C^1-regularity for local graph representations of immersions

We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.


Introduction
An immersion into R n is a differentiable function f : M → R n defined on a differentiable manifold M m , such that for each q ∈ M the mapping f * |T q M is injective. 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.
This concept is used in various geometric contexts; as an example and as motivation we consider the following compactness theorem proved by J. Langer [5]: Let f i : Σ i → R 3 be a sequence of immersed surfaces with uniformly L p -bounded second fundamental form, p > 2, and uniformly bounded area. Then, after passing to a subsequence, there are a limit immersion 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 . The result can be generalized to higher dimensions and codimensions; see [2], [3], [4]. For proving the statement, one uses the Sobolev embedding and shows 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.
This conclusion plays an important role in the proof of the compactness theorem and is just one example of a fundamental principle frequently used in geometric analysis and related fields: For a given global object, that is a manifold embedded or immersed in R nusually of some specific geometric type, for example a minimal surface -one investigates the local graph representations in order to derive further characteristics of the given object. For that one uses the global geometric information and derives specific properties satisfied 1. Introduction by each of the graph functions, for example bounds for specific norms, or particular partial differential equations to be satisfied. For each of the graph functions, it is then possible to apply all the well-known results from real analysis like embedding theorems or regularity theory.
In this paper, we like to take a slightly different point of view. Instead of deriving special kinds of graph representations from specific geometrical settings, we shall take immersions with specific graph representations as our starting point. More precisely, our concept is the following: We consider an immersion and assume that it can be represented at any point over a ball of fixed radius r > 0 as the graph of a function u satisfying some specific properties; now, loosely speaking, we claim that each of the graph functions satisfies much better properties than one would anticipate from the ordinary rules of analysis.
In fact, there is a huge difference between a single graph and a graph coming from an immersion in the way described above. In the latter case, we know as an additional information that such a graph representation is possible at any point of the immersion. In particular, two graphs that are close to each other have overlapping parts and each of the graphs satisfies specific properties, such as a bounded norm. Hence all graphs having one point in common depend on each other. This can be seen as a combinatorial restriction and allows much stronger results than one would expect using only the given properties of each single graph.
Let us first generalize the concept of immersions with bounded norm Du C 0 (Br) ≤ λ for the graph functions u to immersions satisfying only a weaker bound. Again we consider C 1immersions with graph representations u : B r → R k over the affine tangent space, but this time we only assume that u C 0 (Br) ≤ rλ. If such a representation is possible at every point for fixed λ and r, we say that f is a C 0 -(r, λ)-immersion. The factor r on the right hand side is necessary for scale-invariance. A graph function of a C 0 -(r, λ)-immersion does not need to be differentiable; this explains the notation that we use for this kind of immersion. For the precise definitions and further details the reader is referred to Section 2.
Of course it is completely impossible to derive Lipschitz estimates for a single function satisfying only a C 0 -bound, even if the function is known to be smooth or if the C 0 -norm is particularly small. However, as we have claimed above, graph functions coming from immersions in the described way have much better properties than a single function. Denoting by m the dimension of the manifold on which the immersion is defined, we obtain the following theorem: The constant Λ can be given explicitly by Λ(m) := 10 −5 m −2 .
Hence a sufficiently small C 0 -norm implies that each graph function is smooth with small C 0 -norm of Du, that is with small Lipschitz constant. Equivalently, we can say that the space of C 0 -(r, λ)-immersions embeds into the space of (r, λ Λ )-immersions. The statement is true in arbitrary codimension and also for noncompact manifolds.
As here we are assuming only a small C 0 -norm, we obtain all at once whole classes of new embedding theorems -provided the functions come from graph representations as described above. For example one can think of the case of Hölder continuous C 0,α -graphs or the Sobolev border case of W 2,m -graphs in dimension m.
The question arises, whether the result will still be true, if we assume graph representations not over the affine tangent space, but over other appropriately chosen m-spaces. In the appendix we will show that this is not the case.

Notation and definitions
We begin with some general notations: For n = m + k let G n,m denote the Grassmannian of (non-oriented) 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 1immersion. 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 . In this manner we define the tangent 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 Let π : R n → R m be the standard projection onto the first m coordinates.
. 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 [5]): 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 .

