Ann. I. H. Poincaré – AN 29 (2012) 545–572
www.elsevier.com/locate/anihpc
Compactness of immersions with local Lipschitz representation
Patrick Breuning
1Institut für Mathematik, Goethe Universität Frankfurt am Main, Robert-Mayer-Straße 10, D-60325 Frankfurt am Main, Germany Received 19 August 2011; accepted 6 February 2012
Available online 2 March 2012
Abstract
We consider immersions admitting uniform representations as aλ-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixedλ <∞and uniformly bounded volume. The same result is shown in arbitrary codimension forλ14.
©2012 Elsevier Masson SAS. All rights reserved.
1. Introduction
In[14]J. Langer investigated compactness of immersed surfaces inR3admitting uniform bounds on the second fundamental form and the area of the surfaces. For a given sequence fi :Σi →R3, there exist, after passing to a subsequence, a limit surface f :Σ→R3 and diffeomorphismsφi :Σ→Σi, such that fi ◦φi converges in the C1-topology to f. In particular, up to diffeomorphism, there are only finitely many manifolds admitting such an immersion. The finiteness of topological types was generalized by K. Corlette in[6]to immersions of arbitrary di- mension and codimension. Moreover, the compactness theorem was generalized by S. Delladio in[7]to hypersurfaces of arbitrary dimension. The general case, that is compactness in arbitrary dimension and codimension, was proved by the author in[4].
The proof strongly relies on a fundamental principle which we like to describe in the following. A simple con- sequence of the implicit function theorem says that any immersion can locally be written as the graph of a function u:Br →Rk over the affine tangent space. Moreover, for a givenλ >0 we can chooser >0 small enough such that DuC0(Br)λ. If this is possible at any point of the immersion with the same radiusr, we callf an(r, λ)-immersion.
Using the Sobolev embedding it can be shown that a uniformLp-bound for the second fundamental form with p greater than the dimension implies that for anyλ >0 there is an r >0 such that every immersion is an(r, λ)- immersion.
Inspired by this result, it is a natural generalization to investigate compactness properties also for(r, λ)-immersions with fixedrandλ; this is the topic of the present paper. In the proof of the theorem of Langer it is essential thatλcan be chosen very small. Then, using the local graph representation overBr, all immersions are close to each other and nearly flat. These properties are used repeatedly, for example for the construction of the diffeomorphismφi.
E-mail address:breuning@math.uni-frankfurt.de.
1 Supported by the DFG-ForschergruppeNonlinear Partial Differential Equations: Theoretical and Numerical Analysis. The contents of this paper were part of the author’s dissertation, which was written at Universität Freiburg, Germany.
0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2012.02.001
Here, we would first like to show compactness of(r, λ)-immersions in codimension 1 for any fixedλ. We do not re- quire any smallness assumption forλ. Moreover, we do not only consider immersions with graph representations over the affine tangent space, but also over other appropriately chosenm-spaces. LetF1(r, λ)be the set ofC1-immersions f :Mm→Rm+1with 0∈f (M), which may locally be written over anm-space as the graph of aλ-Lipschitz func- tionu:Br→R(the precise definitions of all notations used in this paper are given in Section 2). Here all manifolds are assumed to be compact. Moreover, letF1V(r, λ)be the set of immersions inF1(r, λ)with vol(M)V. Similarly, we define the setF0(r, λ)by replacing C1-immersions in F1(r, λ)by Lipschitz functions. We obtain the following compactness result:
Theorem 1.1 (Compactness of (r, λ)-immersions in codimension one). The set F1V(r, λ) is relatively compact in F0(r, λ)in the following sense:
Letfi:Mi→Rm+1be a sequence inF1V(r, λ). Then, after passing to a subsequence, there exist anf:M→Rm+1 inF0(r, λ)and a sequence of diffeomorphismsφi :M→Mi, such thatfi ◦φi is uniformly Lipschitz bounded and converges uniformly tof.
Here the Lipschitz bound forfi◦φi is shown with respect to the local representations of some finite atlas ofM.
For these representations, we obtain a Lipschitz constant Ldepending only onλ. As an immediate consequence of Theorem1.1we deduce the following corollary:
Corollary 1.2.There are only finitely many manifolds inF1V(r, λ)up to diffeomorphism.
