On the distance between homotopy classes of maps between spheres

On the distance between homotopy classes of maps between spheres Shay Levi and Itai Shafrir February 18, 214 Department of Mathematics, Technion - I.I.T., 32 Haifa, ISRAEL Dedicated with great respect to Haim Brezis on the occasion of his 7th birthday Abstract Certain Sobolev spaces of maps between manifolds can be written as a disjoint union of homotopy classes. Rubinstein and Shafrir computed the distance between homotopy classes in the spaces W 1,p S 1, S 1 ), for different values of p, and in the space W 1,2 Ω, S 1 ), for certain multiply connected two dimensional domains Ω. We generalize some of these results to higher dimensions. Somewhat surprisingly we find that in W 1,p S 2, S 2 ), with p > 2, the distance between any two distinct homotopy classes equals a universal positive constant cp). A similar result holds in W 1,p S n, S n ), for any n 2 and p > n. 1 Introduction Sobolev spaces of maps from a domain or a manifold with values in spheres appear naturally in Geometry and Analysis, especially in the study of harmonic maps. Motivation to study such maps comes from several areas of Physics like Liquid Crystals and Superconductivity. In general, decomposition of the relevant space W 1,p D, S n ) where D is either a domain or a manifold) into homotopy classes is a very subtle issue see [2, 3, 6, 7, 8, 9]). In some cases a decomposition of the form W 1,p D, S n ) = E d, 1.1) d of the space into a disjoint union of homotopy classes, holds, where d is either an integer or a vector of integers. Such partitions for Sobolev spaces of mappings between two Riemannian manifolds were developed by B. White [18]). The existence of such a partition for maps from a 1

two dimensional disk to S 2 for p = 2) was proved by Brezis and Coron who used it to establish existence of certain harmonic maps. Rubinstein and Sternberg [13] used such a partition for maps from the solid torus to S 1 to explain persistent currents in Superconductivity and in a later work, with Kim [1], to predict new structures in Liquid Crystals here again p = 2). The partition 1.1) where D is a multiply connected domain, m = 1 and p = 2, arises naturally in the study by Bethuel, Brezis and Hélein [1] of minimizers of Ginzburg-Landau type energy in this case d is a vector of length n, where n is the number of holes). Usually partitions like 1.1) are used to prove existence of non-trivial p-harmonic maps, as minimizers of the p-energy within a specific homotopy classes. The study of the distance between two distinct homotopy classes seems to be initiated by Rubinstein and Shafrir in [12]. They considered two classes of maps. The first is H 1 S 1, S 1 ) = W 1,2 S 1, S 1 ) = E d = : deg u = d}, d Z d Z{u where for any two integers d 1 d 2 the distance δd 1, d 2 ) between the homotopy classes E d1 E d2 is defined by { } δ 2 d 1, d 2 ) = inf u 1 u 2 ) 2 : u 1 E d1, u 2 E d2. 1.2) S 1 Rubinstein and Shafrir found an explicit formula for δ d 1, d 2 ), namely and δ 2 d 1, d 2 ) = 8d 2 d 1 ) 2. 1.3) π They also proved analogous formula for different values of p. Here we study the distance between homotopy classes of self-maps of spheres in higher dimension. Since the case of maps from S n to S n turns out to be essentially the same for any n 2, we restrict ourselves below to the case n = 2 see Section 3 for details on the n-dimensional case). A well-defined notion of degree for maps in W 1,p S 2, S 2) exists only for p 2 see Section 2 for details). We then have W 1,p S 2, S 2) = d Z E d = d Z {u : deg u = d}. The distance between E d1 and E d2 is defined by { } δp p d 1, d 2 ) = inf u 1 u 2 ) p : u 1 E d1, u 2 E d2. S 2 We compute explicitly δ p d 1, d 2 ), and somewhat surprisingly, the results are quite different from those of [12] for the case n = 1. First, in the case p = 2 we found in Theorem 2 that δ 2 d 1, d 2 ) = for every d 1, d 2 Z. Second, when p > 2, for every pair d 1 d 2, the distance δ p d 1, d 2 ) equals a 2