Notation and definitions
The notion of a C 0 -(r, λ)-immersion: Every (r, λ)-immersion admits a local representation as a graph of a differentiable function u with Du C 0 (Br) ≤ λ. This inequality corresponds to an estimate of the slope of the graph, i.e. to an estimate of the Lipschitz constant of u. It is a natural generalization to consider immersions with graph functions u, which satisfy only a bound for some weaker norm. Any such definition should reasonably be scale-invariant (i.e. if f is an (r, λ)-immersion and c > 0, then cf is a (cr, λ)-immersion).
Assuming only a bound for the C 0 -norm yields the notion of a C 0 -(r, λ)-immersion: It would not be sensible here to assume u C 0 (Br) ≤ λ, as the notion of C 0 -(r, λ)-immersions would not be scale-invariant then. For that reason we require the bound rλ.
Here we require u only to be a continuous function. Note that the assumption on f to be a smooth immersion does not imply that u is differentiable. Surely the implicit function theorem ensures a smooth graph representation over the tangent space. However this representation might only be possible for radii less than r. Over the ball B r one might have a continuous graph representation with a graph which gets vertical in a point. Hence smoothness of f does not guarantee smoothness of u. Obviously every (r, λ)-immersion is also a C 0 -(r, λ)-immersion. Surprisingly, in some sense also the opposite is true: . This is precisely the statement of Theorem 1.1. Moreover, as we have seen above, a graph function u does not need to be smooth in the case of a C 0 -(r, λ)-immersion; for that reason we may interpret Theorem 1.1 also as a higher regularity result.
Reformulation of Theorem 1.1: Theorem 1.1 is a statement for C 0 -(r, λ)-immersions with fixed r and λ. We like to give an alternative formulation which holds for any immersion.
For an immersion f : M → R n let r 1 (f, λ) ≥ 0 be the maximal radius, such that for any Similarly, let r 0 (f, λ) ≥ 0 be the maximal radius, such that for any q ∈ M the set A −1 q •f (U r,q ) is the graph of a C 0 -function with u C 0 (Br) ≤ rλ.
With this notation Theorem 1.1 reads as follows:

Preparations for the proof
The main step of the proof is to compare the position of two tangent spaces at points on the surface that are not too far from each other. For that we have to find a sufficiently large set U ⊂ M , such that f (U ) may be written over both spaces as graph with small C 0 -norm respectively; this will be done in Lemma 3.3. To compare the spaces with each other, we shall use a finite number of comparison points on each space, constructed by means of the immersion piece f (U ). A concrete estimate (in a slightly more general formulation) is deduced in Lemma 3.1. Using this method, we are able to deduce smoothness of the graphs and to estimate the Lipschitz constant. However, due to the limited size of f (U ), this estimate holds only on a smaller radius ̺ < r. Lemma 3.2 shows a method how to enlarge the radius, provided the Lipschitz constant is sufficiently small. This enables us to prove the theorem.
Let us come to the first statement, the comparison of two spaces by distance bounds of finitely many points. The proof consists of elementary geometry and is carried out here in full detail: Proof: First we show that E is a graph over R m × {0}. Suppose E might not be written as a graph over R m × {0}. If π denotes the standard projection from R n = R m × R k onto R m , then and on the other hand with (3.1) and L ≤ 1 for each j The following constructions are carried out within the subspace R m ∼ = R m × {0} ⊂ R n . By As long as s = e ν the points 0, e ν and s constitute a rectangular triangle with hypotenuse in G. The splitting e ν = e ⊤ ν + e ⊥ ν ∈ G ⊕ G ⊥ yields with the Euclidean theorem |e ν | 2 = |e ⊤ ν ||s|, hence also in the case s = e ν . But then |w − e ν | ≥ 1 √ m for all w ∈ G ⊥ and as π(E) ⊂ G ⊥ |w − e ν | ≥ 1 √ m for all w ∈ π(E).
(3.6) But (3.5) is true also for j = ν, a contradiction. This shows that E is a graph over R m × {0}.
We like to estimate the norm of A. For x ∈ R n and j ∈ {1, . . . , m} let x j ∈ R n be the orthogonal projection of x onto span{(e j , 0)} ⊂ R n . With L ≤ 1 and (3.
The second intercept theorem implies With w j j = (e j , 0) we obtain Next choose ν, such that |a j | ≤ |a ν | for all j. Without loss of generality ν = 1. There are λ 1 , . . . , λ m ∈ R with w 1 = m j=1 λ j (e j , a j ). As If f : M → R n is an immersion and q ∈ M , then the Euclidean isometry A q is not uniquely determined (as remarked in Section 2). We say that a Euclidean isometry is admissible for the point q ∈ M , if the origin is mapped to f (q) and the subspace If a statement is true for one admissible isometry, it often is also true for any admissible isometry. This will be used in the proof of the following lemma. Although the statement of the lemma is not very surprising, its proof is quite complex as we have to use the precise Definition 2.1 in the conclusion. Of course the numbers in the lemma are not optimal, but they suffice to prove Theorem 1.1. In Step 2 below we shall apply Lemma 3.1, however the main application of this lemma will be in the proof of Theorem 1.1. Proof: Step 1: Let q ∈ M , p ∈ U r,q and ϕ q := π • A −1 q • f , where A q is an arbitrary but fixed admissible isometry as explained above. Then B 4 5 r (ϕ q (p)) ⊂ ϕ q (U r,p ).

