Immersions with local Lipschitz representation

Transcription

1 ALBERT-LUDWIGS-UNIVERSITÄT FREIBURG FAKULTÄT FÜR MATHEMATIK UND PHYSIK Immersions with local Lipschitz representation Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau vorgelegt von Patrick Breuning betreut durch Prof. Dr. Ernst Kuwert März 2011

2 Dekan: Prof. Dr. Kay Königsmann 1. Gutachter: Prof. Dr. Ernst Kuwert 2. Gutachter: Prof. Dr. Tobias Lamm Datum der mündlichen Prüfung: 08. Juni 2011

3 Contents Introduction iii 1. Foundations Immersions with uniform graph representation Geometry of Grassmann manifolds The Riemannian center of mass Compactness of immersions with bounded second fundamental form Notation and overview of the proof Existence of intersection points by a fixed point argument The L p -bound for the second fundamental form of the limit Compactness of proper immersions on noncompact manifolds Compactness of immersions with local Lipschitz representation Definitions and basics Transversality and tubular neighborhoods Intersection points and definition of ϕ i The limit function lies in F 0 (r, λ) Compactness in higher codimension Lipschitz regularity of local graph representations of immersions Preparations for the proof Proof of the embedding theorem A. Size of tubular neighborhoods 73 Bibliography 79 i

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5 Introduction In [22] J. Langer investigated compactness of immersed surfaces in R 3 admitting uniform bounds on the second fundamental form and the area of the surface. 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 [7] to immersions of arbitrary dimension and codimension. Moreover, the compactness theorem was generalized by S. Delladio in [8] to hypersurfaces of arbitrary dimension. The general case, that is compactness in arbitrary dimension and codimension, was proved in the author s diploma thesis [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 (B r ) 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. This 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 n usually 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 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 parts of this thesis, 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 or a sequence of immersions and assume that any of these 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 iniii

6 Introduction formation 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. Primarily, we shall use such an approach in Chapter 4 where graphs with small C 0 -norm are considered, but to a certain extent also in Chapter 3, where we consider uniformly Lipschitz bounded graphs. We like to describe the contents of the present paper in detail now. The first part after a chapter with foundations is concerned with Langer s compactness theorem and is a continuation and generalization of the author s diploma thesis. For the convenience of the reader the main theorem of [4] is restated on page 11 of this thesis. As described above, we consider immersions admitting a uniform L p -bound for the second fundamental form and a uniform bound for the volume. In the first sections we would like to recall the main arguments of the proof. Several improvements are included. In particular, for the construction of the diffeomorphisms ϕ i one uses a kind of projection in an averaged normal direction ν. To show that ν intersects the image of each (appropriately restricted) immersion f i in exactly one point, we shall use Banach s fixed point theorem here, which is more elegant and replaces the original argument using Brouwer s invariance of domain. Moreover, the definition of ν was based on a smoothing of the limit immersion. Here we present a different possibility involving the Riemannian center of mass (this part is postponed to the subsequent chapter, for which it is more essential). The bound A(f) L p (M) A for the limit immersion in the case of dimension 1 and 1 < p < 2 was omitted in [4]. The proof is included here for the sake of completeness. Finally, as main part of the chapter, we prove a version for smooth proper immersions admitting uniform L -bounds for the second fundamental form A and its covariant derivatives k A. Here the manifolds on which the immersions are defined are not required to be compact. For a proper immersion f : M R n with induced metric g and volume measure µ g on M, let µ = f(µ g ) be the Radon measure on R n defined by µ(e) = µ g (f 1 (E)) for E R n. Abbreviating we write L (B R ) for the L -norm on f 1 (B R ), where B R R n is the open ball of radius R centered at the origin. We obtain the following theorem: Theorem 0.1 (Compactness of immersions with bounded second fundamental form) Let f i : M i R n be a sequence of proper immersions, where M i is an m-manifold without boundary and 0 f i (M i ). With µ i = f i (µ g i) assume µ i (B R ) C(R) for any R > 0, (1) k A i L (B R ) C k (R) for any R > 0 and k N 0. (2) Then there exists a proper immersion f : M R n, where M is again an m-manifold without boundary, such that after passing to a subsequence there are diffeomorphisms ϕ i : U i (f i ) 1 (B i ) M i, where U i M are open sets with U i U i+1 and M = i=1 U i, such that f i ϕ i f C 0 (U i ) 0, and moreover f i ϕ i f locally smoothly on M. iv

