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Compactness of immersions with local Lipschitz representation

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.


Introduction
In [14] J. Langer investigated compactness of immersed surfaces in R 3 admitting uniform bounds on the second fundamental form and the area of the surfaces. For a given sequence f i : Σ i → R 3 , there exist, after passing to a subsequence, a limit surface f : Σ → R 3 and diffeomorphisms φ i : Σ → Σ i , such that f i • φ i converges in the C 1 -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 dimension 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 consequence of the implicit function theorem says that any immersion can locally be written as the graph of a function u : B r → R k over the affine tangent space. Moreover, for a given λ > 0 we can choose r > 0 small enough such that Du C 0 (Br) ≤ λ. If this is possible at any point of the immersion with the same radius r, we call f an (r, λ)-immersion.
Using the Sobolev embedding it can be shown that a uniform L p -bound for the second fundamental form with p greater than the dimension implies that for any λ > 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 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 representation over B r , all immersions are close to each other and nearly flat. These properties are used repeatedly, for example for the construction of the diffeomorphism φ i .
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 F 1 (r, λ) be the set of C 1 -immersions f : M m → R m+1 with 0 ∈ f (M ), which may 1. Introduction locally be written over an m-space as the graph of a λ-Lipschitz function u : B r → 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, let F 1 V (r, λ) be the set of immersions in F 1 (r, λ) with vol(M ) ≤ V. Similarly, we define the set F 0 (r, λ) by replacing C 1 -immersions in F 1 (r, λ) by Lipschitz functions. We obtain the following compactness result: Theorem 1.1 (Compactness of (r, λ)-immersions in codimension one) The set F 1 V (r, λ) is relatively compact in F 0 (r, λ) in the following sense: Let f i : M i → R m+1 be a sequence in F 1 V (r, λ). Then, after passing to a subsequence, there exist an f : M → R m+1 in F 0 (r, λ) and a sequence of diffeomorphisms φ i : M → M i , such that f i • φ i is uniformly Lipschitz bounded and converges uniformly to f .
Here the Lipschitz bound for f i • φ i 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 F 1 V (r, λ) up to diffeomorphism.
The situation is slightly different when considering (r, λ)-immersions in arbitrary codimension. For the construction of the diffeomorphisms φ i one uses a kind of projection in an averaged normal direction ν.
In higher codimension, the averaged normal 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 F 1 V (r, λ) and F 0 (r, λ) be defined as above, but this time for functions with values in R m+k for a fixed k. We obtain the following theorem: Theorem 1.3 (Compactness of (r, λ)-immersions in arbitrary codimension) Let λ ≤ 1 4 . Then F 1 V (r, λ) is relatively compact in F 0 (r, λ) in the sense of Theorem 1.1.
As in Corollary 1.2, we deduce for λ ≤ 1 4 that there are only finitely many manifolds in F 1 V (r, λ) up to diffeomorphism. Surely, the bound λ ≤ 1 4 is 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 with L p -bounded second fundamental form, p > m, is shown to be also a sequence of (r, λ)-immersions (for some fixed r and λ). 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 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 C 0,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.
We begin with some general notations: For n = m + k let G n,m denote the Grassmannian of (nonoriented) m-dimensional subspaces of R n . Unless stated otherwise let B ̺ denote the open ball in R m of radius ̺ > 0 centered at the origin. Now let M be an m-dimensional manifold without boundary and f : M → R n a C 1 -immersion. Let q ∈ M and let T q M be the tangent space at q. Identifying vectors X ∈ T q M with f * X ∈ T f (q) R n , we may consider T q M as an m-dimensional subspace of R n . Let (T q M ) ⊥ denote the orthogonal complement of T q M in R n , that is In this manner we may define a tangent and a normal map The notion of an (r, λ)-immersion: We call a mapping A : R n → R n a Euclidean isometry, if there is a rotation R ∈ SO(n) and a translation T ∈ R n , such that For a given point q ∈ M let A q : R n → R n be a Euclidean isometry, which maps the origin to f (q), and the subspace R m × {0} ⊂ R m × R k onto f (q) + τ f (q). Let π : R n → R m be the standard projection onto the first m coordinates.
Finally let U r,q ⊂ M be the q-component of the set (π • A −1 q • f ) −1 (B r ). Although the isometry A q is not uniquely determined, the set U r,q does not depend on the choice of A q .
We come to the central definition (as first defined in [14]): Here, for any x ∈ B r we have Du(x) ∈ R k×m . In order to define the C 0 -norm for Du, we have to fix a matrix norm for Du(x). Of course all norms on R k×m are equivalent, therefore our results are true for any norm (possibly up to multiplication by some positive constant). Let us agree upon for A = (a 1 , . . . , a m ) ∈ R k×m . For this norm we have A op ≤ A for any A ∈ R k×m and the operator norm · op . Hence the bound Du C 0 (Br) ≤ λ directly implies that u is λ-Lipschitz. Moreover the norm Du C 0 (Br) does not depend on the choice of the isometry A q .

