Compactness of immersions with local Lipschitz representation

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by  Patrick Breuning

Compactness of immersions with local Lipschitz representation
Patrick Breuning 1
Institut fu¨r Mathematik der Goethe Universit¨at Frankfurt am Main
Robert-Mayer-Straße10, D-60325 Frankfurt am Main, Germany
email: [email protected]
2
1
0 Abstract
2
We consider immersions admitting uniform representations as a λ-Lipschitz graph. In

n codimension 1, we show compactness for such immersions for arbitrary fixed λ < ∞ and
a uniformly bounded volume. The same result is shownin arbitrarycodimension for λ≤ 1.
J 4

4
2 1. Introduction

]
G In [14] J. Langer investigated compactness of immersed surfaces in R3 admitting uniform bounds on
the second fundamental form and the area of the surfaces. For a given sequence fi : Σi →R3, there
D
exist, after passing to a subsequence, a limit surface f : Σ → R3 and diffeomorphisms φi : Σ → Σi,
.
h such that fi ◦φi converges in the C1-topology to f. In particular, up to diffeomorphism, there are
at only finitely many manifolds admitting such an immersion. The finiteness of topological types was
m generalizedbyK.Corlettein[6]toimmersionsofarbitrarydimensionandcodimension. 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

1
author in [4].
v
3
9 The proof strongly relies on a fundamental principle which we like to describe in the following. A
9 simple consequence of the implicit function theorem says that any immersion can locally be written
4 as the graph of a function u : B → Rk over the affine tangent space. Moreover, for a given λ > 0
r
.
1 we can choose r > 0 small enough such that kDukC0(Br) ≤ λ. If this is possible at any point of the
0 immersion with the same radius r, we call f an (r,λ)-immersion.
2
1
Using the Sobolev embedding it can be shown that a uniform Lp-bound for the second fundamental
:
v form with p greater than the dimension implies that for any λ> 0 there is an r > 0 such that every
Xi immersion is an (r,λ)-immersion.
r
a Inspired by this result, it is a natural generalization to investigate compactness properties also for
(r,λ)-immersions with fixed r and λ; this is the topic of the present paper. In the proof of the
theorem of Langer it is essential that λ can be chosen very small. Then, using the local graph repre-
sentation over B , all immersions are close to each other and nearly flat. These properties are used
r
repeatedly, for example for the construction of the diffeomorphism φi.
Here, we would first like to show compactness of (r,λ)-immersions in codimension 1 for any fixed
λ. We do not require any smallness assumption for λ. Moreover,we do not only consider immersions
with graph representations over the affine tangent space, but also over other appropriately chosen
m-spaces. Let F1(r,λ) be the set of C1-immersions f : Mm → Rm+1 with 0 ∈ f(M), which may
1P. Breuning was supported by the DFG-Forschergruppe Nonlinear Partial Differential Equations: Theoretical and
NumericalAnalysis. Thecontentsofthispaperwerepartoftheauthor’sdissertation,whichwaswrittenatUniver-
sit¨atFreiburg,Germany.
1
1. Introduction
locally be written over an m-space as the graph of a λ-Lipschitz function u : B → R (the precise
r
definitions of all notations used in this paper are given in Section 2). Here all manifolds are assumed
tobecompact. Moreover,letF1(r,λ)bethesetofimmersionsinF1(r,λ)withvol(M)≤V. Similarly,
V
we define the set F0(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 F1(r,λ) is relatively compact in F0(r,λ) in the following sense:
V
Let fi : Mi → Rm+1 be a sequence in F1(r,λ). Then, after passing to a subsequence, there exist an
V
f : M → Rm+1 in F0(r,λ) and a sequence of diffeomorphisms φi : M → Mi, such that fi ◦φi is
uniformly Lipschitz bounded and converges uniformly to f.
Here the Lipschitz bound for fi◦φi is shown with respect to the local representations of some finite
atlas of M. For these representations, we obtain a Lipschitz constant L depending only on λ. As an
immediate consequence of Theorem 1.1 we deduce the following corollary:
Corollary 1.2 There are only finitely many manifolds in F1(r,λ) up to diffeomorphism.