fixed positive value independently of the degrees) that is given explicitly by 2 C p, where C p = 2π) 1/p π Γ Γ p 2 2p 2 2p 3 2p 2 ) ) 1 1/p see Theorem 3). The constant C p arises as the best constant in a Sobolev type inequality on two dimensional spheres, which is due to Talenti [17]. A brief explanation for the value 2 C p goes as follows. It is not difficult to see that for any two maps, u 1 E d1 and u 2 E d2, the scalar) function v = u 2 u 1 must take both the value and 2 somewhere we assume here for simplicity that d 2 d 1 ). Then, Talenti s inequality applied to v yields, u 1 u 2 ) p v p max v min S2 S2 v = 2. S 2 S C 2 p C p This is essentially the proof of the lower bound of Theorem 3. The proof of the upper bound uses an explicit construction based on the profile of the optimal function in Talenti s inequality. The main difference from the case n = 1 is explained by the extra dimension, that allows us to construct maps possessing k-axial symmetry, i.e., maps of the form uϕ, θ) = sin Φϕ) coskθ), sin Φϕ) sinkθ), cos Φϕ)). The degrees of these maps are the result of rotations around the z-axis that do not affect the distance between the maps, see the proof of Theorem 3 for details. The rather straightforward generalization of the above results to maps between higher dimensional spheres is given in Section 3. Remark 1. The M.Sc. thesis [11] also contains a generalization of the above results for the distance between homotopy classes of maps in W 1,p S 2, Σ ), where Σ = K is a surface which is the boundary of a convex body K R 2 of class C 2 + see [14]). There it is proved that for p > 2 we have δ p d 1, d 2 ) = W C p, where W is the width of the convex body K i.e., the minimal distance between two parallel planes bounding K, see [14] for details). In [11] one can find also generalization of the result of [12] for W 1,p S 1, S 1 ) to the case of the space W 1,p S 1, C), where the closed curve C is the boundary of convex body in R 2. Acknowledgment. The research of I.S. was partially supported by by the Technion V.P.R. Fund. 2 Maps from S 2 to S 2 Let S 2 = { x R 3 : x 2 1 + x 2 2 + x 2 3 = 1 } denote the unit sphere in R 3. Denote the north and south poles by N =,, 1) and S =,, 1). With a slight abuse of notation each v : S 2 R can be also viewed as a map from [, π] [, 2π] to R such that v, θ) and v π, θ) are independent 3

of θ and also v ϕ, ) = v ϕ, 2π) for all < ϕ < π. Hence we can also write any u : S 2 S 2 as u = v 1, v 2, v 3 ) where v i : [, π] [, 2π] R. Here θ longitude) and ϕ colatitude) are geographical coordinates on S 2. Thus x 1 = cos θ sin ϕ, x 2 = sin θ sin ϕ, x 3 = cos ϕ, where θ 2π, ϕ π. Note that H 2 dx) = sin ϕ dϕdθ where x runs over S 2 and H 2 denotes the Hausdorff 2- dimensional measure on S 2. We also have v i = u = vi ) 2 1 + sin ϕ ϕ v 1 2 + v 2 2 + v 3 2. ) 2 v i, i = 1, 2, 3, θ Note that for p > 2, W 1,p S 2, S 2) C 1 2/p S 2, S 2), so that each u W 1,p S 2, S 2) has a well-defined degree. For p = 2 the degree is still well-defined thanks to the density of C 1 S 2, S 2) in the Sobolev space H 1 S 2, S 2) [15]). This is a special case of the VMO degree that was developed by Brezis and Nirenberg in [4]. Thus, for p 2 we may write W 1,p S 2, S 2) = E d = : deg u = d}. d Z d Z{u The distance between E d1 and E d2 is defined by { } δp p d 1, d 2 ) = inf u 1 u 2 ) p : u 1 E d1, u 2 E d2. 2.1) S 2 In this Section we will compute δ p d 1, d 2 ) for any p 2 and d 1, d 2 Z. Interestingly, in contrast with the case of maps between S 1 to S 1 studied in [12], we will show that in dimension two and higher) δ p d 1, d 2 ) is the same for any d 1 d 2. It turns out that the computation of the distance between the homotopy classes is related to the best constant in a certain Sobolev type inequality on the sphere. We recall below the relevant result which is due to Talenti [17]). Theorem 1. Let p > 2. If v W 1,p S 2, R ), then max v min v C p v L S 2 S 2 p S 2 ), 2.2) 4