Definitions and preliminaries
The notion of a generalized (r, λ)-immersion: For any (r, λ)-immersion f : M → R n and any q ∈ M , we have a local graph representation over the affine tangent space f (q) + τ f (q). It is natural to extend this definition to immersions with local graph representations over other appropriately chosen m-spaces in R n .
For a given q ∈ M and a given m-space E ∈ G n,m let A q,E : R n → R n be a Euclidean isometry, which maps the origin to f (q), and the subspace . Again the isometry A q,E is not uniquely determined but the set U E r,q does not depend on the choice of A q,E .
For fixed dimension m and codimension k we denote by F 1 (r, λ) the set of generalized (r, λ)-immersions f : Here the volume of M is measured with respect to the volume measure induced by the metric f * g eucl . Note that M is not fixed in these sets (in order to obtain a set in a strict set theoretical sense one may consider every manifold as embedded in R N for an N = N (m)). The condition 0 ∈ f (M ) can be weakened in many applications to f (M ) ∩ K = ∅ for a compact set K ⊂ R m+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 R n which are not immersed. For a given E ∈ G n,m the set U E r,q can be defined for any continuous function f : M → R n . Moreover the condition Du C 0 (Br) ≤ λ in the smooth case corresponds to a Lipschitz bound of the function u. Hence the following definition can be seen as the natural generalization to continuous functions: is the graph of a Lipschitz continuous function u : B r → R k with Lipschitz constant λ.
We additionally assume here, that E can be chosen such that f is injective on U E r,q . This property is 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 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 Theorem 1.1 and 1.3 will be smooth.

Geometry of Grassmann manifolds
For k, n ∈ N with 0 < k < n let G n,k again be the set of (non-oriented) k-dimensional subspaces of R n .
The set G n,k may be endowed with the structure of a differentiable k(n − k)-dimensional manifold, see e.g. [15]. Moreover there is a Riemannian metric g on G n,k being invariant under the action of O(n) in R n . 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 of SO(n) in R n except for the case G 4,2 ). For more details we refer the reader to [16].
In general, if (M, g) is a Riemannian manifold, the induced distance on M is defined by Here L(γ) := b a | dγ dt (t)| dt denotes the length of γ. If M is complete, by the Theorem of Hopf-Rinow any two points p, q ∈ M can be joined by a geodesic of length d(p, q). This applies to the Grassmannian as G n,k is complete. Now suppose that E, G ∈ G n,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 for all close k-planes E and G, where d denotes the distance corresponding to g, and θ 1 , . . . , θ k the 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: Let K(·, ·) denote the sectional curvature of G n,k and let X, Y ∈ T P G n,k be linearly independent tangent vectors for a P ∈ G n,k . Then
The injectivity radius of G n,k is π 2 (see [2], p. 53). A subset U of a Riemannian manifold (M, g) 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 and lies entirely in U . For the Grassmannian G n,k , any open Riemannian ball B ̺ (P ) around P ∈ G n,k with ̺ < π 4 is convex; see [8], p. 228.

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 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 distance d as in ( where spt µ denotes the support of µ. We define a function 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 ̺ < 1 4 πκ −1/2 , then P is a strictly convex function on B ̺ and has a unique minimum 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.
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 ̺ < 1 4 πκ −1/2 . Let µ 1 , µ 2 be two probability measures on M with spt µ 1 ⊂ B ̺ , spt µ 2 ⊂ B ̺ with centers of mass q 1 , q 2 respectively. Then for a universal constant where |µ 1 − µ 2 | denotes the total variation measure of the signed measure µ 1 − µ 2 .
Using spt µ i ⊂ B ̺ , by Theorem 1.2 in [13] we have where exp −1 y : B ̺ → T y M is considered as a vector valued function.
Moreover, as q 2 is a center of mass, grad P 2 (q 2 ) = 0.
Then with the argumentation of [11], Lemma 4.8.7 (where manifolds of nonpositive sectional curvature are considered) we have where we used | exp −1 q2 (x)| = d(q 2 , x) and spt µ i ⊂ B ̺ in the last line.