7 Moreover, the immersion f also satisfies (1) and (2), that is µ(b R ) C(R) and k A L (B R ) C k (R). The assumption 0 f i (M i ) ensures that the immersions f i do not diverge uniformly. This can be weakened to f i (M i ) K for a fixed compact set K R n. In contrast to the compact case, here the bound A L (B R ) C(R) depends on the radius of the image. Hence the radius of the graph representation depends on p M and is decreasing as f(x) is getting larger. This explains the need of some technical refinements which allow us to handle graphs with different radii. We would like to remark that a similar result is shown by A. Cooper in [6], however the construction of the diffeomorphisms ϕ i is not carried out there (see Remark 2.26 in this thesis). Theorem 0.1 has some important applications such as convergence proofs for geometric flows, for example for the mean curvature flow or the Willmore flow (see e.g. [14], [21], [25]). As a corollary of Theorem 0.1 we prove convergence of the corresponding measures: Corollary 0.2 Let f i and f be as in Theorem 0.1 and let µ i = f i (µ g i), µ = f(µ g ). Then µ i µ in C 0 c (R n ) as i. Inspired by these results, in the next chapter we like to take immersions with uniform graph representations as our starting point. Having proven compactness for immersions with uniformly bounded second fundamental form, and having previously shown that any such immersion is an (r, λ)-immersion, it is very natural to investigate compactness properties also for (r, λ)-immersions with fixed r and λ. 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, nearly flat, and the averaged normal ν is almost perpendicular. These properties are used repeatedly, for example to show that ν intersects the image of each (appropriately restricted) immersion f i in exactly one point. 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 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 Chapter 1.1). 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 0.3 (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. v

8 Introduction 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 0.3 we deduce the following corollary: Corollary 0.4 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. Here the averaged normal ν cannot be constructed as in the case of hypersurfaces. We will give an alternative construction involving the above mentioned 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 0.5 (Compactness of (r, λ)-immersions in arbitrary codimension) Let λ 1 4. Then F1 V (r, λ) is relatively compact in F 0 (r, λ) in the sense of Theorem 0.3. As in Corollary 0.4, we deduce for λ 1 4 that there are only finitely many manifolds in F1 V (r, λ) up to diffeomorphism. Surely, the bound λ 1 4 is not optimal; at the end of Chapter 3 we will discuss some possibilities how to prove the theorem for greater Lipschitz constant. Having investigated immersions with bounded norm Du C 0 (B r ) λ for the graph functions u, we would like to generalize our concept to immersions satisfying only a weaker bound. Again we consider C 1 -immersions with graph representations u : B r R k over the affine tangent space, but this time we only assume that u C 0 (B r ) 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 further details the reader is referred to Chapter 1.1. 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: Theorem 0.6 (Embedding theorem for C 0 -(r, λ)-immersions) For every m N there is a Λ = Λ(m) > 0, such that every C 0 -(r, λ)-immersion with λ Λ is also an (r, λ Λ )-immersion. The constant Λ can be given explicitly by Λ(m) := 10 5 m 2. vi

9 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. In this thesis we have investigated immersions with local Lipschitz and local C 0 -representation, making use of combinatorial properties of overlapping graphs. It seems natural to extend our method to immersions with graphs satisfying a specific kind of partial differential equation. Such immersions frequently appear in various geometric contexts and it seems likely that similar methods can yield useful results. Closer investigations of such immersions will be an interesting topic for future research. Finally, we would like to give a brief outline of this thesis: In the first chapter, the foundations, we settle the notation and introduce all different kinds of immersions that are used in this thesis. Moreover we would like to recall some basic properties of Grassmannians and to introduce the Riemannian center of mass. In the second chapter we consider immersions with bounded second fundamental form. The third chapter is concerned with compactness properties of (r, λ)-immersions, both in codimension 1 and in arbitrary codimension. In the last chapter we will show that C 0 -(r, λ)-immersions are (r, λ Λ )-immersions for λ Λ. Finally, in the appendix, we estimate the size of tubular neighborhoods around a graph which is needed in order to show compactness of (r, λ)-immersions. Acknowledgement I would like to thank my advisor Prof. Dr. Ernst Kuwert for his help, guidance and the excellent support. I also wish to express my gratitude to all members of the working group Geometric Analysis in Freiburg for their support and many stimulating discussions within the last years. Moreover I would like to thank Dr. Manuel Breuning for diligently correcting linguistic and typographic mistakes. From December 2006 to November 2009 I was member of the DFG-Forschergruppe Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis. I would like to thank the DFG for the support. vii