The situation is slightly different when considering(r, λ)-immersions in arbitrary codimension. For the construc- tion of the diffeomorphismsφione uses a kind of projection in an averaged normal directionν. In higher codimension, the averaged normalνcannot be constructed as in the case of hypersurfaces. We will give an alternative construction involving a Riemannian center of mass. However, for doing so we have to assume here thatλis not too large. Let F1V(r, λ)andF0(r, λ)be defined as above, but this time for functions with values inRm+kfor a fixedk. We obtain the following theorem:
Theorem 1.3 (Compactness of(r, λ)-immersions in arbitrary codimension). Letλ 14. ThenF1V(r, λ)is relatively compact inF0(r, λ)in the sense of Theorem1.1.
As in Corollary1.2, we deduce forλ14that there are only finitely many manifolds inF1V(r, λ)up to diffeomor- phism. Surely, the boundλ14is not optimal; at the end of Section 6 we will discuss some possibilities how to prove the theorem for bigger Lipschitz constant.
In[14]and[4]any sequence of immersions withLp-bounded second fundamental form,p > m, is shown to be also a sequence of(r, λ)-immersions (for some fixedrandλ). The same conclusion holds in many other situations, where the geometric data (such as curvature bounds) ensure uniform graph representations with control over the slope of the graphs. Hence it seems natural to unearth the compactness of(r, λ)-immersions as a theorem on its own. In any general situation, where compactness of immersions is desired (e.g. when considering convergence of geometric flows), only the condition of Definition2.2in Section 2 has to be verified. If in addition some bound for higher derivatives of the graph functions is known (or for instance aC0,α-bound forDu), with methods as in[4]one easily derives additional properties of the limit, such as higher order differentiability or curvature bounds. Hence, Theorems1.1and1.3can be seen as the most general kind of compactness theorem in this context.
2. Definitions and preliminaries
We begin with some general notations: For n=m+k letGn,mdenote the Grassmannian of (non-oriented) m- dimensional subspaces ofRn. Unless stated otherwise letBdenote the open ball inRmof radius >0 centered at the origin.
Now letM be anm-dimensional manifold without boundary and f :M→RnaC1-immersion. Letq∈Mand letTqM be the tangent space atq. Identifying vectorsX∈TqMwithf∗X∈Tf (q)Rn, we may considerTqMas an
m-dimensional subspace ofRn. Let(TqM)⊥denote the orthogonal complement ofTqMinRn, that is Rn=TqM⊕(TqM)⊥
and(TqM)⊥is perpendicular toTqM. We may define the tangent-space field τf :M→Gn,m,
q→TqM, (2.1)
and the normal-space field νf :M→Gn,k,
q→(TqM)⊥. (2.2)
2.1. The notion of an(r, λ)-immersion
We call a mappingA:Rn→RnaEuclidean isometry, if there is a rotationR∈SO(n)and a translationT ∈Rn, such thatA(x)=Rx+T for allx∈Rn.
For a given point q ∈M letAq :Rn→Rn be a Euclidean isometry, which maps the origin tof (q), and the subspaceRm× {0} ⊂Rm×Rk ontof (q)+τf(q). Letπ :Rn→Rm be the standard projection onto the firstm coordinates.
Finally letUr,q⊂Mbe theq-component of the set(π◦A−q1◦f )−1(Br). Although the isometryAqis not uniquely determined, the setUr,qdoes not depend on the choice ofAq.
We come to the central definition (as first defined in[14]):
Definition 2.1.An immersionf is called an(r, λ)-immersion, if for each pointq∈Mthe setA−q1◦f (Ur,q)is the graph of a differentiable functionu:Br→Rk withDuC0(Br)λ.
Here, for anyx∈Br we haveDu(x)∈Rk×m. In order to define theC0-norm forDu, we have to fix a matrix norm forDu(x). Of course all norms onRk×mare equivalent, therefore our results are true for any norm (possibly up to multiplication by some positive constant). Let us agree upon
A = m
j=1
|aj|2 12
forA=(a1, . . . , am)∈Rk×m. For this norm we haveAopAfor anyA∈Rk×mand the operator norm · op. Hence the boundDuC0(Br)λdirectly implies that uis λ-Lipschitz. Moreover the normDuC0(Br) does not depend on the choice of the isometryAq.