Proof of Step 1:
Without loss of generality we may assume A q = Id R n . The set A −1 q • f (U r,q ) is the graph of a C 1 -function u : B r → R k . We set w := ϕ q (p) ∈ B r . After a suitable rotation, we may assume that {v 1 , . . . , v m } with v j := (e j ,∂ j u(w)) √ 1+|∂ j u(w)| 2 for 1 ≤ j ≤ m is an orthonormal basis of τ f (p) (and still may assume A q = Id R n ). Let R ∈ SO(n) be a rotation with R(e j , 0) = v j for all j ∈ {1, . . . , m}. In particular the mapping A p : R n → R n , A p (x) := Rx + f (p), is an admissible Euclidean isometry for the point p ∈ M . Therefore A −1 p • f (U r,p ) is the graph of a C 1 -functionũ : B r → R k withũ(0) = 0 and Dũ C 0 (Br ) ≤ λ. We define a mapping g : B 29 30 r (0) → R m , y → y − π • R(y,ũ(y)).
Proof of Step 2: By Step 1 we have π •A −1 q •f (U ) = B 4 5 r (ϕ q (p)). Moreover, as one can replace B 29 30 r (0) in Step 1 by B r−ε (0) for any sufficiently small ε > 0, we deduce with the fixed point argument of Step 1 that A −1 q • f (U ) is a graph over B 4 5 r (ϕ q (p)). Now let p ′ ∈ U r,p . We write A −1 p • f (U r,p ) (where A p is as in Step 1) as graph of the C 1 -functionũ : B r → R k . Then there is a unique x ∈ B r with A −1 p • f (p ′ ) = (x,ũ(x)). With the rotation R of Step 1 we have ∂ 1ũ (x)), . . . , (e m , ∂ mũ (x))}.
Let v j and w be as in Step 1. We note that R(e j , 0) = v j and estimate We apply Lemma 3.1 with E := τ f (p ′ ) ∈ G n,m , v j := R(e j , ∂ jũ (x)), L := 8 √ mλ and conclude that τ f (p ′ ) may be written as a graph over R m × {0}. As this is true for any p ′ ∈ U r,p , an argument similar to the one in the paragraph preceding (4.6) together with the considerations at the beginning of the proof of Step 2 allows us to conclude that A −1 q • f (U ) is the graph of a C 1 -functionû : B 4 5 r (ϕ q (p)) → R k with Dû C 0 (B 4 5 r (ϕ(q))) ≤ 8 √ mλ. In particular ϕ q : U → B 4 5 r (ϕ q (p)) is a diffeomorphism; hence U is connected.