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11 1. Foundations In this chapter we fix the notation and introduce all different types of immersions that are used in the present paper. Moreover we would like to recall some properties of Grassmann manifolds and to introduce the concept of a Riemannian center of mass (which will be applied in Chapter 3.5 to a Grassmann manifold) Immersions with uniform graph representation In the first section we like to explain in detail all different kinds of immersions that are used in this paper. 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 ϱ denote the open ball in R m of radius ϱ > 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 R n = T q M (T q M) and (T q M) is perpendicular to T q M. In this manner we may define a tangent and a normal map and τ f : M G n,m, q T q M, ν f : M G n,k, q (T q M). (1.1) (1.2) 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 A(x) = Rx + T for all x R n. For a given point q M let A q : R n R n be a Euclidean isometry, which maps the 1

12 1. Foundations 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. M m f R n = R m R k f(q) q U r,q π A 1 q f A 1 q ( ) B r 0 R m π Figure 1.1 Local representation as a graph. The subset of M drawn in bold lines represents the pre-image (π A 1 q f) 1 (B r ). We come to the central definition: Definition 1.1 An immersion f is called an (r, λ)-immersion, if for each point q M the set A 1 q f(u r,q ) is the graph of a differentiable function u : B r R k with Du C 0 (B r ) λ. 2

13 1.1. Immersions with uniform graph representation 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 ( )1 m 2 A = a j 2 j=1 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 (B r) λ directly implies that u is λ-lipschitz. Moreover the norm Du C 0 (B r) does not depend on the choice of the isometry A q. Now let M i be a sequence of compact m-manifolds and f i : M i R n a sequence of W 2,p - immersions with A i L p (M i ) A (1.3) as in [4], where A i denotes the second fundamental form of f i, p > m and A <. We like to explain the fundamental difference between a sequence of (r, λ)-immersions as in Theorem 0.3 or 0.5, and a sequence of immersions f i with the property (1.3). For immersions with (1.3), it is shown in [22] and in [4] that for all λ > 0 there is an r > 0 such that every immersion f i is an (r, λ)-immersion. Here the size of the radius r does not depend on the index i. On the other hand a sequence of (r, λ)-immersions as in Theorem 0.3 or 0.5 does not have this property as λ > 0 is fixed here. For any λ with 0 < λ < λ and any i N one can choose an r < r such that f i is an (r, λ )-immersion; however in general r cannot be chosen independently from i. Moreover, sequences with (1.3) satisfy further properties, for example uniform L p -bounds for the second derivatives of the graph functions. Hence a sequence of (r, λ)-immersions generalizes sequences of immersions with L p -bounded second fundamental form. If M is noncompact and f : M R n is a proper immersion with A L (B R ) C 0 (R) for any R > 0 as in Theorem 0.1, it is no longer possible to choose a radius r > 0 independently from q M to obtain a graph representation of the set A 1 q f(u r,q ). In fact the possible size of the radius r depends in this case on the norm of f(q) R n. For this reason we have to adapt the notion of an (r, λ)-immersions by replacing the real number r > 0 by a sequence (r j ) j N. This will be done in detail in Chapter 2.4. 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. 3