where C p = 2π) 1/p π Γ Γ p 2 2p 2 2p 3 2p 2 ) ) 1 1/p. Inequality 2.2) is sharp. Let v be a smooth map from S 2 to R. Without loss of generality we assume v. Let V t) be the level sets of v, V t) := { x S 2 : v x) > t }. The distribution function of v is given by µ t) = H 2 V t)). Let v s) = χ [,µt)] s) dt, denote the decreasing rearrangement of v in the sense of Hardy and Littlewood - i.e., the decreasing right-continuous map from [, 4π] into [, ) which is equidistributed with v. It can be shown that v is locally Lipschitz continuous. The spherical symmetric rearrangement v of v is a function from S 2 to [, ) which is equidistributed with v and whose level sets are concentric spherical caps. Hence, if ϕ is the colatitude of x and B ϕ) is the area of the cap which is intercepted on S 2 by a circular cone having its vertex in the center of S 2 and aperture 2ϕ, then v x) = v B ϕ)) = v 4π sin 2 ϕ 2 ). 2.3) An important property of v is that it does not increase the L p - norm of the gradient. In fact, the following Lemma is a special case of a symmetrization theorem from [16]: Lemma 1. If p 1 then v L p S 2 ) 4π [ ] v = L [s 4π s)] p/2 dv p p S 2 ) ds s) ds We give below a sketch of proof for Theorem 1. For convenience, we assume v. By the definition of v, max S 2 v = v ), min S 2 Hence, Hölder inequality gives max S 2 v min v S 2 v = v 4π ) and max v min v = S 2 S 2 4π [s 4π s)] p/2p 1) ds 1 1/p 4π 1/p [ ] dv ds s) ds. 4π [ ] [s 4π s)] p/2 dv p ds s) ds. 1/p. 2.4) 5

Inequality 2.2) follows from Lemma 1 since the first term on the R.H.S of 2.4) equals C p. Remark 2. An inspection shows that equality holds in 2.2) if and only if v satisfies v s) = c 1 1 [t 1 t)] p/2p 1) dt + c 2, 2.5) s/4π for some constants c 1 and c 2. Using 2.2) we deduce the following corollary. Corollary 1. Equality holds in 2.2) for a radially symmetric function v : S 2 R if and only if v satisfies v x) = c 1 where ϕ is the colatitude of x and c 1, c 2 are constants. For p > 2 we define the function f = f p) : [, π] [, 1] by f ϕ) = π π ϕ sin t) 1/p 1) dt + c 2, 2.6) ϕ 1 sin t) dt) 1/p 1) sin t) 1/p 1) dt. 2.7) Let v x) := f ϕ), where x S 2 and ϕ is the colatitude of x. The function f will be useful later in the proof of the upper bound for δ p. We have v N) = f ) = and v S) = f π) = 1. From Corollary 1 equality holds in 2.2) for the function v. Thus, π v Lp S 2 ) = 2π f ϕ) ) p sin ϕ dϕ = C 1 p. 2.8) Lemma 2. For d 1, d 2 Z, d 1 d 2, p 2, let u 1 and u 2 be two continuous maps in W 1,p S 2, S 2) with deg u i = d i, i = 1, 2. Then, there is a point x S 2 such that u 2 x) = u 1 x). Proof. We claim that there exist x S 2 and t, 1) such that t u 1 x) + 1 t ) u 2 x)) =. 2.9) Indeed, otherwise, the map I : [, 1] S 2 S 2 given by I t, x) = tu 1 x) + 1 t) u 2 x)) tu 1 x) + 1 t) u 2 x)), would be a homotopy between u 1 and u 2. Since dim S 2) is even, deg u 2 ) = d 2. Hence d 1 = 6