2.2. The notion of a generalized(r, λ)-immersion
For any(r, λ)-immersionf :M→Rnand anyq∈M, we have a local graph representation over the affine tangent spacef (q)+τf(q). It is natural to extend this definition to immersions with local graph representations over other appropriately chosenm-spaces inRn.
For a givenq∈Mand a givenm-spaceE∈Gn,mletAq,E:Rn→Rn be a Euclidean isometry, which maps the origin tof (q), and the subspaceRm× {0} ⊂Rm×Rkontof (q)+E.
LetUr,qE ⊂M be the q-component of the set (π◦A−q,E1 ◦f )−1(Br). Again the isometry Aq,E is not uniquely determined but the setUr,qE does not depend on the choice ofAq,E.
Definition 2.2. An immersion f is called a generalized (r, λ)-immersion, if for each point q ∈ M there is an E=E(q)∈Gn,m, such that the set A−q,E1 ◦f (Ur,qE )is the graph of a differentiable function u:Br →Rk with DuC0(Br)λ.
Obviously every (r, λ)-immersion is a generalized (r, λ)-immersion, as we can choose E(q)=τf(q) for any q∈M.
For fixed dimensionmand codimensionkwe denote byF1(r, λ)the set of generalized(r, λ)-immersionsf:M→ Rm+k with 0∈f (M), whereMis any compactm-manifold without boundary. ForV>0 we denote byF1V(r, λ)the set of all immersions inF1(r, λ)with vol(M)V. Here the volume of Mis measured with respect to the volume measure induced by the metricf∗geucl. Note thatMisnotfixed in these sets (in order to obtain a set in a strict set theoretical sense one may consider every manifold as embedded inRN for anN=N (m)). The condition 0∈f (M) can be weakened in many applications tof (M)∩K= ∅for a compact setK⊂Rm+k.
The notion of a generalized (r, λ)-immersion has one major advantage: As the definition does not make use of the existence of a tangent space, it allows us to define similar notions for functions intoRnwhich are not immersed.
For a givenE∈Gn,mthe setUr,qE can be defined foranycontinuous functionf :M→Rn. Moreover the condition DuC0(Br)λin the smooth case corresponds to a Lipschitz bound of the functionu. Hence the following definition can be seen as the natural generalization to continuous functions:
Definition 2.3.A continuous functionf is called an(r, λ)-function, if for each pointq∈Mthere is anE=E(q)∈ Gn,m, such that the setA−q,E1 ◦f (Ur,qE)is the graph of a Lipschitz continuous functionu:Br →Rk with Lipschitz constantλ.
We additionally assume here, thatEcan be chosen such thatf is injective onUr,qE. This property isnotimplied by the preceding definition, if one reads the latter word for word.
We shall always consider (r, λ)-functions defined on compact topological manifolds (without boundary). Using the local Lipschitz graph representation, any such manifold can be endowed with an atlas with bi-Lipschitz change of coordinates. If the Lipschitz constant of the graphs is sufficiently small (and hence the coordinate changes are almost isometric with bi-Lipschitz constant close to 1), by the results in[13]there exists even a smooth atlas. In our case, the limit manifold both in Theorems1.1and1.3will be smooth.
Finally, we define the setF0(r, λ)by replacing generalized(r, λ)-immersions inF1(r, λ)by(r, λ)-functions.
2.3. Geometry of Grassmann manifolds
Fork, n∈Nwith 0< k < nletGn,kagain be the set of (non-oriented)k-dimensional subspaces ofRn.
The setGn,kmay be endowed with the structure of a differentiablek(n−k)-dimensional manifold, see e.g.[15].
Moreover there is a Riemannian metricgonGn,k being invariant under the action ofO(n)inRn. It is unique up to multiplication by a positive constant (and — again up to multiplication by a positive constant — the only metric being invariant under the action ofSO(n)inRnexcept for the caseG4,2). For more details we refer the reader to[16].
In general, if(M, g)is a Riemannian manifold, the induced distance onMis defined by d(p, q)=inf
L(γ )γ:[a, b] →Mpiecewise smooth curve withγ (a)=p, γ (b)=q
. (2.3)
HereL(γ ):=b
a|dγdt(t)|dtdenotes the length ofγ. IfMis complete, by the theorem of Hopf–Rinow any two points p, q∈Mcan be joined by a geodesic of lengthd(p, q). This applies to the Grassmannian asGn,kis complete.