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 → R m+1 and for every q ∈ M we can choose an E q ∈ G m+1,m with the properties of Definition 2.2. This yields a mapping E : M → G m+1,m , q → E q . For every (r, λ)-immersion we choose and fix such a mapping E. So every given (r, λ)-immersion f can be thought of as a pair (f, E), even if E is not explicitly mentioned in the notation. With A q,E(q) and U E(q) r,q as in Definition 2.2, we set In fact this means that A q and U ̺,q also depend on E(q). However, all properties shown below for U ̺,q are true for any admissible choice of E.
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: Proof: a) Pass to the graph representation, use the bound on the C 0 -norm of the derivative of the graph and the triangular inequality.
we have the following important property: is an (r, λ)-immersion and p ∈ M , we may use the local graph representation to conclude that the set f (U r,p ) is homeomorphic to the ball B r . Hence we may choose a continuous unit normal ν p : U r,p → S m with respect to f |U r,p . If q ∈ M is another point and ν q : U r,q → S m a continuous unit normal on U r,q , we note that ν p and ν q do not necessarily coincide on U r,p ∩ U r,q . However, we have the following statement: Lemma 2.9 Let f : M → R m+1 be an (r, λ)-immersion and p, q ∈ M . Let ν p : U δ1,p → S m , ν q : U δ1,q → S m be continuous unit normals. Suppose U δ1,p ∩ U δ1,q = ∅. Then exactly one of the following two statements is true: Proof: Choose a ξ ∈ U δ1,p ∩ U δ1,q . First suppose that ν p (ξ) = ν q (ξ). As U r,p is homeomorphic to B r and connected, there are exactly two continuous unit normals on U r,p . Let ν be the one with ν(ξ) = ν p (ξ).
Then W is a nonempty subset of the connected set U δ1,p . Moreover W is easily seen to be open and closed in U δ1,p . Therefore W = U δ1,p and ν p = ν on U δ1,p . As U δ1,q ⊂ U r,p by (2.6), the preceding argumentation can also be applied to ν q . With 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 sets U and V , each of which is homeomorphic to B r , such that U ∩ V has exactly two components. If we choose continuous unit normals ν 1 , ν 2 on U, V respectively, we have ν 1 = ν 2 on one of the components, and ν 1 = −ν 2 on the other. Such a behavior of the normals is excluded by Lemma 2.9, irrespective whether U δ1,p ∩ U δ1,q is connected or not.
We need the notion of a δ-net:

Definitions and preliminaries
Definition 2.11 Let Q = {q 1 , . . . , q s } be a finite set of points in M and let 0 < δ < r. We say that Note that every δ-net is also a δ ′ -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 a δ-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 in M is covered by the net. More precisely, we have the following lemma: for every fixed p ∈ M.

Proof:
Let q 1 ∈ M be an arbitrary point. Assume we have found points this procedure yields after at most δ −m l+1 vol(M ) steps a cover.
For the second relation let p ∈ M . Let Q = {q 1 , . . . , q s } be the net that we found above. Moreover let Z(p) = {q ∈ Q : p ∈ U δ2,q }. By Lemma 2.8 b) we have Hence we may estimate as above As the immersion is an (r, λ)-immersion, we have Combining (2.7) and (2.8), we estimate which implies the statement.
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 on λ but not on vol(M ).
For ν 1 , ν 2 ∈ R m+1 \{0} let ∢(ν 1 , ν 2 ) denote the non-oriented angle between ν 1 and ν 2 , that is We consider the metric space (S m , d), where S m ⊂ R m+1 is the m-dimensional unit sphere and d the intrinsic metric on S m , that is Moreover let S ⊂ P(S m ) denote the set of closed nonempty subsets of S m . We denote by d H the Hausdorff metric on S, given by 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 and A the set of closed nonempty subsets of X.
Then (A, d H ) is compact, i.e. every sequence in A has a subsequence that converges to an element in 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 We consider a vector bundle E over B ̺ , given by For ε > 0 let Moreover we define a mapping where y ∈ ω(x).

