Taylor formula around s = f0(ε) and s = 1 in conjunctionwith (2.18) yields, for s ∈ [f0(ε), 1],

|J ′ε (s)| ≤

c

εm(s− f0 (ε))

m, (2.19)

|J ′ε (s)| ≤

c

εm(1− s)

m, (2.20)

where c is a constant independent of ε. For s ∈ [1− ε

2 , 1]we use again the

Taylor formula around 1 to obtain

Jε(s) = 1 +J(m+1)ε (θ)

(m+ 1)!(s− 1)m+1, θ ∈ (s, 1),

implying∣∣∣∣∣ J ′ε (s)

(1− Jε (s))1/2

∣∣∣∣∣ ≤ c

εm/2(1− s)(m−1)/2, s ∈

[1− ε

2, 1]. (2.21)

Define the functions Φi = Φ(ε)i : [0, π] → [0, π], i = 1, 2, by

Φ1(ϕ) =

⎧⎨⎩π

2εϕ, [0, ε],

π − arccosF (ϕ), (ε, π],

Φ2(ϕ) =

⎧⎨⎩π

2εϕ, [0, ε],

arccosF (ϕ), (ε, π].

(2.22)

Then, define the maps ui = u(ε)i : S2 → S2, for i = 1, 2, by

ui (ϕ, θ) =(sin Φi (ϕ) cos (kθ) , sin Φi (ϕ) sin (kθ) , cos Φi (ϕ)

), (2.23)

where 0 ≤ ϕ ≤ π, 0 ≤ θ ≤ 2π and k = d2 − d1 �= 0.


Next we prove that u1 ∈ Ed2−d1 and u2 ∈ E0. Computation of the degreesof u1 and u2 gives

deg ui =1

4π

∫S2

ui · uiϕ ∧ uiθ =2πk

4π

∫ π

0

Φ′i (ϕ) sin Φi (ϕ) dϕ

=k

2

[cos Φi (0)− cos Φi (π)

]=

{k, i = 1,

0, i = 2.

We will prove that

ui ∈ W 1,∞ (S2, S2

) ⊂ W 1,p(S2, S2

)by showing that the derivatives of F and

√1− F 2 are bounded. Obviously,

it is enough to consider the intervals [ε, ε+ ε] and [π − ε, π] for some ε > 0.Let ε be such that f0 (π − ε) > 1− ε

2 . On the interval [ε, ε+ ε] we have

f ′0 (ϕ) = (sinϕ)

−1/(p−1) ≤ c ϕ−1/(p−1),

f0 (ϕ) = c

∫ ϕ

0

(sin t)−1/(p−1)

dt ≤ c ϕ(p−2)/(p−1).

From (2.19) we get

F ′ (ϕ) = J ′ε (f0 (ϕ)) f

′0 (ϕ) ≤

c

εm(f0 (ϕ)− f0 (ε))

mf ′0 (ϕ)

≤ c

εm(f0 (ϕ))

mf ′0 (ϕ) ≤

c

εmϕm((p−2)/(p−1))−1/(p−1) ≤ c

εm.

In the last inequality we used that

mp− 2

p− 1− 1

p− 1=

p− 2

p− 1

[m− 1

p− 2

]> 0.

For the function√1− F 2 we simply have∣∣∣∣(√1− F 2

)′(ϕ)

∣∣∣∣ = FF ′√1− F 2

≤ cF ′ ≤ c

εm.

On the interval [π − ε, π] the functions f0 and f ′0 satisfy

f ′0 (ϕ) = (sin (π − ϕ))

−1/(p−1) ≤ c (π − ϕ)−1/(p−1)

,

1− f0 (ϕ) = c

∫ π

π−ϕ

(sin t)−1/(p−1)

dt ≤ c (π − ϕ)(p−2)/(p−1)

.

Hence, using (2.20),

F ′ (ϕ) = J ′ε (f0 (ϕ)) f

′0 (ϕ) ≤

c

εm(1− f0 (ϕ))

mf ′0 (ϕ)

≤ c

εm(π − ϕ)

m[(p−2)/(p−1)]−1/(p−1) ≤ c

εm.


Since f0 (π − ε) > 1− ε2 , from (2.21) we get(√

1− F 2)′

(ϕ) ≤ c

∣∣∣∣∣ J ′εf

′0

(1− Jε)1/2

∣∣∣∣∣ ≤ c

εm/2(1− f0 (ϕ))

(m−1)/2f ′0 (ϕ)

≤ c

εm/2(π − ϕ)

[(m−1)/2]·[(p−2)/(p−1)]−1/(p−1) ≤ c

εm/2.