V
Thesituationisslightlydifferentwhenconsidering(r,λ)-immersionsinarbitrarycodimension. Forthe
constructionofthediffeomorphismsφi oneusesakindofprojectioninanaveragednormaldirectionν.
In higher codimension, the averagednormal ν cannot 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 that λ is not too large. Let F1(r,λ) and F0(r,λ) be defined as above, but
V
this time for functions with values in Rm+k for a fixed k. We obtain the following theorem:
Theorem 1.3 (Compactness of (r,λ)-immersions in arbitrary codimension)
Let λ≤ 1 . Then F1(r,λ) is relatively compact in F0(r,λ) in the sense of Theorem 1.1.
4 V
As in Corollary1.2, we deduce forλ≤ 1 that there areonlyfinitely manymanifolds inF1(r,λ) up to
4 V
diffeomorphism. Surely, the bound λ≤ 1 is not optimal; at the end of Section 6 we will discuss some
4
possibilities how to prove the theorem for bigger Lipschitz constant.
In [14] and [4] any sequence of immersions with Lp-bounded second fundamental form, p > m, is
shownto be also a sequence of(r,λ)-immersions (for some fixedr andλ). The same conclusionholds
in many other situations, where the geometric data (such as curvature bounds) ensure uniform graph
representationswith controloverthe slope ofthe graphs. Hence it seems naturalto unearththe com-
pactness of (r,λ)-immersions as a theorem on its own. In any general situation, where compactness
of immersions is desired (e.g. when considering convergenceof geometric flows), only the condition of
Definition 2.2 in Section 2 has to be verified. If in addition some bound for higher derivatives of the
graph functions is known (or for instance a C0,α-bound for Du), with methods as in [4] one easily
derives additional properties of the limit, such as higher order differentiability or curvature bounds.
Hence, Theorems 1.1 and 1.3 can be seen as the most general kind of compactness theorem in this
context.
Acknowledgement: I would like to thank my advisor Ernst Kuwert for his support. Moreover I
would like to thank Manuel Breuning for proofreading my dissertation [3], where the results of this
paper were established first.
2
2. Definitions and preliminaries
2. Definitions and preliminaries
We begin with some general notations: For n = m+k let G denote the Grassmannian of (non-
n,m
oriented) m-dimensional subspaces of Rn. Unless stated otherwise let B denote the open ball in Rm
̺
of radius ̺>0 centered at the origin.
Now let M be an m-dimensional manifold without boundary and f : M → Rn a C1-immersion.
Letq ∈M andletT M be the tangentspaceatq. Identifying vectorsX ∈T M withf X ∈T Rn,
q q ∗ f(q)
we may consider T M as an m-dimensional subspace of Rn. Let (T M)⊥ denote the orthogonal
q q
complement of T M in Rn, that is
q
Rn =T M ⊕(T M)⊥
q q
and (T M)⊥ is perpendicular to T M. In this manner we may define a tangent and a normal map
q q
τ :M → G ,
f n,m (2.1)
q 7→ T M,
q
and
ν :M → G ,
f n,k
(2.2)
q 7→ (T M)⊥.
q
The notion of an (r,λ)-immersion:
We call a mapping A : Rn → Rn a Euclidean isometry, if there is a rotation R ∈ SO(n) and a
translation T ∈Rn, such that A(x)=Rx+T for all x∈Rn.
For a given point q ∈ M let A : Rn → Rn be a Euclidean isometry, which maps the origin to
q
f(q), and the subspace Rm ×{0} ⊂ Rm×Rk onto f(q)+τ (q). Let π : Rn → Rm be the standard
f
projection onto the first m coordinates.