Lemma 2.15 (Size of tubular neighborhoods)
Let γ < π 2 . With the notation as above, assume that Then the following is true: The trivial but long proof is carried out in detail in the appendix. Figure 2.1 Tubular neighborhood around a graph.
Finally we like to define again a metric for graph systems. First of all let Every Euclidean isometry A : R m+1 → R m+1 splits uniquely into a rotation R ∈ SO(m + 1) and a translation T ∈ R m+1 . If · denotes the operator norm and if Γ = ( This makes (G s , d) a metric space.

Transversality and tubular neighborhoods
In this section we like to construct lines in R m+1 , that intersect each (appropriately restricted) immersion f i transversally -even in the case, that the Lipschitz constant λ of the graph functions is large. This yields local tubular neighborhoods around f i and is the crucial step in the proof. Let vol(M i ) elements respectively and with As vol(M i ) ≤ V, we may pass to a subsequence such that each net has exactly s points for a fixed s ∈ N.
To simplify the notation, for 0 Moreover, we choose for every i ∈ N and every j ∈ {1, . . . , s} a continuous unit normal ν i j : U i r,j → S m with respect to f i |U i r,j . Let these normal mappings be fixed from now on.
For S ⊂ S m let S be the closure of S with respect to the metric d defined in (2.9). We set For each fixed j, this yields a sequence (S i j ) i∈N in S. By Lemma 2.14 we can pass successively to subsequences in order to obtain a sequence with In particular for every j By (3.3) we may choose another subsequence such that for every j To each q i j ∈ Q i we may assign a neighborhood U i r,j , a Euclidean isometry A i j and a differentiable function u i j : B r → R as in Definition 2.2. This yields the corresponding graph systems (3.5) Let constants L, γ and σ be defined by By (3.5) we may pass to another subsequence such that For i = 1 we sometimes suppress the index 1 and write for instance q j and u j instead of q 1 j and u 1 j . For the immersion f 1 , let E 1 : M 1 → G m+1,m be a mapping as explained in the beginning of Chapter 3.1. We set E j := E 1 (q 1 j ) ∈ G m+1,m (this means E j is an m-space for the point q 1 j ∈ M 1 as in Definition 2.2).
Our next task is to find a mapping ω : M 1 → G m+1,1 , which defines the direction in which we project from f 1 (M 1 ) onto f i (M i ) in order to construct diffeomorphisms φ i : M 1 → M i . First we would like to give a local construction. In Lemma 3.5 we will show that ω is even globally well-defined. The construction is similar to that in [14], but more involved.
Using the triangular inequality, we conclude with (3.14) and (3.15) that By Lemma 3.1 a) the mapping S does not vanish on U 1 δ3,j . We define T by normalizing S, that is Identifying U 1 δ3,j with B δ3 by means of the diffeomorphism π • A −1 j • f 1 : U 1 δ3,j → B δ3 , we may consider T and S as mappings defined on the ball B δ3 . We show, that T considered as mapping on B δ3 is Lipschitz with respect to the Euclidean norm: = 0 by the definition of g.
Using the preceding considerations, g ′ C 0 (R ≥0 ) ≤ 2 and |Z(p)| ≤ [3(1 + λ)] 6m , |Z(q)| ≤ [3(1 + λ)] 6m , we estimate as follows: Remark 3.3 Of course, T is also Lipschitz as a mapping on U 1 δ3,j with respect to the metric induced by f 1 . The estimate of the Lipschitz constant gets even better in this case. Moreover, we note that in the preceding lemma L depends on r. However, we will see that the Lipschitz constant of f i • φ i does not depend on r in the end.
We set which is well-defined as S(q) = 0 by Lemma 3.1 a).
We like to explain how ω locally forms a tubular neighborhood around f 1 : For that we consider the mapping As g is smooth and g(t) = 0 for t ≥ 1, it is easily seen that g k can be extended to a smooth function g k : M 1 → R by settingḡ k = 0 outside U 1 δ2,k . This implies that S : U 1 δ3,j → R m+1 is differentiable, even if the sum in the definition of S depends on Z(q). Hence also T = S |S| is differentiable. Moreover Lemma 3.2 says that T is L-Lipschitz with L = [3(1 + λ)] 6m+4 r −1 and by Lemma 3.1 b) we have ∢(T (p), ν 1 j (q)) ≤ π 4 + 1 2 arctan λ for all p, q ∈ U 1 δ3,j .

