D. On the homotopy role of the Lq-norm of the differential

1)/

on H for q

<

N. Let us look at the maps

f

with

IIDf

I

HllLq

~ c for some q < N. Now, we do not have the full finiteness-contractibility result but one obtains a weaker homotopy conclusion by restricting

f

to the k-skeleton Vk C V brought to a sufficiently general position with respect to

f.

Then the bound

IIDf I HllLq

^~ c implies a similar bound on V^{k ,} namely

where c' = const ·c, where the "tangent bundle" T(Vk) is understood as the set of the vectors tangent to the (smooth!) simplices in V^k and where the Lq-norm of the differential V

f

on the intersection T(Vk) n H is obtained by integration over V k. Now, if the intersection T(Vk) n H induces a C-C structure of formal Hausdorff dimension N' ~ q on each k-face of Vk , then the bound (*)' has non-trivial homotopy effect on

f' = f I

Vk and hence on

f·

Namely, if

IIVf I HliLq

is sufficiently small, then the restriction

f

I Vk is contractible whenever Vk can be stably brought into a position where dimHau (Vk, T(Vk) n H)) ~ q. Here the stability means the existence of a measure J-L on the space of embeddings (Le. positions) V^k in V having dimHau ~ q and such that the push-forward of the measure J-Lxdv^k to V is absolutely continuous with respect to Lebesgue (or equivalently C-C Hausdorff) measure in V.

If V is Riemannian (i.e. dim V

=

dimHau V) then the above stability is trivially satisfied with N' = k and so every smooth map

f :

V ^----+ W with

IIVfllLq

small is null-homotopic on some (and hence every) k-skeleton V^k C V with k ~ q. In fact the converse is also known to be true. Every smooth map

f :

V ^----+ W sending a small neighbourhood of V^k to a point can be composed with a suitable diffeotopy 'Pt : V ----+ V, 0 ~ t

<

^00,

such that

IIV f

^{0 'Pt}

IlL

_{q t---+(X)} ^----+ 0 for every q

<

k

+

^{1. (If V}

= sn

^and

V^O C V is the south pole, one uses the north pole south pole push for 'Pt. In general one uses such pushes in the cells in V - Vk . First every n-cell B is radially pushed from the center bo E B toward the boundary, so that in the limit for t ----+ 00 all of B - {bo} ^{goes to aBo Then one}

2.5 C-C SPACES SEEN FROM WITHIN 181

composes the above push toward V^{n - 1} with a similar push of a small neighbourhood of vn-1 toward V^{n - 2} and so on, see p.388 in [E-L] and references therein. Probably, a similar construction can be carried over for certain C-C manifolds, e.g. the contact ones.

Now, let us look at the maps f with IIDf

I

HllLq

<

^C with a possibly large c and observe that there are at most finitely many homotopy classes of restrictions of

f

to V^k in the following two cases, (i) q

>

N' and (ii) q = N' and 7r1 (W) acts trivially on 7rk(W), where N' is, as earlier, the minimal integer so that Vk can be made stably of formal Hausdorff dimension N' for the C-C metric associated to T(Vk)

n

H. For example, N' = k in the Riemannian case. If V is a contact C-C manifold, then N'

=

k for 2k

<

dim V and N'

=

k

+

^{1 for 2k ~} dim V as follows from the discussion in 3.4.B, and see 4.? for the general C-C case.

The space Fq,c' Let us look at the homotopy property of the space Fq,c of the maps f with

liD

f I HllLq ~ c. We have just seen that the zero-dimensional homotopy (i.e. connected components) of Fq,c are strongly affected by q and c but, probably, there is no additional link between the geometry of maps

f

(encoded into q and c) and higher homotopies of spaces of these f for q

<

N. Namely, if fa : V -* W is a family of smooth map parametrized by a compact polyhedron A :3 a such that each fa can be individually contracted to Fq,c then, conjecturally, the whole family can be continuously moved to Fq,c (or, possibly, to Fq,c^l for c' = c'(c, A)) in the case where q

<

N. This appears easy in the Riemannian case.

For example, the above diffeotopy 'Pt of V works for families of maps fa : V --t W which send a fixed (i.e. independent of a) skeleton V^k to a point. In general, if IIDfallLq is small for all a, we can only have fa almost constant on Yak depending on a. In fact we can make Yak constant in a on each simplex of a suitable subdivision of A and then 'Pa,t can be probably build using some induction on skeletons (or partition of unity) in A.