d 2, contradicting our initial assumption. From 2.9) we get t u 1 x) = 1 t ) u 2 x)). Therefore, t = 1 2 and the result follows from 2.9). Note that the continuity assumption is needed only for p = 2 since for p > 2 every u W 1,p S 2, S 2) has a continuous representative. Lemma 3. If d 1 d 2, p 2, then δ p d 1 + k, d 2 + k) δ p d 1, d 2 ), k Z. Proof. Take any u 1 E d1, u 2 E d2. We may assume without loss of generality that u 1, u 2 are smooth maps. Since d 1 d 2, by Lemma 2 there is a point x S 2 such that u 1 x) = u 2 x). We may choose the coordinates axes in the domain and in the range of the maps u i such that x = S and u 1 x) = u 2 x) = S. Thus, u 1 S) = u 2 S) = S. For a small > define the maps ũ i = ũ ) i, i = 1, 2, on S 2 by ) π u i π ũ i ϕ, θ) = ϕ, θ [ sin π π ϕ)] cos kθ), sin [ π π ϕ)] sin kθ), cos [ π π ϕ)]) ϕ The maps ũ i belong to E di+k, i = 1, 2, and satisfy S 2 ũ 2 ũ 1 ) p = 2π dθ = π π π 2π π u 2 π ϕ, θ) u π 1 π ϕ, θ)) p sin ϕ dϕ dθ π u 2 ϕ, θ) u 1 ϕ, θ)) p sin π π Since the maps u 1, u 2 are smooth and sin π π ϕ) sin ϕ) + for ϕ π, we get ũ 2 ũ 1 ) p 2π 2 M p + u 2 u 1 ) p, S 2 S 2 where M := u 2 u 1 ) L S 2 ) <. Hence lim ũ 2 ũ 1 ) p S 2 u 2 u 1 ) p. S 2 The result follows since u i can be chosen arbitrarily in E di. Now we treat the case p = 2. Theorem 2. For every d 1, d 2 Z we have δ 2 d 1, d 2 ) =. ϕ [, π ], π, π]. 2.1) ϕ ) d ϕ. Proof. We begin with a brief description of our strategy. It would be enough to deal with the case where one of the degrees is zero, and then use Lemma 3 to deduce the general case. We shall construct two maps, both with m axial symmetry, one of degree zero and the second one of degree m = d 2 d 1. On a small sphere of order on S 2, centered at the north pole, the two maps are identical, each covering the upper hemisphere. On the remaining much larger) part of S 2 one of 7

the maps "goes back" from the equator to the north pole, so its degree is zero. On the other hand, the values taken by the second map on that part of S 2 are just the reflection w.r.t. the xy plane of the values taken by the first map. The degree of the second map equals therefore to m and the difference between the two maps has a nonzero component only in the z-direction. Using the fact that a point has zero 2 capacity in dimension two, we can arrange to have arbitrarily small energy contribution from that component. The detailed construction is given below. For any small > define the maps Φ ) i : [, π] [, π] by Φ i ϕ) = Φ ) i ϕ) = π 2 where i = 1, 2. A direct computation yields, We define the maps u i = u ) i π lim π 2 from S 2 to S 2 by ϕ ) ϕ [, ], i log ϕ log 1 1) ϕ, π], log π log 2.11) Φ i ϕ) ) 2 sin ϕ dϕ =. 2.12) u i ϕ, θ) = sin Φ i ϕ) sin mθ), sin Φ i ϕ) cos mθ), cos Φ i ϕ)), i = 1, 2, where ϕ π, θ 2π and m = d 2 d 1. Since u i < c, i = 1, 2, c is a positive constant, independent of ) the maps belong to W 1, S 2, S 2 ) and satisfy u 1 E d2 d 1 and u 2 E. We have Thus,,, ) ϕ [, ], u 2 u 1 =,, 2 cos Φ 2 ϕ)) ϕ, π]. S 2 u 2 u 1 ) 2 2 3 π From 2.12), 2.13) and Lemma 3 applied to and d 2 d 1 ) we get π δ 2 d 1, d 2 ) δ 2 d 2 d 1, ) =. Φ 2 ϕ) ) 2 sin ϕ dϕ. 2.13) Remark 3. The above theorem is analogous to a result from [4]see Lemma 6 and Remark 6 there) which states that the distance between homotopy classes in H 1/2 S 1, S 1 ) is always zero. Next we turn to the case p > 2. The lower-bound is given by the following lemma. 8