In the last inequality we used that

m− 1

2· p− 2

p− 1− 1

p− 1=

p− 2

2 (p− 1)

[m− 1− 2

p− 2

]≥ 0.

Our next step will be to prove that

limε→0

∫S2

|∇ (u2 − u1) |p ≤(

2

Cp

)p

. (2.24)

Note that

u2 − u1 =

{(0, 0, 0), ϕ ∈ [0, ε],

(0, 0, 2F (ϕ)), ϕ ∈ (ε, π].

Therefore, ∫S2

|∇ (u2 − u1) |p ≤ 2p · 2π∫ π

ε

(F ′(ϕ)

)psinϕdϕ. (2.25)

Set ε = 2f0 (ε). Note that

ε ≤ c ε1−1/(p−1),

f0 ([ε, π − ε]) ⊂ [2f0 (ε) , 1− ε].(2.26)

Since ∫ ε+ε

ε

(F ′(ϕ)

)psinϕdϕ ≤ c

∫ ε+ε

ε

(f ′0(ϕ)

)psinϕdϕ

≤ c

∫ ε+ε

ε

ϕ−p/(p−1)ϕdϕ

≤ c ε1−1/(p−1) ≤ c ε[1−1/(p−1)]2 ,

we have

limε→0

∫ ε+ε

ε

(F ′(ϕ)

)psinϕdϕ = 0. (2.27)

Moreover,∫ π

π−ε

(F ′(ϕ)

)psinϕdϕ ≤ c

∫ π

π−ε

(f ′0(ϕ)

)psinϕdϕ

≤ c

∫ π

π−ε

(π − ϕ)−p/(p−1)

(π − ϕ) dϕ

= c ε1−1/(p−1) ≤ c ε[1−1/(p−1)]2 ,

implying that

limε→0

∫ π

π−ε

(F ′(ϕ)

)psinϕdϕ = 0. (2.28)


Finally, on [ε + ε, π − ε] we have by (2.26) and (2.18) that F (ϕ) = f0(ϕ).From (2.8) we obtain∫ π−ε

ε+ε

(F ′(ϕ)

)psinϕdϕ ≤

∫ π

0

(f ′0(ϕ)

)psinϕdϕ =

1

2π(Cp)

−p. (2.29)

From (2.25) and (2.27)–(2.29) we deduce (2.24). Hence

δpp (d2 − d1, 0) ≤(

2

Cp

)p

.

Lemma 2.6 (applied to 0 and d2 − d1) yields (2.16). �

Next we turn to the question of attainability of δp(d1, d2).

Theorem 2.11. For p > 2, δp(d1, d2) is not attained for all d1 �= d2.

Proof. Assume by negation that there exist maps u1 ∈ Ed1 and u2 ∈ Ed2 suchthat

‖∇ (u2 − u1)‖Lp(S2) = δp(d1, d2).

We may assume, without loss of generality, that d2 �= 0. As in the proof ofLemma 2.9 we can find a point x1 such that u2(x1) = −u1(x1), and bychanging the axes we may assume that u2(x1) = N . Denote

u2 − u1 = (v1, v2, v3).

Since d2 �= 0, there exists x2 ∈ S2 such that u2(x2) = S, implying thatv3(x2) ≤ 0. Since δp is attained, equalities hold in all the inequalities in (2.14).Thus,

‖∇v1‖Lp(S2) = ‖∇v2‖Lp(S2) = 0,

‖∇v3‖Lp(S2) =2

Cp,

minS2

v3 = 0, maxS2

v3 = 2.

Since v3(x1) = 2, we deduce that v1(x1) = v2(x1) = 0. Therefore, v1 ≡ 0,v2 ≡ 0 and

|uz1 (x)| = |uz

2 (x)| for all x ∈ S2, (2.30)

where uz1 and uz

2 denote the z-components of u1 and u2, respectively. Sinced2 �= 0 and v3 ≥ 0, we deduce from (2.30) that the set {v3 = 0} has positivemeasure (we must have v3(x) = 0 at points x where ux3

2 (x) ≤ 0). Hence, thedistribution function of v3 satisfies μ(0) < 4π, and the decreasing rearrange-ment of v3 satisfies v∗3(s) = 0 on the interval (μ(0), 4π). This contradictsRemark 2.3 for the function v3. �


3. Generalization to dimension n

In this short section we shall show how to generalize the results of Section 2to maps from Sn to Sn for every n ≥ 3. Set

Sn ={x ∈ R

n+1 : x21 + x2

2 + · · ·+ x2n+1 = 1

}.