Transversality and tubular neighborhoods
Finally, after a rotation and a translation, f (U 1 δ3,j ) may be written as the graph of a C 1 -function u 1 j : B δ3 → R. Let us introduce some more notation: We consider a vector bundleÊ j over U 1 δ3,j , given bŷ with bundle projectionπ. We may identify the zero section ofÊ j with U 1 δ3,j . For ε > 0 let Finally we define a mapping where y ∈ ω(x). . Then the following is true: • for each fibreÊ q =π −1 (q) it holds F j (Ê q ) = ω(q).
Moreover for σ = cos 2 γ 2L(1+λ) we have the inclusion Proof: This is just a reformulation of Lemma 2.15. Note that Up to this point we have constructed for each j ∈ {1, . . . , s} a tubular neighborhood locally around f (U 1 δ3,j ). Since the mapping S depends on j, we should write more accurately S j instead of S. In the same way we should write ω j instead of ω. However, we can show that ω is globally well-defined. More precisely we have the following lemma: In particular there is a smooth mapping ω : M 1 → G m+1,1 with ω|U 1 δ3,j = ω j for each j ∈ {1, . . . , s}.
Since this is true for all l ∈ Z(q), we conclude S j (q) = S k (q) and hence ω j (q) = ω k (q).

Intersection points and definition of φ i
In this section we like to show that for p ∈ M 1 the line f 1 (p) + ω(p) intersects each appropriately restricted immersion f i (M i ) in exactly one point. Using this, we are able to give a definition of the mappings φ i : M 1 → M i . Each φ i will be shown to be a diffeomorphism. Moreover, it will be shown that f i • φ i is uniformly Lipschitz bounded.
We note here that ω(p) does not depend on x. Let the function F be defined by where y ∈ ω(p). With an argumentation as in Lemma 3.4, using (4.1) and the fact that ω(p) is constant, we conclude that F (G) forms a tubular neighborhood around f i (U i δ2,j ), and moreover We would like to show that i.e. f 1 (U 1 δ3,j ) lies within the tubular neighborhood defined above. But this means that there is a q ∈ U i δ2,j such that It remains to show that f 1 (p) + ω(p) intersects f i (U i δ1,j ) in not more than one point. By (4.1) we have ∢(T (p), ν i j (q)) < π 2 for every q ∈ U i δ1,j . By the definition of ω this implies R m+1 = τ f i (q) ⊕ ω(p) for every q ∈ U i δ1,j . As f i is an (r, λ)-immersion, we conclude with Lemma A.1 in the appendix that f 1 (p) + ω(p) intersects f i (U i δ1,j ) in at most one point.
Now we are able to define the mappings φ i : We like to show that each φ i is a diffeomorphism. For that we follow in parts the argumentation of [4]: With (3.9) and an estimation completely analogous to that in the proof of ,j )). Hence, for every q ∈ U i δ4,j there is a p ∈ U 1 δ3,j with f i (q) ∈ f 1 (p) + ω(p). By the definition of φ i this yields φ i (p) = q.

Proof:
First we note that for every j ∈ {1, . . . , s} we have φ i (U 1 δ5,j ) ⊂ U i δ4,j . This is shown by the same argumentation as in Lemma 4.1. Moreover, by the proof of Lemma 4.3, we know that For showing global injectivity, let x, y ∈ M 1 with x = y. As Q 1 is a δ 5 -net for f 1 , there are j, k By the considerations at the beginning of this proof, we have .
By the considerations of above, φ i is injective on U 1 δ3,j . Again we conclude φ i (x) = φ i (y).
for every x ∈ U 1 δ3,j . Using a trivialization of the trivial bundleÊ j , one easily concludes that f i • φ i : M 1 → R m+1 is an immersion (see also [4]). Moreover, the mapping φ i is surjective by Lemma 4.3, and injective by Lemma 4.4. We conclude that φ i is a diffeomorphism.
Finally we would like to prove that the reparametrizations f i • φ i are uniformly Lipschitz bounded. As above, for j ∈ {1, . . . , s} we can consider f i • φ i |U 1 δ3,j also as a mapping defined on B δ3 . This mapping shall be denoted byf i : B δ3 → R m+1 .
Remark 4.7 If we choose some of the constants more carefully, we can give a better bound for Λ in the preceding lemma. Choosing the right hand side in (3.4) extremely small, we can replace γ by a numberγ which is slightly greater than arctan λ. Moreover, we can choose ε with |µ 1 |, |µ 2 | < ε so small, that the term εL can almost be neglected. With these constants, we finally obtain Λ = (1 + tanγ)(1 + λ + εL) < 2(1 + λ) 2 . In particular, Λ does not depend on the dimension m here, although L depends on m.
Finally, with Lemma 4.6, we may pass to a subsequence such that f i • φ i converges uniformly to a limit function f : M 1 → R m+1 . As limit manifold we define M := M 1 . Thus the limit manifold is a compact differentiable m-manifold.