Lemma 4. If d 1 d 2 then for p > 2 we have: δ p d 1, d 2 ) 2 C p. Proof. Take any u 1 E d1, u 2 E d2. Since d 1 d 2 Lemma 2 applied to u 1 and u 2 implies that there is a point x 1 S 2 such that u 2 x 1 ) = u 1 x 1 ). We may assume W.l.o.g. that d 2. We choose the coordinates axes in the range so that u 2 x 1 ) = N. Since d 2, it follows that there is x 2 S 2 such that u 2 x 2 ) = S. Let v 3 : S 2 R be the third component of u 2 u 1. The function v 3 belongs to W 1,p S 2, R ) and satisfies v 3 x 1 ) = 2, v 3 x 2 ). From Theorem 1 we get u 2 u 1 ) L p S 2 ) v 3 L p S 2 ) max S 2 v 3 min S 2 v 3 C p 2 C p. 2.14) Next we prove the main result of this Section. Theorem 3. If d 1 d 2 then for p > 2 we have: δ p d 1, d 2 ) = 2 C p. 2.15) Proof. Thanks to the lower bound of Lemma 4, it is enough to prove that the following upper bound holds, δ p d 1, d 2 ) 2, 2.16) C p for all d 1 d 2. The construction of pairs of maps that realize 2.16) in the limit shares some similarities with the construction used in the proof of Theorem 2. Indeed, once again it is enough to consider the case where one of the degrees is zero and both maps are taken to be equal on a small sphere, whose image by each of the maps is the upper hemisphere. On the remaining part of S 2 one map the one of zero degree) covers again the upper hemisphere while the second map is just the reflection w.r.t. the xy-plane of the first one. This time however we arrange so that the difference between the two maps which is in the direction of the z-axis) is equal approximately to f see 2.7)-2.8)) which is the profile of the minimizer in Theorem 1. For any small > consider the following approximation F = F ) : [, π] [, 1] of the function f defined in 2.7) ): F ) ϕ) = J ɛ f ϕ)), 2.17) 9

where the map J : [f ), 1] [, 1] is a C m+1 -map satisfying the following properties: c 2 m < J 1) = 1, J 1) = = J m) 1) =, J f )) =, J f )) = = J m) f )) =, J s) = s, 2f ) s 1, J s) < c, f ) s 1, s) < c1, f m ) s 1, J m+1) s) < c1, 1 m 2 < s 1, J m+1) 2.18) for some constants c, c 1, c 2 independent of ) and m an integer that satisfies m 1 + 2 p 2. In fact, we need to construct a function J ɛ on the two intervals [f ), 2f )] and [1, 1] "connecting" the values J f )) = and J 1 ) = 1 to the values J s) = s on the interval [2f ), 1 ], that satisfies the estimates in 2.18). This requires a change of order f ) for J on the interval [f ), 2f )] and of order on the interval [1, 1]. An appropriate J will then have a derivative of order O1). But now the change of order 1 between J f )) and J 2f )) and between J 1 ) = 1 and J 1) = requires J 1 of order max f, 1 ) ) = 1 since f ) p 2 k) p 1 ). Similarly, for higher order derivatives we will get J s) c. Since k 1 what we are requiring is just interpolation between certain given values of the function and some of its derivatives at two pairs of points, it is clear that we can even take a polynomial for J. Using Taylor formula around s = f ) and s = 1 in conjunction with 2.18) yields for s [f ), 1], J s) c m s f )) m, 2.19) J s) c m 1 s)m, 2.2) where c is a constant independent of. For s [ 1 2, 1] we use again Taylor formula around 1 to obtain, implying Define the functions Φ i = J s) = 1 + J m+1) θ) m + 1)! s 1)m+1, θ s, 1), J s) 1 J s)) 1/2 c 1 m/2 s)m 1)/2, s [1, 1]. 2.21) 2 Φ ) i : [, π] [, π], i = 1, 2, by π 2ϕ [, ], π 2ϕ [, ], Φ 1 ϕ) =, Φ2 ϕ) =. 2.22) π arccos F ϕ), π], arccos F ϕ), π], 1