Exercise. Determine the homotopy structure of the space of maps

f :

8¹x[O,1] -* 8² with

IIDfllLq ~f

UIIDfllq)t

~

c for given c

>

^{0 and}

1 ~ q

<

2.

182 MIKHAEL GROMOV § 2 corresponding result in the C-C category remains conjectural. A closely related (apparently more global) question is that of the homotopy content of FfI and the homotopy structure of the inclusions Coo n FqH C FfI and Coo nF,;;c C F,;;c C FfI where F,;;c = {f ^E FqH

IIIDI I HllLq ::;;

c}, where the space FfI is given L;J -topology arizing from the norm

IIV f I HllLq

^via

an embedding of W into some Euclidean space. One asks in this regard, for example, which homotopy invariants of smooth maps extends to

frH

and are FfI -continuous (or continuous in some weaker topology on Fq ).

2.5.E'. Examples of

F,JI

-non-density of smooth maps in

F,JI.

(7) corollary in 2.5.A applied to concentric spheres in the ball B

=

Bn.

Now let V be a smooth (topological) ball around the origin in a nilpo-tent Lie group with a one-parametric self-similarity such that each orbit of this self-similarity transversally meets sn-1

=

aV at a single point.

Then the radial projection fo : V ~ sn-1 along the orbits is in FfI for all q

<

N = dimHau V and neither this fo nor any

I

^{= cp 0} fo for a non-null-homotopic cp can be approximated by smooth maps in Ff!-l (under the standing assumption of (codim I)-stability of V).

7 Compare [Sh-Uhl], [Beth] and [HajASM].

2.5 C-C SPACES SEEN FROM WITHIN 183

2.5.F. Space

FJ!.

Let us look at the homotopy structure of a map

f

E FJ!. We start with the (well known) Riemannian case (where H =

T(V)) and observe that if f E F,JI then the restriction f I V' has IIVf I TV'IILq(VI)

<

⁰⁰ for a generic hypersurface V' in V and so for q

>

dim V-1 this restriction in continuous. However, the homotopy class of

f

I V' may jump under small perturbations of V' in V for q

<

^dim V. Let us show this does not happen for q

=

ⁿ

=

dim V. We observe that for each point v E V and every small positive c there is a sphere 88 around v of radius 8 in the interval c ::s; 8 ::s; 2c, such that the Ln-norm of

f

on this sphere 88 with the normalized metric (= dist /8) is bounded by the Ln-norm of

f

on the ball B2e (by integrating IIDflln over the annulas between 8e and 82e ).

It follows, that Diamf(88) is bounded in terms of

fB2E

IIVf(v)lIn dv, as we, in fact, have seen earlier and therefore for every c

>

0 one can cover V by balls of radii between c and 2c, such that every ball among these at most v = v( n) neighbours. We assume without loss of generality that the union ~ of the boundary spheres of these balls is connected and we partition V into the connected components of the complement V ^-~, say V = UUi , where the boundary of each Ui has Diamf(8Ui ) ::s; vDiamf(88 )

which uniformly (in i) goes to zero as c -+ O. Then one can regularize

f

by using some standard continuous extension of

f

I 8Ui for each i to all of Ui within the (small) ball of radius p

=

Diamf(8Ui ) in W. Furthermore, given a hypersurface V' C V, one can do the same to the (finer) partition into the connected components U: of V - (~ U V') as all Diamf(UD are necessarily small for small (now, depending also on V') c. Thus the (continuous) restriction

f

I V' : V' -+ W admits a continuous extension to all of V which, in particular, imply the contractibility of

f

I V' for small spheres in V (which are contractible in V). Then, obviously, the homotopy type of f I V'is invariant under the deformations of V' in V and also under homotopies of

f

in the space F!! for H = T(V).