Finally let U ⊂ M be the q-component of the set (π ◦A−1 ◦f)−1(B ). Although the isometry
r,q q r
A is not uniquely determined, the set U does not depend on the choice of A .
q r,q q
We come to the central definition (as first defined in [14]):
Definition 2.1 An immersion f is called an (r,λ)-immersion, if for each point q ∈M the set A−1◦
q
f(U ) is the graph of a differentiable function u:B →Rk with kDuk ≤λ.
r,q r C0(Br)
Here, for any x∈B we have Du(x)∈Rk×m. In order to define the C0-norm for Du, we have to fix
r
a matrix norm for Du(x). Of course all norms on Rk×m are equivalent, therefore our results are true
for any norm (possibly up to multiplication by some positive constant). Let us agree upon
1
m 2
kAk = |a |2
j
!
j=1
X
for A=(a ,...,a )∈Rk×m. For this norm we have kAk ≤ kAk for any A∈Rk×m and the oper-
1 m op
ator norm k·k . Hence the bound kDuk ≤λ directly implies that u is λ-Lipschitz. Moreover
op C0(Br)
the norm kDuk does not depend on the choice of the isometry A .
C0(Br) q
3
2. Definitions and preliminaries
The notion of a generalized (r,λ)-immersion:
For any (r,λ)-immersion f : M → Rn and any q ∈ M, we have a local graph representation over
the affine tangent space f(q)+τ (q). It is natural to extend this definition to immersions with local
f
graph representations over other appropriately chosen m-spaces in Rn.
For a given q ∈ M and a given m-space E ∈ G let A : Rn → Rn be a Euclidean isometry,
n,m q,E
which maps the origin to f(q), and the subspace Rm×{0}⊂Rm×Rk onto f(q)+E.
Let UE ⊂ M be the q-component of the set (π ◦ A−1 ◦ f)−1(B ). Again the isometry A is
r,q q,E r q,E
not uniquely determined but the set UE does not depend on the choice of A .
r,q q,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) ∈ G , such that the set A−1 ◦f(UE ) is the graph of a differen-
n,m q,E r,q
tiable function u:B →Rk with kDuk ≤λ.
r C0(Br)
Obviously every (r,λ)-immersion is a generalized(r,λ)-immersion, as we can choose E(q)=τ (q) for
f
any q ∈M.
ForfixeddimensionmandcodimensionkwedenotebyF1(r,λ)thesetofgeneralized(r,λ)-immersions
f : M → Rm+k with 0 ∈ f(M), where M is any compact m-manifold without boundary. For V > 0
we denote by F1(r,λ) the set of all immersions in F1(r,λ) with vol(M) ≤ V. Here the volume of M
V
is measured with respect to the volume measure induced by the metric f∗g . Note that M is not
eucl
fixed in these sets (in order to obtain a set in a strict set theoretical sense one may consider every
manifold as embedded in RN for an N =N(m)). The condition 0∈f(M) can be weakened in many
applications to f(M)∩K 6=∅ for a compact set K ⊂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 into
Rn which are not immersed. For a given E ∈ G the set UE can be defined for any continuous
n,m r,q
function f : M → Rn. Moreover the condition kDuk ≤ λ in the smooth case corresponds
C0(Br)
to a Lipschitz bound of the function u. Hence the following definition can be seen as the natural
generalization to continuous functions:
Definition 2.3 A continuous function f is called an (r,λ)-function, if for each point q ∈M there is
an E =E(q)∈G , such that the set A−1 ◦f(UE ) is the graph of a Lipschitz continuous function
n,m q,E r,q
u:B →Rk with Lipschitz constant λ.
r
We additionally assume here, that E can be chosen such that f is injective on UE . This property is
r,q
not implied 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 bound-
ary). Using the localLipschitzgraphrepresentation,anysuchmanifoldcanbe endowedwithanatlas
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 Theorem 1.1
and 1.3 will be smooth.
Finally, we define the set F0(r,λ) by replacing generalized (r,λ)-immersions in F1(r,λ) by (r,λ)-
functions.
4
2. Definitions and preliminaries
Geometry of Grassmann manifolds
For k,n ∈ N with 0 < k < n let G again be the set of (non-oriented) k-dimensional subspaces
n,k
of Rn.