Proof of Step 3:
Let ϕ q and A q be as in Step 1. For every x ∈ ∂B 19 20 r there is exactly one p x ∈ U r,q with ϕ q (p x ) = x. For each x ∈ ∂B 19 20 r set U x := U r,px ∩ ϕ −1 q (B 4 5 r (x)). Moreover set V q := U r,q ∪ x∈∂B 19 20 r U x . By Step 1 we have = B 7 4 r (0). Each set U x is connected and we have p x ∈ U r,q ∩ U x . Therefore also V q is connected, and we have q ∈ V q . Now let R > r be the greatest radius, such that A −1 q • f (U R,q ) is the graph of a C 1 -function u : B R → R k . Suppose R < 7 4 r. As R > r, we have ̺ := R − 4 5 r > 0. Define sets U x as above, but here for x ∈ ∂B ̺ . Set W q := U r,q ∪ x∈∂B̺ U x . Analogous to the considerations above, W q is a connected set containing q, and it holds ϕ q (W q ) = B R (0). We deduce W q ⊂ U R,q . As we assumed here, that A −1 q • f (U R,q ) is a graph over B R (0), and as ϕ q (W q ) = B R (0), we conclude W q = U R,q . As R is maximal, we deduce Du C 0 (B R ) = ∞. But this contradicts Step 2, saying that Du C 0 (B 4 5 r (x)) ≤ 8 √ mλ for all x ∈ ∂B ̺ . Hence it holds R ≥ 7 4 .
Using the preceding considerations, we conclude V q = U 7 4 r,q (in particular V q does not depend on the choice of A q ) and A −1 q • f (U 7 4 r,q ) is the graph of a C 1 -function u : As this is true for any point q ∈ M , the function f is a ( 7 4 r, 8 √ mλ)-immersion.
b) If λ ≤ 1 10 and p ∈ U 2 5 r,q , then U 2 5 r,q ⊂ U r,p . Proof: a) Pass to the graph representation, use the bound on the C 0 -norm and the triangular inequality.
b) Let x ∈ U 2 5 r,q and ϕ p = π • A −1 p • f . With part a) we estimate Hence U 2 5 r,q ⊂ ϕ −1 p (B r ). But U 2 5 r,q is a connected set containing p, hence included in the p-component of ϕ −1 p (B r ), that is in U r,p . Hence U 2 5 r,q ⊂ U r,p . Now let λ ≤ Λ, r > 0 and f : M m → R n be a given C 0 -(r, λ)-immersion. We set ̺ := r 5 . Moreover let q ∈ M be an arbitrary point. As 2̺ < r, the set f (U 2̺,q ) may be written over f (q) + τ f (q) as the graph of a function u : B 2̺ → R k with u C 0 (B 2̺ ) ≤ rλ.

Proof of the embedding theorem
As the argumentation of this proof is invariant under rotations and translations, we may assume without loss of generality that A q = Id R n (where A q : R n → R n is an admissible isometry for the point q ∈ M ). In particular f (q) = 0 and τ f (q) = R m ×{0} ⊂ R m ×R k = R n .
Therefore the set f (U 2̺,q ) may be written also over f (p) + τ f (p) as graph of a function with small C 0 -norm -more precisely there exists a functionũ : B r → R k with ũ C 0 (Br) ≤ rλ and f (U 2̺,q ) ⊂ {A p (y,ũ(y)) : y ∈ B r }.
Let {e 1 , . . . , e m } be the standard basis of R m . For 1 ≤ j ≤ m define As x ∈ B ̺ we have x j ∈ B 2̺ for each j. Hence for each j there is exactly one p j ∈ U 2̺,q with f (p j ) = A q (x j , u(x j )) = (x j , u(x j )). (4.3) As p j ∈ U 2̺,q and U 2̺,q ⊂ U r,p , there are also unique y j ∈ B r with f (p j ) = A p (y j ,ũ(y j )). We wonder whether there is a Λ > 0, such that each generalized C 0 -(r, λ)-immersion with λ ≤ Λ is also a generalized (r, λ Λ )-immersion. The following figure shows, that this is not the case: ✒ ✿ here we choose graph representations over the affine tangent space ✗ ❖ ( ) here in any point we choose graph representations over the horizontal line Figure 5.1 This example shows, that a generalized C 0 -(r, λ)-immersion with very small λ does not need to be a generalized (r, λ Λ )-immersion.
Moreover, in the figure above, the part of the immersion over the horizontal line cannot be represented over any other line (with the same radius). This shows that we require graph representations over the affine tangent space for our theorem.