Now suppose that E, G∈Gn,k are two close k-planes; this means that the projection of each onto the other is non-degenerate. Applying a transformation to principal axes, there are orthonormal bases {v1, . . . , vk}of Eand {w1, . . . , wk}ofGsuch that
vi, wj =δijcosθi withθi∈ 0,π 2
for 1i, j k. For givenk-spacesE andG, theθ1, . . . , θk are uniquely determined (up to the order) and called theprincipal anglesbetweenEandG. Under all metrics onGn,k being invariant under the action ofO(n), there is exactly one metricgwith
d(E, G)= k
i=1
θi2 1
2
for all closek-planesEandG, whered denotes the distance corresponding tog, andθ1, . . . , θk the principal angles betweenE andGas defined above; see[2]and the references given there. We shall always use this distinguished metric.
We will need the following estimate for the sectional curvatures of a Grassmannian:
Lemma 2.4.Let max{k, n−k}2. Let K(·,·)denote the sectional curvature ofGn,k and letX, Y∈TPGn,k be linearly independent tangent vectors for aP ∈Gn,k. Then
0K(X, Y )2.
Proof. For min{k, n−k} =1 all sectional curvatures are constant withK(X, Y )=1. For a proof see[16, p. 351]. For min{k, n−k}2 we have 0K(X, Y )2 by[17, Theorem 3]. 2
The injectivity radius ofGn,k is π2 (see[2, p. 53]). A subsetU of a Riemannian manifold(M, g)is said to be convex,if and only if for eachp, q∈U the shortest geodesic fromptoq is unique inMand lies entirely inU. For the GrassmannianGn,k, any open Riemannian ballB(P )aroundP ∈Gn,k with <π4 is convex; see[8, p. 228].
2.4. The Riemannian center of mass
The well-known Euclidean center of mass may be generalized to aRiemannian center of masson Riemannian manifolds. This was introduced by K. Grove and H. Karcher in [9]. A simplified treatment is given in [13]. See also[11]. We like to give a short sketch of this concept.
Let(M, g)be a complete Riemannian manifold with induced distanced as in (2.3). Letμbe a probability measure onM, i.e. a nonnegative measure with
μ(M)=
M
dμ=1.
Letqbe a point inMandB=B(q)a convex open ball of radiusaroundq inM. Suppose sptμ⊂B,
where sptμdenotes the support ofμ. We define a function P :B→R,
P (p)=
M
d(p, x)2dμ(x).
Definition 2.5.Aq∈Bis called a center of mass forμif P (q)= inf
p∈B
M
d(p, x)2dμ(x).
The following theorem asserts the existence and uniqueness of a center of mass:
Theorem 2.6.If the sectional curvatures ofMinBare at mostκwith0< κ <∞and ifis small enough such that <14π κ−1/2, thenP is a strictly convex function onBand has a unique minimum point inBwhich lies inBand is the unique center of mass forμ.
Proof. See[13, Theorem 1.2]and the following pages there. 2
In the preceding theorem, we do not require the boundκ to be attained; in particular all sectional curvatures are also allowed to be less than or equal to 0. The same applies to the following lemma:
Lemma 2.7.Assume that the sectional curvatures ofMinB are at mostκwith0< κ <∞and <14π κ−1/2. Let μ1, μ2 be two probability measures onM withsptμ1⊂B,sptμ2⊂B with centers of massq1, q2 respectively.
Then for a universal constantC=C(κ, ) <∞ d(q1, q2)C
M
d(q2, x) d|μ1−μ2|(x),
where|μ1−μ2|denotes the total variation measure of the signed measureμ1−μ2. Proof. LetPi(p)=12
Md(p, x)2dμi(x)fori=1,2. By Theorem 1.5.1 in[13], with C=C(κ, ):=1+
κ1/2−1
tan 2κ1/2
, (2.4)
we have for ally∈Bthe estimate d(q1, y)CgradP1(y).
Using sptμi⊂B, by Theorem 1.2 in[13]we have gradPi(y)= −
B
exp−y1(x) dμi(x), (2.5)
where exp−y1:B→TyMis considered as a vector valued function.
Moreover, asq2is a center of mass, gradP2(q2)=0.