ThesetG maybeendowedwiththestructureofadifferentiablek(n−k)-dimensionalmanifold,see
n,k
e.g. [15]. Moreover there is a Riemannian metric g on G being invariant under the action of O(n)
n,k
in Rn. It is unique up to multiplication by a positive constant (and — again up to multiplication by
a positive constant— the only metric being invariantunder the actionof SO(n) in Rn except for the
case G ). For more details we refer the reader to [16].
4,2
In general, if (M,g) is a Riemannian manifold, the induced distance on M is defined by
d(p,q)=inf{L(γ)| γ:[a,b]→M piecewise smooth curve with
(2.3)
γ(a)=p, γ(b)=q}.
Here L(γ):= b|dγ(t)|dt denotes the length of γ. If M is complete, by the Theorem of Hopf-Rinow
a dt
any two points p,q ∈M can be joined by a geodesic of length d(p,q). This applies to the Grassman-
R
nian as G is complete.
n,k
Nowsuppose thatE,G∈G aretwoclosek-planes;this meansthatthe projectionofeachontothe
n,k
other is non-degenerate. Applying a transformation to principal axes, there are orthonormal bases
{v ,...,v } of E and {w ,...,w } of G such that
1 k 1 k
π
hv ,w i=δ cosθ with θ ∈ 0,
i j ij i i
2
h (cid:17)
for1≤i,j ≤k. Forgivenk-spacesE andG,the θ ,...,θ areuniquelydetermined(up to the order)
1 k
and called the principal angles between E and G. Under all metrics on G being invariant under
n,k
the action of O(n), there is exactly one metric g with
1
k 2
d(E,G)= θ2
i
!
i=1
X
for all close k-planes E and G, where d denotes the distance corresponding to g, and θ ,...,θ the
1 k
principal angles between E and G as 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 of G and let
n,k
X,Y ∈T G be linearly independent tangent vectors for a P ∈G . Then
P n,k n,k
0≤K(X,Y)≤2.
Proof:
For min{k,n−k}=1 allsectionalcurvaturesare constantwith K(X,Y)=1. For a proofsee [16], p.
351. For min{k,n−k}≥2 we have 0≤K(X,Y)≤2 by [17], Theorem 3. (cid:3)
The injectivity radius of G is π (see [2], p. 53). A subset U of a Riemannian manifold (M,g)
n,k 2
is said to be convex, if and only if for each p,q ∈U the shortest geodesic from p to q is unique in M
andliesentirelyinU. FortheGrassmannianG ,anyopenRiemannianballB (P)aroundP ∈G
n,k ̺ n,k
with ̺< π is convex; see [8], p. 228.
4
5
2. Definitions and preliminaries
The Riemannian center of mass
The well-known Euclidean center of mass may be generalized to a Riemannian center of mass on
Riemannian manifolds. This was introduced by K. Grove and H. Karcher in [9]. A simplified treat-
ment 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 distance d as in (2.3). Let µ be a
probability measure on M, i.e. a nonnegative measure with
µ(M)= dµ=1.
ZM
Let q be a point in M and B =B (q) a convex open ball of radius ̺ around q in M. Suppose
̺ ̺
spt µ⊂B ,
̺
where spt µ denotes the support of µ. We define a function
P :B → R,
̺
P(p) = d(p,x)2dµ(x).
ZM
Definition 2.5 A q ∈B is called a center of mass for µ if
̺
P(q)= inf d(p,x)2dµ(x).
p∈B̺ZM
The following theorem asserts the existence and uniqueness of a center of mass:
Theorem 2.6 If the sectional curvatures of M in B are at most κ with 0<κ<∞ and if ̺ is small
̺
enough such that ̺< 1πκ−1/2, then P is a strictly convex function on B and has a unique minimum
4 ̺
point in B which lies in B and is the unique center of mass for µ.
̺ ̺
Proof:
See [13], Theorem 1.2 and the following pages there. (cid:3)
In the preceding theorem, we do not require the bound κ 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 of M in B are at most κ with 0 < κ < ∞ and
̺
̺< 1πκ−1/2. Let µ ,µ betwo probability measures on M with spt µ ⊂B , spt µ ⊂B with centers
4 1 2 1 ̺ 2 ̺
of mass q ,q respectively. Then for a universal constant C =C(κ,̺)<∞
1 2
d(q ,q )≤C d(q ,x)d|µ −µ |(x),
1 2 2 1 2
ZM
where |µ −µ | denotes the total variation measure of the signed measure µ −µ .