The limit function lies in F 0 (r, λ)
Up to this point we have found a subsequence and diffeomorphisms φ i : M 1 → M i , such that f i • φ i is uniformly Lipschitz bounded and converges uniformly to an f : M 1 → R m+1 . In this section we will show that the limit function f lies in F 0 (r, λ).
For that we have to show, that for each point q ∈ M 1 there is an E = E(q) ∈ G m+1,m , such that f is injective on U E r,q and the set A −1 q,E • f (U E r,q ) is the graph of a Lipschitz continuous function u : B r → R with Lipschitz constant λ.
Passing to another subsequence, we may assume as i → ∞, where u : B r → R and E ∈ G m+1,m . In particular, u is Lipschitz continuous with Lipschitz constant λ.
Let A q,E be a Euclidean isometry, which maps the origin to f (q), and the subspace R m ×{0} ⊂ R m ×R onto f (q) + E. Then we have in any case A q,E ({(x, u(x)) : x ∈ B r }) ⊂ f (M 1 ).
To finish the proof, we show that f is injective on U E r,q and that A −1 q,E • f (U E r,q ) is the graph of the function u. This is true, if and only if for every ̺ with 0 < ̺ < r the function f is injective on U E ̺,q and the set A −1 q,E • f (U E ̺,q ) is the graph of the function u|B ̺ .
As this is true for every ε > 0 with ε < min{̺, r − ̺}, we conclude with the definition of U E ̺,q that This is the desired graph property.
Similarly, one shows that f is injective on q,E • f and hence also f is injective on U E ̺,q .
This shows that the limit function f lies in F 0 (r, λ).