Then by the arguments of[11, Lemma 4.8.7](where manifolds of nonpositive sectional curvature are considered), we have
d(q1, q2)CgradP1(q2)
=C
B
exp−q21(x) dμ1(x)
=C
B
exp−q1
2(x) dμ1(x)−
B
exp−q1
2 (x) dμ2(x) C
M
d(q2, x) d|μ1−μ2|(x),
where we used|exp−q1
2 (x)| =d(q2, x)and sptμi⊂Bin the last line. 2 2.5. Basics for the proof
We like to fix some further notation and to deduce some basic facts that are needed in the proof.
First of all let us simplify the notation. For a given(r, λ)-immersionf :M→Rm+1and for everyq∈Mwe can choose anEq∈Gm+1,mwith the properties of Definition2.2. This yields a mappingE:M→Gm+1,m,q→Eq. For every(r, λ)-immersion we choose and fix such a mappingE. So every given(r, λ)-immersionf can be thought of as a pair(f,E), even ifEis not explicitly mentioned in the notation. WithAq,E(q)andUr,qE(q)as in Definition2.2, we set
Aq:=Aq,E(q) and for 0< r
U,q:=U,qE(q).
However, all properties shown below forU,qare true for any admissible choice ofE.
As an analogue to Lemma 3.1 in[14]we obtain the following statement, wheref is assumed to be ageneralized (r, λ)-immersion here:
Lemma 2.8.Letf :M→Rm+1be an(r, λ)-immersion andp, q∈M.
a) If0< randp∈U,q, then|f (q)−f (p)|< (1+λ).
b) If0< randδ= [3(1+λ)]−1andUδ,q∩Uδ,p= ∅, thenUδ,p⊂U,q.
Proof. a) Pass to the graph representation, use the bound on the C0-norm of the derivative of the graph and the triangular inequality.
b) Letx∈Uδ,pandy∈Uδ,q∩Uδ,p. Withϕq:=π◦A−q1◦f we have ϕq(x)f (x)−f (q)
f (x)−f (p)+f (p)−f (y)+f (y)−f (q)
<3(1+λ)δ
=.
Hence Uδ,p⊂ϕq−1(B). ButUδ,p∪Uδ,q is a connected set containing q, hence included in the q-component of ϕq−1(B), that is inU,q. We concludeUδ,p⊂U,q. 2
Now letr, λ >0 be given. Forl∈N0defineδl:= [3(1+λ)]−lr. For an(r, λ)-immersion f :M→Rm+1, by Lemma2.8b) we have the following important property:
Ifp, q∈MandUδl+1,q∩Uδl+1,p= ∅, thenUδl+1,p⊂Uδl,q. (2.6) Iff :M→Rm+1is an(r, λ)-immersion andp∈M, we may use the local graph representation to conclude that the setf (Ur,p)is homeomorphic to the ball Br. Hence we may choose a continuous unit normalνp:Ur,p→Sm with respect tof|Ur,p. Ifq∈Mis another point andνq:Ur,q→Sma continuous unit normal onUr,q, we note thatνp
andνqdo not necessarily coincide onUr,p∩Ur,q. However, we have the following statement:
Lemma 2.9. Letf :M→Rm+1be an (r, λ)-immersion andp, q∈M. Letνp:Uδ1,p→Sm,νq:Uδ1,q →Sm be continuous unit normals. SupposeUδ1,p∩Uδ1,q= ∅. Then exactly one of the following two statements is true:
• νp(x)=νq(x)for everyx∈Uδ1,p∩Uδ1,q,
• νp(x)= −νq(x)for everyx∈Uδ1,p∩Uδ1,q.
Proof. Choose a ξ ∈Uδ1,p∩Uδ1,q. First suppose thatνp(ξ )=νq(ξ ). As Ur,p is homeomorphic to Br and con- nected, there are exactly two continuous unit normals onUr,p. Let ν be the one withν(ξ )=νp(ξ ). Let W= {x∈ Uδ1,p:ν(x)=νp(x)}. ThenW is a nonempty subset of the connected setUδ1,p. MoreoverWis easily seen to be open and closed inUδ1,p. Therefore W=Uδ1,p andνp=ν onUδ1,p. AsUδ1,q ⊂Ur,p by (2.6), the preceding argumen- tation can also be applied toνq. Withν(ξ )=νp(ξ )=νq(ξ )we concludeνq=νonUδ1,q. Henceνp=ν=νq on Uδ1,p∩Uδ1,q, as in the claim above. Ifνp(ξ )= −νq(ξ ), a similar argument yieldsνp= −νq onUδ1,p∩Uδ1,q. 2 Remark 2.10.The statement of the preceding lemma might seem to be obvious at first sight. However one can think of a Möbius strip covered by two open setsU andV, each of which is homeomorphic toBr, such thatU∩V has exactly two components. If we choose continuous unit normalsν1,ν2onU, V respectively, we haveν1=ν2on one of the components, andν1= −ν2on the other. Such a behavior of the normals is excluded by Lemma2.9, irrespective whetherUδ1,p∩Uδ1,q is connected or not.