1 2 1 2
Proof:
Let P (p)= 1 d(p,x)2dµ (x) for i=1,2. By Theorem 1.5.1 in [13], with
i 2 M i
R C =C(κ,̺):=1+(κ1/2̺)−1tan(2κ1/2̺), (2.4)
we have for all y ∈B the estimate
̺
d(q ,y)≤C|grad P (y)|.
1 1
6
2. Definitions and preliminaries
Using spt µ ⊂B , by Theorem 1.2 in [13] we have
i ̺
grad P (y)=− exp−1(x)dµ (x), (2.5)
i y i
ZB̺
where exp−1 :B →T M is considered as a vector valued function.
y ̺ y
Moreover,as q is a center of mass,
2
grad P (q )=0.
2 2
Thenwiththeargumentationof[11],Lemma4.8.7(wheremanifoldsofnonpositivesectionalcurvature
are considered) we have
d(q ,q ) ≤ C|grad P (q )|
1 2 1 2
= C exp−1(x)dµ (x)
q2 1
(cid:12)ZB̺ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
= C(cid:12) exp−q21(x)dµ1(x)(cid:12)− exp−q21(x)dµ2(x)
(cid:12)ZB̺ ZB̺ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
≤ C(cid:12) d(q2,x)d|µ1−µ2|(x), (cid:12)
ZM
where we used |exp−1(x)|=d(q ,x) and spt µ ⊂B in the last line. (cid:3)
q2 2 i ̺
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)-immersion f : M → Rm+1 and for ev-
ery q ∈ M we can choose an E ∈ G with the properties of Definition 2.2. This yields a
q m+1,m
mapping E : M → G , q 7→ E . For every (r,λ)-immersion we choose and fix such a mapping
m+1,m q
E. So every given (r,λ)-immersion f can be thought of as a pair (f,E), even if E is not explicitly
E(q)
mentioned in the notation. With A and U as in Definition 2.2, we set
q,E(q) r,q
A :=A
q q,E(q)
and for 0<̺≤r
U :=UE(q).
̺,q ̺,q
In fact this means that A and U also depend on E(q). However, all properties shown below for
q ̺,q
U are true for any admissible choice of E.
̺,q
As an analogue to Lemma 3.1 in [14] we obtain the following statement, where f is assumed to be a
generalized (r,λ)-immersion here:
Lemma 2.8 Let f :M →Rm+1 be an (r,λ)-immersion and p,q ∈M.
a) If 0<̺≤r and p∈U , then |f(q)−f(p)|<(1+λ)̺.
̺,q
b) If 0<̺≤r and δ =[3(1+λ)]−1̺ and U ∩U 6=∅, then U ⊂U .
δ,q δ,p δ,p ̺,q
7
2. Definitions and preliminaries
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) Let x∈U and y ∈U ∩U . With ϕ :=π◦A−1◦f we have
δ,p δ,q δ,p q q
|ϕ (x)| ≤ |f(x)−f(q)|
q
≤ |f(x)−f(p)|+|f(p)−f(y)|+|f(y)−f(q)|
< 3(1+λ)δ
= ̺.
Hence U ⊂ ϕ−1(B ). But U ∪U is a connected set containing q, hence included in the
δ,p q ̺ δ,p δ,q
q-component of ϕ−1(B ), that is in U . We conclude U ⊂U . (cid:3)
q ̺ ̺,q δ,p ̺,q
Nowletr,λ>0begiven. Forl∈N defineδ :=[3(1+λ)]−lr. Foran(r,λ)-immersionf :M →Rm+1,
0 l
by Lemma 2.8 b) we have the following important property:
If p,q ∈M and U ∩U 6=∅, then U ⊂U . (2.6)
δl+1,q δl+1,p δl+1,p δl,q
If f : M → Rm+1 is an (r,λ)-immersion and p ∈ M, we may use the local graph representation to
conclude that the set f(U ) is homeomorphic to the ball B . Hence we may choose a continuous
r,p r
unit normal ν : U → Sm with respect to f|U . If q ∈ M is another point and ν : U → Sm
p r,p r,p q r,q
a continuous unit normal on U , we note that ν and ν do not necessarily coincide on U ∩U .