Compactness in higher codimension
In the final section we want to prove Theorem 1.3, that is compactness of (r, λ)-immersions in higher codimension with λ ≤ 1 4 . Our main task here is to give an analogous construction of the averaged normal projection for arbitrary codimension. For that we shall use a Riemannian center of mass, which was introduced above.
So let f i be a sequence as in Theorem 1.3 with λ ≤ 1 4 . For all objects of the preceding sections that are defined also in arbitrary codimension, we shall use precisely the same notation. We note that Lemmas 2.8 and 2.12 are true also in higher codimension. For q ∈ M 1 we set As in the proof of Lemma 3.1 a) we conclude that there is a k ∈ Z(q) with λ q k = 1. For each j ∈ {1, . . . , s} let N j ∈ G n,k be the k-space perpendicular to E j . We define for each q ∈ M 1 a probability measure µ q on G n,k by where δ N denotes the Dirac measure on G n,k supported at N ∈ G n,k .
Moreover, let be the normal map of f 1 as defined in (2.2), and τ : M 1 → G n,m the corresponding tangent map as in (2.1). Now consider P : B π 6 (ν(q)) → R, where B π 6 (ν(q)) ⊂ G n,k is the closed ball of radius π 6 around ν(q). Here the radius is measured with respect to the canonical distance d on G n,k as defined in (2.3).
In particular we have spt µ q ⊂ B π 6 (ν(q)). Hence we conclude with Lemma 2.4 and Theorem 2.6, that there is exactly one center of mass N (q) ∈ B π 6 (ν(q)) ⊂ G n,k for µ q . In this way we may define a mapping An important property of the averaged normal N constructed in this way is its differentiability. It is needed in order to obtain diffeomorphisms φ i : M 1 → M i . We will show that N is in C k if the function f 1 is in C k (here we denote by k the degree of differentiability, and by k the codimension). First, for functions defined on manifolds, we need the following variation of the implicit function theorem: is in C 2 (N ) and is strictly convex. Let k ≥ 1 be an integer. Denoting by grad h x the gradient of the fixed function h x defined above, assume that Then there are open neighborhoods U ⊂ M of x 0 and V ⊂ N of y 0 , and moreover a function Proof: Let ϕ 1 : U 1 → ϕ(U 1 ) be a coordinate chart of M with x 0 ∈ U 1 , and let ϕ 2 : V 1 → ϕ 2 (V 1 ) a coordinate chart of N with y 0 ∈ V 1 . For fixed x ∈ M , in the local coordinates ϕ 2 we have and, with the corresponding Christoffel symbols Γ k ij = 1 2 n l=1 g kl (∂ i g jl +∂ j g il −∂ l g ij ), the components of the Hessian D 2 If we assume ϕ 2 to be Riemannian normal coordinates centered in y 0 , we obtain Let us now consider the local representations of h x and f in the coordinates ϕ 2 and ϕ 1 ×ϕ 2 respectively. We denote these representations simply by h x and f again. Moreover, we identify x 0 and y 0 with ϕ 1 (x 0 ) and ϕ 2 (y 0 ) respectively. The condition on h x to be strictly convex means that the Hessian D 2 h x is positive definite in every point. Hence, by (6.2), the Hessian matrix D 2 h x (y 0 ) of the local representation is positive definite, in particular 3) The Jacobian Df may be considered as a mapping Df : Ω → R m+n , where Ω = ϕ 1 (U 1 ) × ϕ 2 (V 1 ) ⊂ R m × R n . We write Df = (D x f, D y f ) ∈ R m × R n and consider the mapping D y f : Ω → R n .
Similarly, for the Jacobian of D y f , we write D(D y f ) = (D x (D y f ), D y (D y f )) ∈ R n×m × R n×n . As D y f (x 0 , y 0 ) = Dh x0 (y 0 ) and as H(x 0 , y 0 ) = 0, we conclude Similarly, as D y (D y f )(x 0 , y 0 ) = D 2 h x0 (y 0 ), we know by (6.3) that The assumption on H to be in C k implies with (6.1) that also D y f : Ω → R n is in C k . Hence we may use (6.4), (6.5) and apply the usual implicit function theorem to the function D y f . From this we deduce the statement.
Using the preceding lemma, we are able to deduce that the mapping N is differentiable: Let N : M 1 → G n,k be the averaged normal corresponding to f 1 : M 1 → R n , as constructed above. Assume that f 1 ∈ C k (M 1 , R n ) for a k ≥ 1. Then N ∈ C k (M 1 , G n,k ).