Then, define the maps u i = u ) i : S 2 S 2, for i = 1, 2, by u i ϕ, θ) = sin Φ i ϕ) cos kθ), sin Φ i ϕ) sin kθ), cos Φ ) i ϕ), 2.23) where ϕ π, θ 2π and k = d 2 d 1. Next we prove that u 1 E d2 d 1 and u 2 E. Computation of the degrees of u 1 and u 2 gives deg u i = 1 u i u iϕ u iθ = 2πk π Φ i ϕ) sin 4π S 4π Φ i ϕ) dϕ 2 = k [ cos 2 Φ i ) cos Φ ] k i = 1, i π) = i = 2. We will prove that u i W 1, S 2, S 2) W 1,p S 2, S 2) by showing that the derivatives of F and 1 F 2 are bounded. Obviously it is enough to consider the intervals [, + ] and [π, π] for some >. Let be such that f π ) > 1 2. On the interval [, + ] we have From 2.19) we get f ϕ) = sin ϕ) 1/p 1) cϕ 1/p 1), ϕ f ϕ) = c sin t) 1/p 1) dt cϕ p 2)/p 1). F ϕ) = J ɛ f ϕ)) f ϕ) c m f ϕ) f )) m f ϕ) c m f ϕ)) m f ϕ) In the last inequality we used that For the function 1 F 2 we simply have c m ϕmp 2)/p 1)) 1/p 1) m p 2 p 1 1 p 1 = p 2 [ m 1 ] >. p 1 p 2 1 F 2) ϕ) = F F cf c 1 F 2 m. On the interval [π, π] the functions f and f satisfy c m. f ϕ) = sin π ϕ)) 1/p 1) c π ϕ) 1/p 1), π 1 f ϕ) = c sin t) 1/p 1) dt c π ϕ) p 2)/p 1). π ϕ 11

Hence, using 2.2) F ϕ) = J ɛ f ϕ)) f ϕ) c m 1 f ϕ)) m f ϕ) c m π ϕ)m[p 2)/p 1)] 1/p 1) c m. Since f π ) > 1 2, from 2.21) we get ) 1 F 2 ϕ) J c f 1 J ) 1/2 c 1 f m/2 ϕ)) m 1)/2 f ϕ) c π ϕ)[m 1)/2] [p 2)/p 1)] 1/p 1) c m/2. m/2 In the last inequality we used that m 1 2 Our next step will be to prove that p 2 p 1 1 p 1 = p 2 [ m 1 2 ]. 2 p 1) p 2 ) p 2 lim u 2 u 1 ) p. 2.24) S C 2 p Note that,, ) ϕ [, ], u 2 u 1 =,, 2F ϕ)) ϕ, π]. Therefore, u 2 u 1 ) p 2 p 2π S 2 Set = 2f ). Note that π F ϕ) ) p sin ϕ dϕ. 2.25) c 1 1/p 1), f [, π ]) [2f ), 1 ]. 2.26) Since + F ϕ) ) + p sin ϕ dϕ c f ϕ) ) + p sin ϕ dϕ c ϕ p/p 1) ϕ dϕ c 1 1/p 1) c [1 1/p 1)]2, 12