r,q p q r,p r,q
However, we have the following statement:
Lemma 2.9 Let f : M → Rm+1 be an (r,λ)-immersion and p,q ∈ M. Let ν : U → Sm,
p δ1,p
ν : U → Sm be continuous unit normals. Suppose U ∩U 6= ∅. Then exactly one of the
q δ1,q δ1,p δ1,q
following two statements is true:
• ν (x)=ν (x) for every x∈U ∩U ,
p q δ1,p δ1,q
• ν (x)=−ν (x) for every x∈U ∩U .
p q δ1,p δ1,q
Proof:
Choose a ξ ∈ U ∩U . First suppose that ν (ξ) = ν (ξ). As U is homeomorphic to B and
δ1,p δ1,q p q r,p r
connected,thereareexactlytwocontinuousunitnormalsonU . Letν bethe onewithν(ξ)=ν (ξ).
r,p p
Let W = {x ∈ U : ν(x) = ν (x)}. Then W is a nonempty subset of the connected set U .
δ1,p p δ1,p
Moreover W is easily seen to be open and closed in U . Therefore W = U and ν = ν on
δ1,p δ1,p p
U . As U ⊂ U by (2.6), the preceding argumentation can also be applied to ν . With
δ1,p δ1,q r,p q
ν(ξ)=ν (ξ)=ν (ξ) we conclude ν =ν onU . Hence ν =ν =ν on U ∩U , as in the claim
p q q δ1,q p q δ1,p δ1,q
above. If ν (ξ)=−ν (ξ), a similar argumentation yields ν =−ν on U ∩U . (cid:3)
p q p q δ1,p δ1,q
Remark 2.10 The statementofthepreceding lemmamight seem tobeobvious at firstsight. However
one can think of a M¨obius strip covered by two open sets U and V, each of which is homeomorphic to
B , such that U∩V has exactly two components. If we choose continuous unit normals ν , ν on U,V
r 1 2
respectively, we have ν =ν on one of the components, and ν =−ν on the other. Such a behavior
1 2 1 2
of the normals is excluded by Lemma 2.9, irrespective whether U ∩U is connected or not.
δ1,p δ1,q
We need the notion of a δ-net:
8
2. Definitions and preliminaries
Definition 2.11 Let Q = {q ,...,q } be a finite set of points in M and let 0 < δ < r. We say that
1 s
s
Q is a δ-net for f, if M = U .
δ,qj
j=1
S
Note that every δ-net is also a δ′-net if 0<δ <δ′ <r.
ThefollowingstatementisabitstrongerthanLemma3.2in[14]. Itboundsthenumberofelementsin
aδ-netbyanargumentationsimilarto thatinthe proofofVitali’s coveringtheorem. Simultaneously,
similarly to Besicovitch’s covering theorem, it gives a bound (which does not depend on the volume)
how often any fixed point in M is covered by the net. More precisely, we have the following lemma:
Lemma 2.12 For l ∈N, every (r,λ)-immersion on a compact m-manifold M admits a δ -net Q with
l
|Q| ≤ δ−mvol(M),
l+1
|{q∈Q:p∈U }| ≤ [3(1+λ)](l+1)m for every fixed p∈M.