Proof:
Let q 0 ∈ M 1 be a point. We show that N is C k in a neighborhood of q 0 . Let W ⊂ M 1 be an open neighborhood of q 0 with ν(W ) ⊂ B π 12 (ν(q 0 )). With Lemma 6.1 we have spt µ q ⊂ B π 6 (ν(q 0 )) for every q ∈ W ; this will be implicitly used in the following argumentation. Let Moreover, for fixed q ∈ W let h q : B π 6 (ν(q 0 )) → R, h q := G(q, · ). By this definition, h q is smooth on B π 6 (ν(q 0 )) and by Theorem 2.6 strictly convex. We denote by grad h q the gradient of the fixed function h q , and define H : W × B π 6 (ν(q 0 )) → T B π 6 (ν(q 0 )), (q, p) → grad h q (p). With (2.5) and the definition of µ q , we calculate and by the definition of g, the mapping q → λ q j is in C k if f is in C k . Moreover, as for every j ∈ Z(q) the mapping p → exp −1 p (N j ) is smooth, we conclude that H is in C k . Note that g is smooth with g(1) = 0, hence H is C k even if the sums in (6.6) depend on Z(q).
As N (q) ∈ B π 6 (ν(q 0 )) is the center of mass for µ q , we have H(q, N (q)) = 0 for every q ∈ W , in particular H(q 0 , N (q 0 )) = 0. Now we are in a position to apply Lemma 6.2. We conclude that there are open neighborhoods U ⊂ W of q 0 , V ⊂ B π 6 (ν(q 0 )) of N (q 0 ), and a function F ∈ C k (U, V ) with {(x, y) ∈ U × V : H(x, y) = 0} = {(x, F (x)) : x ∈ U }. With Theorem 2.6 we deduce, that N coincides with F on U . Hence N is in C k on U .
Remark 6.4 In particular, the preceding lemma shows that the averaged normal N can be used for the projection in the case of immersions with L p -bounded second fundamental form, which was the case considered in [4]. For an (r, λ)-immersion f ∈ C k , the normal ν f is in C k−1 , while the averaged normal N is in C k . In particular, the averaged normal of a C 1 -immersion is differentiable and forms locally a tubular neighborhood around the immersion. Thus it is possible to construct diffeomorphisms φ i : M 1 → M i using the averaged normal. However, if one likes to show convergence as in [14] and in [4], we require N even to be in C 2 . For that purpose, an additional smoothing of f is unavoidable; this was also performed by Langer (see the first paragraph on p. 229 in [14], where a C 1 -perturbation is made in order to smooth the immersion). On the other hand, a pure smoothing argument would not suffice to prove Theorems 1.1 and 1.3. As in general the limit is not even differentiable, one has to project from f i0 (M i0 ) for a fixed and sufficiently large i 0 . The averaged normal is needed then in order to estimate the size of the tubular neighborhood.
Now the rest of the proof is analogous to the case of codimension 1. First we note that with the preceding lemma one easily derives an estimate for the size of the tubular neighborhood around f 1 formed by N . This is done by using elementary geometry in much the same way as in the appendix (where the case of codimension 1 is considered); as we assumed λ to be small and hence N nearly to be perpendicular to f 1 , it is even easier here as we can estimate rather roughly (and do not need an analogue of Lemma 3.1 b) for that). Moreover, we can show the existence and uniqueness of intersection points of f 1 (p)+ N (p) with an appropriate restriction of f i (M i ) by the fixed point argument of [4]. To show surjectivity of φ i one uses the estimate for the size of the tubular neighborhood and shows that f i (M i ) lies within this neighborhood. The rest of the proof is the same as in the case of codimension 1.
The question arises, whether compactness in higher codimension, that is Theorem 1.3, can also be shown for an arbitrary Lipschitz constant λ (as in the case of codimension 1). Surely, the bound λ ≤ 1 4 is not optimal. One could try to find the largest possible bound for λ, and -in the case that it is finite -to give a counterexample for immersions exceeding this bound. We would like to suggest two possibilities for extending the construction in this section to immersions with Lipschitz constant larger than the ones considered here: First, as proposed in the remark on p. 511 in [13], one could use another definition for the center of mass, which allows one to define centers in larger balls. The second is to find a center of mass not in a convex ball, but in a larger convex subset of G n,k . Such kind of subsets of Grassmannians have been detected by J. Jost and Y.L. Xin in [12].

A. Size of tubular neighborhoods
In this appendix we like to prove Lemma 2.15, that is we estimate the size of a tubular neighborhood around a graph depending on different quantities such as angles and Lipschitz constants. We shall use the notations introduced in the paragraph preceding Lemma 2.15. For a general treatise on the existence of tubular neighborhoods see [5] and [10]. Moreover, in Lemma A.1 we will show a result needed for proving that the projection in Section 4 has at most one point of intersection with an appropriate subset of f i (M i ).
Proof of Lemma 2.15: a) We like to start with the following initial consideration: Let q ∈ B ̺ . Let f (x) = (x, u(x)) and τ f (q) ∈ G m+1,m be the tangent space at q as in (2.1). In particular τ f (q) is an m-space in R m+1 perpendicular to ν(q). Furthermore let K ⊂ τ f (q) be a 1-dimensional subspace of τ f (q). Let p ∈ B ̺ and let α ≤ π 2 be the smaller angle enclosed by the lines ω(p) and K. From (2.11) we deduce Now let us come to the main part of the proof: Let x, y ∈ B ̺ with x = y. Without loss of generality we may assume x − y ∈ R 1 × {0} ⊂ R m . By the mean value theorem there is a z ∈ {(1 − t)x + ty : t ∈ (0, 1)} ⊂ B ̺ with where x 1 , y 1 are the first coordinate of the vectors x, y respectively. Let {e 1 , . . . , e m } be the standard basis of R m . We set K := span {(e 1 , ∂ 1 u(z))} ⊂ τ f (z).