For p ≥ n each u ∈ W 1,p(Sn, Sn) has a well-defined degree and we may writeagain

W 1,p (Sn, Sn) =⋃d∈Z

Ed =⋃d∈Z

{u : deg u = d}.

Indeed, for p > n the maps in W 1,p(Sn, Sn) are continuous, while in the lim-iting case p = n we refer to the VMO degree (see [5]). The distance betweenEd1 and Ed2 is defined naturally by

δpp (d1, d2) = inf

{∫Sn

|∇ (u1 − u2) |p : u1 ∈ Ed1 , u2 ∈ Ed2

}. (3.1)

Denote by

ωn =2π(n+1)/2

Γ(n+12

)the n-dimensional area of the unit n-sphere. The generalization of Theorem2.1 for higher-dimensional spheres was given by Cianchi in [6].

Theorem 3.1. Let p > n. If v ∈ W 1,p(Sn,R), then

maxSn

v −minSn

v ≤ C(n)p ‖∇v‖Lp(Sn) , (3.2)

where

C(n)p = (ωn−1)

−1/p

⎡⎣√πΓ(

p−n2p−2

)Γ(

2p−n−12p−2

)⎤⎦1−1/p

.

Inequality (3.2) is sharp.

Note that in the proof of Lemma 2.5 we used the fact that the dimensionof S2 is even. Thus, in the generalizations of Lemmas 2.5 and 2.6 to arbitraryn ≥ 2 we should take into account the parity of n. The proof of the nextlemma requires an obvious modification of the one of Lemma 2.5.

Lemma 3.2. Let d1, d2 ∈ Z, p ≥ n and let u1, u2 be two continuous maps inW 1,p(Sn, Sn) such that deg ui = di, i = 1, 2. Then, there is a point x ∈ Sn

such that u2(x) = u1(x) in the following cases:

(i) if n is even and d1 �= −d2,(ii) if n is odd and d1 �= d2.

Next we state a generalization of Lemma 2.6.

Lemma 3.3. For p ≥ n we have

(i) if n is even and d1 �= −d2, then δp(d + k, d2 + k) ≤ δp(d1, d2) for allk ∈ Z;

(ii) if n is odd, then δp(d1 + k, d2 + k) = δp(d1, d2) for all k ∈ Z.


Sketch of the proof. We begin with some notation. On the n-dimensionalsphere

Sn ={x ∈ R

n+1 : x21 + x2

2 + · · ·+ x2n+1 = 1

}define the spherical coordinates ϕi ∈ [0, π], i = 1, 2, . . . , n−1, and θ ∈ [0, 2π],where φi denotes the angle between x and ei+2. Thus,

x1 = cos θ sinϕ1 · · · sinϕn−1,

x2 = sin θ sinϕ1 · · · sinϕn−1,

x3 = cosϕ1 sinϕ2 · · · sinϕn−1,

...

xn = cosϕn−2 sinϕn−1,

xn+1 = cosϕn−1.

(i) Take any u1 ∈ Ed1 , u2 ∈ Ed2 that can be both assumed smooth,without loss of generality. Since d1 �= −d2, Lemma 3.2(i) implies that thereis a point x ∈ Sn such that u1(x) = u2(x). We may assume, without lossof generality, that u1(S) = u2(S) = S. For any small ε > 0 define the maps

ui = u(ε)i , i = 1, 2, on Sn by generalizing the definition in (2.10) as follows:

ui (ϕ1, ϕ2, . . . , ϕn−2, ϕn−1, θ)

=

⎧⎨⎩ui

(ϕ1, ϕ2, . . . , ϕn−2,

π

π − εϕn−1, θ

), ϕn−1 ∈ [0, π − ε],

(v1, v2, . . . , vn+1), ϕn−1 ∈ (π − ε, π],

(3.3)

where vj = vj(ϕ1, ϕ2, . . . , ϕn−1, θ), j = 1, 2, . . . , n+ 1, are defined by

v1 = cos (−kθ) sinϕ1 · · · sinϕn−2 sin[πε(π − ϕn−1)

],

v2 = sin (−kθ) sinϕ1 · · · sinϕn−2 sin[πε(π − ϕn−1)

],

v3 = cosϕ1 sinϕ2 · · · sinϕn−2 sin[πε(π − ϕn−1)

],

...

vn = cosϕn−2 sin[πε(π − ϕn−1)

],

vn+1 = cos[πε(π − ϕn−1)

].