we have Moreover, lim + F ϕ) ) p sin ϕ dϕ =. 2.27) π π F ϕ) ) π p sin ϕ dϕ c π f ϕ) ) π p sin ϕ dϕ c = c 1 1/p 1) c [1 1/p 1)]2, π π ϕ) p/p 1) π ϕ) dϕ implying that lim π π F ϕ) ) p sin ϕ dϕ =. 2.28) Finally, on [ +, π ], we have by 2.26) and 2.18) that F ϕ) = f ϕ). From 2.8) we obtain π + F ϕ) ) π p sin ϕ dϕ f ϕ) ) p sin ϕ dϕ = 1 2π C p) p. 2.29) From 2.25) and 2.27)-2.29) we deduce 2.24). Hence δ p p d 2 d 1, ) 2 C p ) p. Lemma 3 applied to and d 2 d 1 ) yields 2.16). Next we turn to the question of attainability of δ p d 1, d 2 ). Theorem 4. For p > 2, δ p d 1, d 2 ) is not attained for all d 1 d 2. Proof. Assume by negation that there exist maps u 1 E d1 and u 2 E d2 such that u 2 u 1 ) Lp S 2 ) = δ p d 1, d 2 ). We may assume without loss of generality that d 2. As in the proof of Lemma 4 we can find a point x 1 such that u 2 x 1 ) = u 1 x 1 ), and by changing the axes we may assume that u 2 x 1 ) = N. Denote u 2 u 1 = v 1, v 2, v 3). Since d 2, there exists x 2 S 2 such that u 2 x 2 ) = S, implying that v 3 x 2 ). Since δ p is attained, equalities hold in all the inequalities in 2.14). Thus, v 1 Lp S 2 ) = v 2 Lp S 2 ) =, v 3 Lp S 2 ) = 2 C p and min S 2 v 3 =, max S 2 v 3 = 2. Since v 3 x 1 ) = 2 we deduce that v 1 x 1 ) = v 2 x 1 ) =. Therefore, v 1, v 2 and u z 1 x) = u z 2 x) x S 2, 2.3) where u z 1 and u z 2 denote the z-components of u 1 and u 2, respectively. Since d 2 and v 3, we deduce from 2.3) that the set {v 3 = } has positive measure we must have v 3 x) = at points x where u x3 2 x) ). Hence, the distribution function of v 3 satisfies µ ) < 4π, and the decreasing 13

rearrangement of v 3 satisfies v 3 s) = on the interval µ ), 4π). This contradicts Remark 2 for the function v 3. 3 Generalization to dimension n In this short section we shall show how to generalize the results of Section 2 to maps from S n to S n, for every n 3. Set S n = { x R n+1 : x 2 1 + x 2 2 +... + x 2 n+1 = 1 }. For p n each u W 1,p S n, S n ) has a well defined degree and we may write again W 1,p S n, S n ) = E d = : deg u = d}. d Z d Z{u Indeed, for p > n the maps in W 1,p S n, S n ) are continuous, while in the limiting case p = n we refer to the VMO degree see [4]). The distance between E d1 and E d2 is defined naturally by { } δp p d 1, d 2 ) = inf u 1 u 2 ) p : u 1 E d1, u 2 E d2. 3.1) S n Denote by ω n = 2πn+1)/2 Γ n+1 2 ) the n-dimensional area of the unit n-sphere. Theorem 1 for higher dimensional spheres was given by Cianchi in [5]. The generalization of Theorem 5. Let p > n. If v W 1,p S n, R), then max S n v min v S Cn) n p v L p S n ), 3.2) where C n) p = ω n 1 ) 1/p π Γ Γ p n ) 2p 2 2p n 1 2p 2 ) 1 1/p. Inequality 3.2) is sharp. Note that in the proof of Lemma 2 we used the fact that the dimension of S 2 is even. Thus, in the generalizations of Lemma 2 and Lemma 3 to arbitrary n 2 we should take into account the parity of n. The proof of the next Lemma requires an obvious modification of the one of Lemma 2. Lemma 5. Let d 1, d 2 Z, p n and u 1, u 2 two continuous maps in W 1,p S n, S n ) such that deg u i = d i, i = 1, 2. Then, there is a point x S n such that u 2 x) = u 1 x) in the following cases: i) If n is even and d 1 d 2. ii) If n is odd and d 1 d 2. 14