14 1. Foundations 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 R m {0} R m R k onto f(q)+e. Let Ur,q E M be the q-component of the set (π A 1 q,e f) 1 (B r ). Again the isometry A q,e is not uniquely determined but the set Ur,q E does not depend on the choice of A q,e. Definition 1.2 An immersion f is called a generalized (r, λ)-immersion, if for each point q M there is an E = E(q) G n,m, such that the set A 1 q,e f(u E r,q) is the graph of a differentiable function u : B r R k with Du C 0 (B r) λ. Obviously every (r, λ)-immersion is a generalized (r, λ)-immersion, as we can choose E(q) = τ f (q) for any q M. For fixed dimension m and codimension k we denote by F 1 (r, λ) the set of generalized (r, λ)-immersions f : M R m+k with 0 f(m), where M is any compact m-manifold without boundary. For V > 0 we denote by F 1 V (r, λ) the set of all immersions in F 1 (r, λ) with vol(m) V. 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 (B r ) 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: Definition 1.3 A continuous function f is called an (r, λ)-function, if for each point q M there is an E = E(q) G n,m, such that the set A 1 q,e f(u E r,q) 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 [19] there exists even a smooth atlas. In our case, the limit manifold both in Theorem 0.3 and 0.5 will be smooth. Finally, we define the set F 0 (r, λ) by replacing generalized (r, λ)-immersions in F 1 (r, λ) by (r, λ)-functions. 4

15 1.1. Immersions with uniform graph representation 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 (B r ) 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 another 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: Definition 1.4 An immersion f is called a C 0 -(r, λ)-immersion, if for each point q M the set A 1 q f(u r,q ) is the graph of a continuous function u : B r R k with u C 0 (B r) rλ. It would not be sensible here to assume u C 0 (B r ) 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. M f f(u r,q ) ( ) graph gets vertical in one point zoom (}{{} ) r f(q) Figure 1.2 A simple example which shows how a graph function of a smooth C 0 -(r, λ)- immersion fails to be differentiable (here e.g. λ = 2). 5

16 1. Foundations Obviously every (r, λ)-immersion is also a C 0 -(r, λ)-immersion. Surprisingly, in some sense also the opposite is true: Every C 0 -(r, λ)-immersion is also an (r, λ Λ )-immersion if λ Λ = Λ(m). This is precisely the statement of Theorem 0.6. 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 0.6 also as a higher regularity result. The notion of a generalized C 0 -(r, λ)-immersion: For the sake of completeness we like to remark that also the notion of a C 0 -(r, λ)-immersion can be generalized. We obtain the following definition: Definition 1.5 An immersion f is called a generalized C 0 -(r, λ)-immersion, if for each point q M there is an E = E(q) G n,m, such that the set A 1 q,e f(u E r,q) is the graph of a continuous function u : B r R k with u C 0 (B r ) rλ. Of course the preceding definition can be generalized further to the notion of a C 0 -(r, λ)- function as in Definition 1.3. We will not make use of generalized C 0 -(r, λ)-immersions or -functions in this paper, however these notions might be of their own interest. Reformulation of Theorem 0.6: Theorem 0.6 is a statement for C 0 -(r, λ)-immersions with fixed r and λ. an alternative formulation which holds for any immersion. We like to give For an immersion f : M R n let r 1 (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 1 -function u : B r R k with Du C 0 (B r ) λ. 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 (B r ) rλ. Obviously r 1 (f, λ) r 0 (f, λ). With this notation Theorem 0.6 reads as follows: Theorem 1.6 (Reformulation of Theorem 0.6) For every m N there is a Λ = Λ(m) > 0, such that for every immersion f : M m R n and all λ Λ the inequality r 1 (f, λ/λ) r 0 (f, λ) holds. The constant Λ can be given explicitly by Λ(m) := 10 5 m 2. 6

17 1.2. Geometry of Grassmann manifolds 1.2. 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. [23]. 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 [24]. 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 γ(a) = p, γ(b) = q}. (1.4) 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 {v 1,..., v k } of E and {w 1,..., w k } of G such that [ v i, w j = δ ij cos θ i with θ i 0, π ) 2 for 1 i, j k. For given k-spaces E and G, the θ 1,..., θ k are uniquely determined (up to the order) and called the principal angles between E and G. Under all metrics on G n,k being invariant under the action of O(n), there is exactly one metric g with d(e, G) = ( k θi 2 i=1 ) 1 2 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 [3] 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 1.7 Let max{k, n k} 2. 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 0 K(X, Y ) 2. Proof: For min{k, n k} = 1 all sectional curvatures are constant with K(X, Y ) = 1. For a proof see [24], p For min{k, n k} 2 we have 0 K(X, Y ) 2 by [31], Theorem 3. 7