Hence ui ∈ Edi+k, i = 1, 2, and a direct computation, as in the proof ofLemma 2.6, yields

limε→0

∫Sn

|∇ (u2 − u1) |p ≤∫Sn

|∇ (u2 − u1) |p.

The result follows since ui can be chosen arbitrarily in Edi .(ii) Clearly, we may assume that d1 �= d2. By Lemma 3.2(ii) there is

a point x ∈ Sn such that u1(x) = u2(x). As in the proof of (i) we get


δp(d1 + k, d2 + k) ≤ δp(d1, d2) for all k ∈ Z. Since d1 + k �= d2 + k, we canapply again the proof of (i) to obtain

δp(d1 + k − k, d2 + k − k) ≤ δp(d1 + k, d2 + k). �

Our main result for general n ≥ 2, generalizing Theorems 2.7, 2.10and 2.11, is the following.

Theorem 3.4. The distance between homotopy classes in W 1,p(Sn, Sn), p ≥ n,satisfies

(i) δn(d1, d2) = 0 for every d1, d2 ∈ Z;(ii) if p > n, then δp(d1, d2) =

2

C(n)p

for every d1 �= d2 and δp(d1, d2) is not

attained.

Sketch of the proof. (i) It is enough to show that δn(0,m) = 0 for any m,and then apply Lemma 3.3 to get the result for any pair d1, d2. As in theproof of Lemma 3.3 above, we slightly modify the construction in the proofof Theorem 2.7 by letting only the θ and ϕn−1 coordinates to be “active.”For any small ε > 0 define the functions

Φi = Φ(ε)i : [0, π] → [0, π], i = 1, 2,

by (2.11). It is easy to verify that

limε→0

∫ π

ε

(Φ′

i(ϕ))n

sinn−1 ϕdϕ = 0. (3.4)

Using these functions define the maps ui = u(ε)i , i = 1, 2, from Sn to Sn by

ui =(v(i)1 , v

(i)2 , . . . , v

(i)n+1

),

where v(i)j = v

(i)j (ϕ1, ϕ2, . . . , ϕn−1, θ) are defined by

v(i)1 = cos (mθ) sinϕ1 · · · sinϕn−2 sinΦi(ϕn−1),

v(i)2 = sin (mθ) sinϕ1 · · · sinϕn−2 sinΦi(ϕn−1),

v(i)3 = cosϕ1 sinϕ2 · · · sinϕn−2 sinΦi(ϕn−1),

...

v(i)n = cosϕn−2 sinΦi(ϕn−1),

v(i)n+1 = cosΦi(ϕn−1).

Using (3.4) we can easily verify that

limε→0

∫Sn

∣∣∣∇(u(ε)1 − u

(ε)2

)∣∣∣n = 0,

and the result of (i) follows.(ii) For the proof of the lower bound take any u1 ∈ Ed1 , u2 ∈ Ed2 . Since

d1 �= d2, Lemma 3.2 applied to u1 and −u2 implies that there is a point


x1 ∈ Sn such that u2(x1) = −u1(x1) (alternatively, we can see directly thatsuch a point exists because otherwise the map I : [0, 1]× Sn → Sn given by

I (t, x) =tu1 (x) + (1− t)u2 (x)

‖tu1 (x) + (1− t)u2 (x)‖would be a homotopy between u1 and u2). The rest of the proof is the sameas in the proof of Lemma 2.9. The proof that δp(d1, d2) is not attained usesthe same argument as in the proof of Theorem 2.11. �

Acknowledgment

The research of I. Shafrir was partially supported by the Technion V.P.R.Fund.

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Shay LeviDepartment of MathematicsTechnion - Israel Institute of Technology (IIT)Haifa 32000Israele-mail: [email protected]

Itai ShafrirDepartment of MathematicsTechnion - Israel Institute of Technology (IIT)Haifa 32000Israele-mail: [email protected]

on the distance between homotopy classes of maps between spheres

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