Next we state a generalization of Lemma 3. Lemma 6. For p n we have: i) If n is even and d 1 d 2 then δ p d + k, d 2 + k) δ p d 1, d 2 ), k Z. ii) If n is odd then δ p d 1 + k, d 2 + k) = δ p d 1, d 2 ), k Z. Sketch of Proof. We start with some notation. On the n-dimensional sphere S n = { x R n+1 : x 2 1 + x 2 2 +... + x 2 n+1 = 1 } define the spherical coordinates ϕ i [, π], i = 1, 2,..., n 1 and θ [, 2π], where φ i denotes the angle between x and e i+2. Thus, x 1 = cos θ sin ϕ 1 sin ϕ n 1, x 2 = sin θ sin ϕ 1 sin ϕ n 1, x 3 = cos ϕ 1 sin ϕ 2 sin ϕ n 1,. x n = cos ϕ n 2 sin ϕ n 1, x n+1 = cos ϕ n 1. i) Take any u 1 E d1, u 2 E d2 that can be both assumed smooth, without loss of generality. Since d 1 d 2, Lemma 5i) implies that there is a point x S n such that u 1 x) = u 2 x). We may assume without loss of generality that u 1 S) = u 2 S) = S. For any small > define the maps ũ i = ũ ) i, i = 1, 2, on S n by generalizing the definition in 2.1) as follows: ) π u i ϕ 1, ϕ 2,..., ϕ n 2, π ũ i ϕ 1, ϕ 2,..., ϕ n 2, ϕ n 1, θ) = ϕ n 1, θ ϕ n 1 [, π ], v 1, v 2,..., v n+1 ) ϕ n 1 π, π], 3.3) where v j = v j ϕ 1, ϕ 2,..., ϕ n 1, θ), j = 1, 2,..., n + 1, are defined by [ π ] v 1 = cos kθ) sin ϕ 1 sin ϕ n 2 sin π ϕ n 1), [ π ] v 2 = sin kθ) sin ϕ 1 sin ϕ n 2 sin π ϕ n 1), [ π ] v 3 = cos ϕ 1 sin ϕ 2 sin ϕ n 2 sin π ϕ n 1),. [ π ] v n = cos ϕ n 2 sin π ϕ n 1), [ π ] v n+1 = cos π ϕ n 1). 15

Hence ũ i E di+k, i = 1, 2, and a direct computation, as in the proof of Lemma 3, yields lim ũ 2 ũ 1 ) p S n u 2 u 1 ) p. S n The result follows since u i can be chosen arbitrarily in E di. ii) Clearly, we may assume that d 1 d 2. By Lemma 5 ii) there is a point x S n such that u 1 x) = u 2 x). As in the proof of i) we get δ p d 1 + k, d 2 + k) δ p d 1, d 2 ), k Z. Since d 1 + k d 2 + k, we can apply again the proof of i) to obtain δ p d 1 + k k, d 2 + k k) δ p d 1 + k, d 2 + k). Our main result for general n 2, generalizing Theorem 2, Theorem 3 and Theorem 4 is: Theorem 6. The distance between homotopy classes in the space W 1,p S n, S n ) p n), satisfies: i) δ n d 1, d 2 ) = for every d 1, d 2 Z. ii) If p > n then δ p d 1, d 2 ) = 2 C n) p for every d 1 d 2 and δ p d 1, d 2 ) is not attained. Sketch of Proof. i) It is enough to show that δ n, m) = for any m, and then apply Lemma 6 to get the result for any pair d 1, d 2. As in the proof of Lemma 6 above, we slightly modify the construction in the proof of Theorem 2 by letting only the θ and ϕ n 1 coordinates to be active. For any small > define the functions Φ i = Φ ) i : [, π] [, π], i = 1, 2, by 2.11). It is easy to verify that π Φ i ϕ) ) n sin n 1 ϕ dϕ =. 3.4) lim Using these functions define the maps u i = u ) i, i = 1, 2, from S n to S n by u i = v i) 1, vi) 2,..., vi) where v i) j = v i) j ϕ 1, ϕ 2,..., ϕ n 1, θ) are defined by v i) 1 = cos mθ) sin ϕ 1 sin ϕ n 2 sin Φ i ϕ n 1 ), v i) 2 = sin mθ) sin ϕ 1 sin ϕ n 2 sin Φ i ϕ n 1 ), v i) 3 = cos ϕ 1 sin ϕ 2 sin ϕ n 2 sin Φ i ϕ n 1 ),. v i) n = cos ϕ n 2 sin Φ i ϕ n 1 ), v i) n+1 = cos Φ i ϕ n 1 ). Using 3.4) we can easy verify that lim u ) S n 1 u ) 2 ) n =, and the result of i) follows. ii) For the proof of the lower bound take any u 1 E d1, u 2 E d2. Since d 1 d 2 Lemma 5 applied to u 1 and u 2 implies that there is a point x 1 S n such that u 2 x 1 ) = u 1 x 1 ) alternatively, we can see directly that such a point exists because otherwise the map I : [, 1] S n S n given by I t, x) = tu 1 x) + 1 t) u 2 x) tu 1 x) + 1 t) u 2 x) 16 n+1 )

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