The same reasoning applies to (codim I)-stable C-C manifolds V. For example, if V is a contact C-C manifold, then every map

f

E FJ!, N = n

+

1, has a well defined homotopy class of the restriction of

f

to the (n - I)-skeleton of V, provided n

=

dim V ~ 5. (It is unclear what happens for 3-dimensional contact manifolds.)

184 MIKHAEL GROMOV § 2

2.5.F'. Regularization of

F{:f

-maps. The above process of filling small Ui-holes by continuous maps (extending f I aUi ) allows us to ap-proximate every

f

E FfJ by a continuous map, say

f' :

V ^----7 W, such that any two such approximation to

I

are mutually homotopic. More precisely, we have the following proposition (which is well known in the Riemannian case and is due, I believe, to K. Uhlenbeck).

Let V be a compact (codim 1 )-stable C-C manifold (i.e. with sufficiently many (N - 1)- dimensional hypersurfaces for N

=

dimHau V, e.g. V is Riemannian or contact of dimension;) 5) and f : V ----7 W be a map with

IIV

f

I

HIILN :( c

<

^00. Then for every E:

>

0 there is a decomposition of V, say V

=

VE U V1 - E with the following three properties.

(1) VE is an open subset in V of ^meSN VE :( ^E: and V1 - E ⁼ V - V1 - E;

furthermore, each connected component U of VE has Diam(U) :( D where D ^----7 0 for E: ----7 O.

(2) The restriction I

I

V1 -E is a continuous, moreover, C/3-Holder map for

f3

= liN (where the implied Holder constant may depend on E).

Furthermore the image f(aU), of the boundary of every component U ofVE, has Diamf(aU) :( D' where D' ----7 0 for E: ----7 O.

(3) IfW has locally bounded geometry (e.g. compact) and E is sufficiently small, then f I V1- E admits a continuous extension say flO : V ----7 W which is contained in FfJ and, moreover, has

IIV

IE

I

HIILN :(

IIV

I

I

HIILN" (In fact this IE may be chosen taut on VE relative to the boundary aVe C VI-E.)

The proof follows by our earlier argument and is left to the reader.

Notice that the maps

f

converge to

f

in FfJ for E: ----7 0 and so they are all mutually homotopic for small E: by the weak homotopy stability observed in 2.5.A. Also notice that the regularization f ^f--+ flO applies to families of maps and shows that the space FfJ is homotopy equivalent to the space of continuous maps V ^----7 W (where V and Ware compact and V is (codim 1 )-stable, i.e. has many "nice" hypersurfaces as we always assume). In fact the inclusion of the space C¹ of smooth maps V ^----7 W into FfJ is a homotopy equivalence. Furthermore the space FfJ (as well as C¹ C FfJ with the induced topology) is locally contractible. (Of course, this all is well known for Riemannian manifolds V.)

2.5 C-C SPACES SEEN FROM WITHIN 185

Example. Let V be homeomorphic to the sphere

sn

^and W be an n-dimensional Riemannian manifold with locally bounded geometry. Sup-pose there exists a L~-map

f : sn

^---+ W which has degree 1 in the

(a) The degree 1 condition can be more succinctly expressed if V is Rie-mannian by

Iv

^{f*(w) =} 1 for the normalized oriented volume form w on W but I do not know how to do this in the general C-C case.

(b) Here and in future N refers to the Hausdorff dimension of V if it is equiregular and to the formal dimension max ~i (ni - ni-l) otherwise.

But in fact many of our results hold true for N = dimHau under milder (genericity) assumptions than equiregularity.

2.5.G. Restriction of

LIJ

-maps to k-dimensional submanifolds in codim-stable manifolds for q

<

Nand k

<

q. If V is Riemannian then the restriction

f

I V is obviously continuous for generic V' C V of dimension k

<

^{q (where}

f

E F[(V)). Furthermore if k ~ q - 1, then the homotopy type of this restriction is well defined. This is derived from the case codim V'

=

1 as follows. First, let co dim V'

=

2, take two V' which are transversal intersections of hypersurfaces, e.g. to those with trivial normal bundles and then we localize in a standard fashion to make the intersection trick work for all V'. Moreover, one easily shows with a properly localized intersection argument that

f

defines the homotopy type of the restriction of

f

to the k-skeleton of V for k = ent(q - 1) (i.e.

k is biggest integer ~ q - 1).