We need the notion of aδ-net:
Definition 2.11.LetQ= {q1, . . . , qs}be a finite set of points inMand let 0< δ < r. We say thatQis aδ-net forf, ifM=s
j=1Uδ,qj.
Note that everyδ-net is also aδ-net if 0< δ < δ< r.
The following statement is a bit stronger than Lemma 3.2 in[14]. It bounds the number of elements in aδ-net by an argumentation similar to that in the proof of Vitali’s covering theorem. Simultaneously, similarly to Besicovitch’s covering theorem, it gives a bound (which does not depend on the volume) how often any fixed point inMis covered by the net. More precisely, we have the following lemma:
Lemma 2.12.Forl∈N, every(r, λ)-immersion on a compactm-manifoldMadmits aδl-netQwith
|Q|δ−l+m1vol(M), {q∈Q:p∈Uδ2,q}
3(1+λ)(l+1)m
for every fixedp∈M.
Proof. Letq1∈Mbe an arbitrary point. Assume we have found points{q1, . . . , qν}inMwith the propertyUδl+1,qj∩ Uδl+1,qk= ∅forj =k. SupposeUδl,q1∪· · ·∪Uδl,qν does not coverM. Then choose a pointqν+1from the complement.
ThenUδl+1,qk∩Uδl+1,qν+1= ∅forkν, as otherwiseUδl+1,qν+1⊂Uδl,qk by (2.6). As vol(M)
s
j=1
vol(Uδl+1,qj)
s
j=1
Lm(Bδl+1) sδlm+1,
this procedure yields after at mostδ−l+m1vol(M)steps a cover.
For the second relation let p∈M. LetQ= {q1, . . . , qs}be the net that we found above. Moreover letZ(p)= {q∈Q: p∈Uδ2,q}. By Lemma2.8b) we have
q∈Z(p)
Uδ2,q⊂Uδ1,p. Hence we may estimate as above
vol(Uδ1,p)
q∈Z(p)
vol(Uδl+1,q)
Z(p)δml+1Lm(B1). (2.7)
As the immersion is an(r, λ)-immersion, we have
vol(Uδ1,p)(1+λ)mδ1mLm(B1). (2.8)
Combining (2.7) and (2.8), we estimate Z(p)(1+λ)mδ1mδ−l+m1
=3lm(1+λ)(l+1)m, which implies the statement. 2
We would like to emphasize that the second estimate in the preceding lemma does not depend on the volume vol(M). This will be necessary in order to obtain estimates for Lipschitz constants and for angles between different spaces depending only onλbut not on vol(M).
Definition 2.13.Letf:M→Rm+1be an(r, λ)-immersion. Letl∈Nand letQ= {q1, . . . , qs}be aδl-net forf. For ι∈ {0,1, . . . , l}andj ∈ {1, . . . , s}we define
Zι(j ):= {1ks: Uδι,qj ∩Uδι,qk= ∅}.
Forν1, ν2∈Rm+1\{0}let(ν1, ν2)denote the non-oriented angle betweenν1andν2, that is 0 (ν1, ν2)π,
(ν1, ν2)=arccosν1, ν2
|ν1||ν2|.
We consider the metric space(Sm, d), whereSm⊂Rm+1is them-dimensional unit sphere andd the intrinsic metric onSm, that is
d(·,·)=(·,·). (2.9)
For A⊂Sm andx ∈Sm let dist(x, A)=inf{d(x, y): y ∈A}. For >0 let B(A)= {x∈Sm: dist(x, A) < }. Moreover letS⊂P(Sm)denote the set of closed nonempty subsets ofSm. We denote bydH the Hausdorff metric onS, given by
dH:S×S→R0, (S1, S2)→inf
>0:S1⊂B(S2), S2⊂B(S1) .