18 1. Foundations The injectivity radius of G n,k is π 2 (see [3], 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 ϱ (P ) around P G n,k with ϱ < π 4 is convex; this follows from Theorem 4 in [18] (see also [10], 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 [11]. A simplified treatment is given in [19]. See also [16]. We like to give a short sketch of this concept. Let (M, g) be a complete Riemannian manifold with induced distance d as in (1.4). µ be a probability measure on M, i.e. a nonnegative measure with µ(m) = dµ = 1. M Let q be a point in M and B ϱ = B ϱ (q) a convex open ball of radius ϱ around q in M. Suppose spt µ B ϱ, where spt µ denotes the support of µ. We define a function P : B ϱ R, P (p) = d(p, x) 2 dµ(x). M Let Definition 1.8 A q B ϱ is called a center of mass for µ if P (q) = inf d(p, x) 2 dµ(x). p B ϱ The following theorem asserts the existence and uniqueness of a center of mass: M Theorem 1.9 If the sectional curvatures of M in B ϱ are at most κ with 0 < κ < and if ϱ is small enough such that ϱ < 1 4 πκ 1/2, then P is a strictly convex function on B ϱ and has a unique minimum point in B ϱ which lies in B ϱ and is the unique center of mass for µ. Proof: See [19], 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: 8

19 1.3. The Riemannian center of mass Lemma 1.10 Assume that the sectional curvatures of M in B ϱ are at most κ with 0 < κ < and ϱ < 1 4 πκ 1/2. Let µ 1, µ 2 be two probability measures on M with spt µ 1 B ϱ, spt µ 2 B ϱ with centers of mass q 1, q 2 respectively. Then for a universal constant C = C(κ, ϱ) < d(q 1, q 2 ) C d(q 2, x) d µ 1 µ 2 (x), M where µ 1 µ 2 denotes the total variation measure of the signed measure µ 1 µ 2. Proof: Let P i (p) = 1 2 M d(p, x)2 dµ i (x) for i = 1, 2. By Theorem in [19], with we have for all y B ϱ the estimate C = C(κ, ϱ) := 1 + (κ 1/2 ϱ) 1 tan(2κ 1/2 ϱ), (1.5) d(q 1, y) C grad P 1 (y). Using spt µ i B ϱ, by Theorem 1.2 in [19] we have grad P i (y) = exp 1 y (x) dµ i (x), (1.6) B ϱ where exp 1 y : B ϱ 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 [16], Lemma (where manifolds of nonpositive sectional curvature are considered) we have d(q 1, q 2 ) C grad P 1 (q 2 ) = C exp 1 q 2 (x) dµ 1 (x) B ϱ = C exp 1 q 2 (x) dµ 1 (x) exp 1 q 2 (x) dµ 2 (x) B ϱ B ϱ C d(q 2, x) d µ 1 µ 2 (x), M where we used exp 1 q 2 (x) = d(q 2, x) and spt µ i B ϱ in the last line. 9

20

21 2. Compactness of immersions with bounded second fundamental form The second chapter is a continuation of the author s diploma thesis and deals with a compactness theorem for immersions with L p -bounded second fundamental form. The following theorem was proved in [4], generalizing results in [22], [7] and [8]: Theorem 2.1 (Compactness theorem for immersions) Let N be a complete Riemannian manifold (without boundary) and let q be a point in N, m a positive integer, p > m, and A, V > 0 constants. Let F be the set of all mappings f : M N with the following properties: M is an m-dimensional, compact manifold (without boundary) f is an immersion in W 2,p (M, N) with A(f) L p (M) A (2.1) vol(m) V (2.2) q f(m). (2.3) Then for every sequence f i : M i N in F there exists a subsequence f j, a mapping f : M N in F, and a sequence of diffeomorphisms ϕ j : M M j, such that f j ϕ j converges in the C 1 -topology to f. We like to replace some of the arguments of the proof by more elegant ones, and as main part of this chapter give a generalization to proper immersions on noncompact manifolds. More precisely Chapter 2 is made up of the following parts: In the first section we will fix the notation and recall some of the results of [4]. In particular we will introduce the notion of graph convergence and explain how to construct the limit. The second section is concerned with the construction of the diffeomorphisms ϕ i. For that it is necessary to show existence of intersection points of some k-spaces with f i (M i ). This will be done by using Banach s fixed point theorem, which is an elegant method and replaces the original argument using Brouwer s invariance of domain. Section 3 shows the bound A(f) L p (M) A for the limit in the case of a 1-dimensional manifold M and 1 < p < 2. This case was omitted in [4] and shall be included here for the sake of completeness. Finally in Section 4 we will show the main result of this chapter, a generalization to proper immersions on noncompact manifolds. 11