Co dim-stability. Call V stable in codimension n - k if for each tangent k-plane T' C T(V) there exists a germ of smooth sub manifold V' tangent to T' such that the formal Hausdorff dimension of V' at most equals that of V minus n - k (compare 2.4.B). This means, intuitively, that generic k-dimensional submanifolds V' have codimHau V' = codimtop V'.

186 MIKHAEL GROMOV §2

Example. Contact manifolds are (n - k )-stable for k ~ 3.

If V is (n - k )-stable then the restriction of

f

E F: to generic V' of dimension k

<

q is continuous and if k ::;; q - 1 the homotopy class of this restriction is well defined by the above Riemannian argument. This applies in particular to k-dimensional sub manifolds in contact manifolds for k ~ 3 (where the situation for k ::;; 2 remains unclear).

Restriction to horizontal submanifolds V' in a contact V. Such V' are plentiful for k

<

^{n/2, n} = dim V, and the restriction

f I

^V'is

continuous for generic horizontal V' provided q

>

k (compare 3.1). Now we claim that if k ::;; q - 1 and k

+

¹

<

dim V /2, then the homotopy class of this (continuous) restriction

f I

V'is stable under (horizontalf) deformations of V'. This implies (via the horizontal triangulation of V, see 3.4.B) that each

f

E

F:

^has well defined homotopy class of the restriction of

f

to the k-skeleton of V for every k

<

^(dim V /2) - 1 and

q~k+1.

Proof. To grasp the idea we start with the Riemannian case and indicate another way of reducing the case codim V'

>

1 to that of co dim V'

=

1.

For example, let V' be a small k-dimensional sphere in V and let us show that, generically, the restriction

f

I V', which is continuous for q

>

k, is contractible for q ~ k

+

1. We make this sphere V' the boundary of a small (k

+

I)-ball V{

c

V and observe that the restriction

f

I V{ is genericly contained in the space F;revD on V{. Then we restrict further, to a generic sphere in V{ concentric to aV{, and obtain contractible (as well as continuous)

f

on a generic small k-sphere in V. This argument easily generalizes to all (generic) V' C V and then it extends into the contact framework with the provisions of 3.1, 3.4.A.

2.5.G/. On singularities of

f

E F:. One can imagine, following Karen Uhlenbeck, every map

f

E

F:

as being regular (e.g. continuous) away from a certain (pole-like) singularity ~ f C V which, in the case where V is Riemannian, has "dim" ~f ::;; ent(n - q). This is justified by the following three facts

(1) generic V' of dimension k

<

^{q misses ~f as}

f I

V'is continuous;

(2) if k ::;; q - 1, then generic I-parameter families of V's miss ~f and so the homotopy class of

f I

V'is well defined;

2.5 C-C SPACES SEEN FROM WITHIN 187

(3) the blow-up construction in 2.5.E' applied to the balls B^k+¹ xb E Bk+l xBn-k-l

=

V, b E Bn-k+1, gives us

J

E F_{q ,} with q

=

k

+

^{1- c}

for all c

>

0, and with (n - k - I)-dimensional singularity.

Now, in the C-C case, one should think of "£f as some virtual subset (in V or in some auxiliary jet space over V) so that generic submanifolds V' and generic horizontal submanifolds of the same topological dimen-sion have different chances to meet "£f. (Notice that both, "generic" and

"generic horizontal" submanifolds V' of Hausdorff dimension.e, miss "£j,

J

E F:, if .e

<

q, and the restriction

J

I Viis homotopically sound if

.e~q-l.)