We will need the following well-known version of the theorem of Arzelà–Ascoli for the Hausdorff metric (see[1, p. 125]):
Lemma 2.14.Let(X, d)be a compact metric space andAthe set of closed nonempty subsets ofX. Then(A, dH)is compact, i.e. every sequence inAhas a subsequence that converges to an element inA.
We will have to estimate the size of some tubular neighborhoods. To do this we need to introduce some more notation. Suppose we are given >0 andu∈C1(B)withDuC0(B)λ. Moreover letT ∈C1(B,Rm+1)be a mapping with|T (x)| =1 for allx∈B. Suppose thatT isL-Lipschitz for anLwith 0< L <∞. Letω:B→ Gm+1,1, q→span{T (q)}. Finally, let ν:B→Sm be a continuous unit normal with respect to the graph x → (x, u(x)). We consider a vector bundleEoverB, given by
E=
(x, y)∈B×Rm+1: y∈ω(x) . Forε >0 let
Eε=
(x, y)∈E: |y|< ε
⊂E.
Moreover we define a mapping F :E→Rm+1,
(x, y)→ x, u(x)
+y, (2.10)
wherey∈ω(x).
Lemma 2.15(Size of tubular neighborhoods). Letγ <π2 andε=L1cosγ. With the notation as above, assume that
T (p), ν(q)
γ for everyp, q∈B. (2.11)
Then the following are true:
a) The mappingF|Eεis a diffeomorphism onto an open neighborhood of{(x, u(x))∈Rm×R: x∈B}.
b) Letσ:=min{2cosγ ,2L(1cos2+γλ)}. Then Bσ
x, u(x)
∈Rm×R: x∈B
2
⊂F Eε
,
whereBσ(A)= {x∈Rm+1: dist(x, A) < σ}forA⊂Rm+1withdistthe Euclidean distance.
Fig. 2.1. Tubular neighborhood around a graph.
The trivial but long proof is carried out in detail in Appendix A. (See Fig.2.1.) Finally we like to define a metric for graph systems. First of all let
Gs=
(Aj, uj)sj=1: Aj:Rm+1→Rm+1is a Euclidean isometry,uj∈C1(Br) .
Every Euclidean isometry A:Rm+1→Rm+1 splits uniquely into a rotationR∈SO(m+1)and a translationT ∈ Rm+1. If · denotes the operator norm and ifΓ =(Aj, uj)sj=1∈Gs,Γ˜=(A˜j,u˜j)sj=1∈Gs, we set
d(·,·):Gs×Gs→R, d(Γ,Γ )˜ =
s
j=1
Rj− ˜Rj + |Tj− ˜Tj| + uj− ˜ujC0(Br)
. (2.12)
This makes(Gs,d)a metric space.
3. Transversality and tubular neighborhoods
In this section we like to construct lines in Rm+1, that intersect each (appropriately restricted) immersion fi transversally — even in the case, that the Lipschitz constantλof the graph functions is large. This yields local tubular neighborhoods aroundfi and is the crucial step in the proof.
Let r >0 and λ,V <∞. Let fi :Mi →Rm+1 be a sequence of (r, λ)-immersions as in Theorem 1.1. By Lemma2.12we can chooseδ5-netsQi= {q1i, . . . , qi
si}forMi withsiδ6−mvol(Mi)and with q∈Qi: p∈Uδi
2,q
3(1+λ)6m
for every fixedp∈Mi. (3.1)
As vol(Mi)V, we may pass to a subsequence such that each net has exactlyspoints for a fixeds∈N.
For everyi∈N,ι∈ {0,1, . . . ,5}andj∈ {1, . . . , s}we have Zιi(j )⊂P
{1, . . . , s} .
Hence, by successively passing to subsequences, we may assume thatZιi(j )does not depend oni. Denote it byZι(j ).
To simplify the notation, for 0< rwe setU,ji :=Ui
,qij.
Moreover, we choose for everyi∈Nand everyj ∈ {1, . . . , s}a continuous unit normalνji :Ur,ji →Smwith respect tofi|Ur,ji . Let these normal mappings be fixed from now on.
We set Sji :=νji
Uδi
1,j
⊂Sm,
where the closure is taken with respect to the metric defined in (2.9).