δ2,q
Proof:
Let q ∈M be an arbitrarypoint. Assume we have found points {q ,...,q } in M with the property
1 1 ν
U ∩U =∅ for j 6=k. Suppose U ∪...∪U does not cover M. Then choose a point
δl+1,qj δl+1,qk δl,q1 δl,qν
q from the complement. Then U ∩U = ∅ for k ≤ ν, as otherwise U ⊂U
ν+1 δl+1,qk δl+1,qν+1 δl+1,qν+1 δl,qk
by (2.6). As
s
vol(M) ≥ vol(U )
δl+1,qj
j=1
X
s
≥ Lm(B )
δl+1
j=1
X
≥ sδm ,
l+1
this procedure yields after at most δ−mvol(M) steps a cover.
l+1
For the second relation let p ∈ M. Let Q = {q ,...,q } be the net that we found above. More-
1 s
over let Z(p)={q ∈Q:p∈U }. By Lemma 2.8 b) we have
δ2,q
U ⊂U .
δ2,q δ1,p
q∈[Z(p)
Hence we may estimate as above
vol(U ) ≥ vol(U )
δ1,p δl+1,q
q∈XZ(p) (2.7)
≥ |Z(p)|δm Lm(B ).
l+1 1
As the immersion is an (r,λ)-immersion, we have
vol(U )≤(1+λ)mδmLm(B ). (2.8)
δ1,p 1 1
Combining (2.7) and (2.8), we estimate
|Z(p)| ≤ (1+λ)mδmδ−m
1 l+1
= 3lm(1+λ)(l+1)m,
which implies the statement. (cid:3)
9
2. Definitions and preliminaries
We wouldlike to emphasize that the secondestimate in the preceding lemma does not depend onthe
volume vol(M). This will be necessary in order to obtain estimates for Lipschitz constants and for
angles between different spaces depending only on λ but not on vol(M).
Definition 2.13 Let f :M →Rm+1 be an (r,λ)-immersion. Let l∈N and let Q={q ,...,q } be a
1 s
δ -net for f. For ι∈{0,1,...,l} and j ∈{1,...,s} we define
l
Z (j):={1≤k ≤s:U ∩U 6=∅}.
ι δι,qj δι,qk
For ν ,ν ∈Rm+1\{0} let ∢(ν ,ν ) denote the non-oriented angle between ν and ν , that is
1 2 1 2 1 2
0 ≤ ∢(ν ,ν ) ≤ π,
1 2
hν ,ν i
∢(ν ,ν ) =arccos 1 2 .
1 2
|ν ||ν |
1 2
We consider the metric space (Sm,d), where Sm ⊂Rm+1 is the m-dimensional unit sphere and d the
intrinsic metric on Sm, that is
d(·,·)= ∢(·,·). (2.9)
For A ⊂ Sm and x ∈ Sm let dist(x,A) = inf{d(x,y) : y ∈ A}. For ̺ > 0 let B (A) =
̺
{x ∈ Sm : dist(x,A) < ̺}. Moreover let S ⊂ P(Sm) denote the set of closed nonempty subsets
of Sm. We denote by d the Hausdorff metric on S, given by
H
d : S×S → R ,
H ≥0
(S ,S ) 7→ inf{̺>0: S ⊂B (S ), S ⊂B (S )}.
1 2 1 ̺ 2 2 ̺ 1
Wewillneedthefollowingwell-knownversionofthetheoremofArzela`-AscolifortheHausdorffmetric
(see [1], p. 125):
Lemma 2.14 Let (X,d) be a compact metric space and A the set of closed nonempty subsets of X.
Then (A,d ) is compact, i.e. every sequence in A has a subsequence that converges to an element in
H
A.
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 and u ∈ C1(B ) with kDuk ≤ λ. Moreover
̺ C0(B̺)
let T ∈ C1(B ,Rm+1) with |T(x)| = 1 for all x ∈ B . Suppose that T is L-Lipschitz for an L with
̺ ̺
0 < L < ∞. Let ω : B →G , q 7→span {T(q)}. Finally, let ν : B →Sm be a continuous unit
̺ m+1,1 ̺
normal with respect to the graph x7→(x,u(x)). We consider a vector bundle E over B , given by
̺
E ={(x,y)∈B ×Rm+1 :y ∈ω(x)}.
̺
For ε>0 let
Eε ={(x,y)∈E :|y|<ε}⊂E.
Moreover we define a mapping
F : E → Rm+1,
(2.10)
(x,y) 7→ (x,u(x))+y,
where y ∈ω(x).
10