We conclude by observing that the restriction problem is still not solved in full generality for C-C manifolds. Namely, when does every map

J

E

FqH restrict to a continuous map on a generic submanifold V' E V of dimension k? When is the homotopy class of the restriction of

J

to the k-skeleton of V well defined? When does there exist at least one submanifold V' ^C V of dimension k for which

J

I Vi is continuous?

2.5.H. On local estimates for taut L~-maps. Consider a taut map J on an co-ball B(co) C V where

IB

IIDJ(v) I HIIN dv ~ Co for a small Co

>

0 and let us evaluate the diameters of the J-images of the concen-tric c-ball B(c) for c ~ 1/2 co. Since

J

is taut, this diameter does not exceed the infimum of those for the concentric spheres S(p) for p E [co,c]

and Diam J(S(p)) is bounded by constp-l Is(p) IIDJ(v) I HIIN dv, pro-vided the induced C-C geometry on these spheres has formal Hausdorff dimension N - 1. (In fact, the C-C spheres are usually non-smooth and so the formal dimension makes no sense. The true condition we need is, of course, (codim I)-stability which allows us to approximate the spheres by piecewise smooth hypersurfaces of formal dimension N - 1.) Then we integrate over p E [co, c] and conclude to the uniform continuity of Jon B(c^O /2) with the logarithmic modulus of continuity,

dist(f(v),J(v' )) ~ consto co1(-10gdist(v,v'))-k for all v, Vi in B(cO /2).

188 MIKHAEL GROMOV §2

2.5.H'. HOlder estimates. Let us additionally assume that our

f

min-imizes the energy

f

^I--t IIDf I HllfN and prove that then

f

is Holder. It is well known (and obvious by the previous discussion) that the Holder bound for taut maps issues from the following monotonicity inequality

r

^{IIVf(v) I HIIN dv}

^~

^A

r

^{IIVf(v) I HIIN dv (*)}

} B(2e) } B(e)

for 10 :::;; 100/2 and some A> 1 independent of c. Now, since

f

is minimizing, no extension of f from the sphere 8(210) = oB(2c) to a map f' on B(2c) may have IB(2e) IIVf'(v) I HIIN dv

<

IB(2e) IIVf(v) I HIIN dv, and so the following lemma yields (*).

Modification lemma. Every taut map fo : B(2c) -+ W which lands in a (small) ball within the range of the convexity radius of W can be modified to a map f' : B(2c) -+ W agreeing with fo on 8(2c), landing in the same (small) ball in Wand having

r

IID!,(v) I HIIN dv:::;; C

r

^{IIDfo(v) I HIIN dv, (*+)}

} B(2e) } A(e)

where A(c) denotes the annulus B(2c) - B(c) in V.

Proof. Let Bo be the minimal (convex!) ball in W containing the image fo (B (~ c)) and Wo be the center of Woo Observe that the radius of Bo

1

is bounded by Ro

=

Rad Bo :::;; Co (IA(e) IIVfo(v) I HIIN dV) N by our earlier argument (using (codim I)-stability and the tautness of fo). Then we use the notation W I--t 8w, 8 E [0,1], for the geodesic scaling of the ball Bo toward the center (i.e. 8w stands for the geodesic convex combination 8w

+

(1- 8)wo) and define f' on B(2c) by "compressing" fo by means of the cut-off distance function. Namely, take

{ I for v E B (2c) - B (~ c) , d(v)

=

(c/2)-1 dist(v,B(c)) for v E B(~ c)

and f' (v) = d( v) fo (v). The horizontal differential of f' clearly satisfies IIV!'(v) I HII :::;; C 1 Ro

+

IIVfo(v) I HII

all v E A(c) and Vf'(v) = 0 on B(c). Thus (*+) follows from the above bound on Ro and the volume bound meSN B(2c) ;S ION.

2.5 C-C SPACES SEEN FROM WITHIN 189

Remarks

(a) The minimizing property of

f

can be replaced by "quasi-minimizing".

This means that, for every relative compact domain U C V and every map

l' :

V ---+ W obtained by a homotopy of V fixed outside U, the total energy of

l'

^{on U} can not significantly smaller than that of

f,

i.e.

Nel documento Progress in Mathematics (pagine 189-198)