For each fixedj, this yields a sequence(Sji)i∈N inS. By Lemma2.14, passing to a further subsequence (if need be), we can assume
Sji →Sj in(S, dH)asi→ ∞
for each fixedj∈ {1, . . . , s}, whereSj ∈S. In particular for everyj Sji
i∈N is a Cauchy sequence in(S, dH). (3.2)
By (3.2) we may choose another subsequence such that for everyj dH
Sjk, Sjl
<π 4 −1
2arctanλ for allk, l∈N. (3.3)
To each qji ∈Qi we may assign a neighborhood Ur,ji , a Euclidean isometry Aij and a differentiable function uij:Br → R as in Definition 2.2. This yields the corresponding graph systems Γi =(Aij, uij)sj=1∈ Gs. As DuijC0(Br)λ and asfi(Mi)is uniformly bounded, a subsequence of (Γi)i∈N converges in(Gs,d). In partic- ular
Γi
i∈N is a Cauchy sequence in Gs,d
. (3.4)
Let constantsL, γ andσ be defined by L:=
3(1+λ)6m+4
r−1, (3.5)
γ:=π 4 +1
2arctanλ, (3.6)
σ:= cos2γ
2L(1+λ). (3.7)
By (3.4) we may pass to another subsequence such that d
Γk, Γl
<
3(1+λ)(1+r)−1
σ for allk, l∈N. (3.8)
Fori=1 we sometimes suppress the index 1 and write for instanceqj anduj instead ofqj1andu1j. For the immer- sionf1, letE1:M1→Gm+1,mbe a mapping as explained above. We setEj:=E1(qj1)∈Gm+1,m(this meansEjis anm-space for the pointqj1∈M1as in Definition2.2).
Our next task is to find a mappingω:M1→Gm+1,1, which defines the direction in which we project fromf1(M1) ontofi(Mi)in order to construct diffeomorphismsφi :M1→Mi. First we would like to give a local construction.
In Lemma3.5we will show thatωis even globally well defined. The construction is similar to that in[14], but more involved.
We choose aC∞-functiong:R0→Rwith the following properties:
• g(t)=1 fort <δr1,
• 0g(t)1 fort∈ [δr1,1],
• g(t)=0 fort >1,
• −2g(t)0 for allt >0.
We note thatδr1 = [3(1+λ)]−113, hence such a functiongexists.
Let
Z:M1→P
{1, . . . , s} , q→
1ks: q∈Uδ1
2,k
. By (3.1) we have
Z(q)
3(1+λ)6m
for everyq∈M1. (3.9)
For everyk∈ {1, . . . , s}we choose a unit vectorwk that is perpendicular to the subspaceEkdefined above. Let these vectorsw1, . . . , ws be fixed from now on.
Now letj∈ {1, . . . , s},q∈Uδ1
3,jandk∈Z(q). Lemma2.8b) yields Uδ1
1,j⊂Ur,k1 . In particularf1(Uδ1
1,j)is the graph of aλ-Lipschitz function on a subset ofEk. This implies either
wk, νj1(qj)
arctanλ or
−wk, νj1(qj)
arctanλ. (3.10)
Set νk:=
wk, if(wk, νj1(qj))arctanλ,
−wk, otherwise. (3.11)
If we replace the pointqj by any other pointp∈Uδ1
1,j, the relation (3.10) will still be true. Asνj1is continuous and Uδ1
1,j is connected, we easily conclude
νk, νj1(p)
arctanλ for everyp∈Uδ1
1,j, (3.12)
whereνkis the fixed vector defined in (3.11). We finally define a function S:Uδ1
3,j→Rm+1,
q→
k∈Z(q)
g
|f1(q)−f1(qk)| δ2
νk.
Lemma 3.1.The following inequalities hold:
a) |S(q)|(1+λ)−1for everyq∈Uδ1
3,j.
b) (S(q), νji(p))π4 +12arctanλfor everyq∈Uδ1
3,j and everyp∈Uδi
1,j. Proof. a) Letq∈Uδ1
3,j. AsQ1is aδ4-net forf1, there is ak∈ {1, . . . , s}withq∈Uδ1
4,k. By Lemma2.8a) we have
|f1(q)−f1(qk)|< δ3, hence
|f1(q)−f1(qk)| δ2 <δ3
δ2=δ1 r. By the definition ofgthis yields
g
|f1(q)−f1(qk)| δ2
=1.
Now letl∈Z(q). By (3.12) we have(νl, νj1(q))arctanλ. Hence