22 2. Compactness of immersions with bounded second fundamental form 2.1. Notation and overview of the proof In the first section we like to recall the main results of [4]. This is necessary as we will make frequent use of these results. For the proof we will need the notion of an (r, λ)-immersion as in Definition 1.1. However, in the whole second chapter we shall write α instead of λ and therefore speak of (r, α)- immersions. This is due to consistency with the notations in [22] and [4]. In the subsequent chapters (which are independent of the present one) we shall write λ in order to emphasize the connection with Lipschitz graphs. The theorem is proven first for immersions with values in R n. The general case, that is the case of complete Riemannian manifolds as target, is then deduced using the Nash embedding theorem. Local representation as a graph We like to start with one of the central theorems (Theorem 2.5 in [4]): Theorem 2.2 Let p > m and 0 < α 1. Then there exists a universal constant c = c(m, k, p) > 0 such that every immersion f W 2,p (M, R n ) on a compact m-manifold M is an (r, α)-immersion for all r > 0 with r 1 m p cα A(f) 1 L p (M). (2.4) Here k denotes the codimension. The following lemma (Lemma 2.8 in [4]) is crucial for the proof and will also be needed (in a variation) for the noncompact case: Lemma 2.3 Let f : M R n be an (r, α)-immersion and p, q M. a) If p U r,q, then f(p) f(q) (1 + α 2 )r. b) If α 2 < 1 3 and U r 4,q U r 4,p, then U r 4,p U r,q. For (r, α)-immersions f : M R n we define the notion of the δ-net: Definition 2.4 Let Q = {q 1,..., q s } be a finite set of points in M and let 0 < δ < r. We say that Q is a δ-net for f if M = s j=1 U δ,q j. The number of elements of a δ-net can be bounded by above (Lemma 2.10 in [4]): Lemma 2.5 Assume α 2 < 1 3 and 0 < δ < r. Then every (r, α)-immersion f : M Rn admits a δ-net with at most ( ) 4 m δ vol(m) points. 12

23 2.1. Notation and overview of the proof Here the volume vol(m) is measured with respect to the volume measure induced by the metric f g eucl. Convergence of graph systems First we like to explain how to represent an immersion in F as a system of graphs. that we define the space of graph systems with s elements by G s = {(A j, u j ) s j=1 : A j : R n R n is a Euclidean isometry, u j W 2,p (B r, R k )}. For (2.5) Every Euclidean isometry A : R n R n splits uniquely into a rotation R SO(n) and a translation T R n, such that A(x) = Rx + T for all x R n. If denotes the operator norm and if Γ = (A j, u j ) s j=1 Gs, Γ = (Ãj, ũ j ) s j=1 Gs, we set d(, ) : G s G s R, d(γ, Γ) = s ( R j R j + T j T j + u j ũ j C 1 (B r)). j=1 (2.6) This makes (G s, d) a metric space. Now let f : M R n be an (r, α)-immersion and Q = {q 1,..., q s } a δ-net for f with s elements. To each q j Q we may assign a neighborhood U r,qj, a Euclidean isometry A j, and a C 1 -function u j : B r R k as described above. Hence, to given f, r and Q, we may assign a graph system Γ = Γ(f) = (A j, u j ) s j=1 G s. The isometries A j and functions u j are not uniquely determined, but we always have u j (0) = 0 and Du j (0) = 0. For any j {1,..., s} we finally set Z(j) := {1 k s : U δ,qj U δ,qk }. With the preceding notations we are able to define a notion of convergence for graph systems: Definition 2.6 (Convergence in the sense of graph systems) Let a sequence f i : M i R n of immersions be given. We say f i is convergent in the sense of graph systems, if there are fixed α, r, δ > 0 with r > δ, and s N, such that the following properties are satisfied: Each f i is an (r, α)-immersion. For each f i there exists a δ-net with s points, for which the following holds: Z i (j) = Z(j) for fixed sets Z(j) independent of i. There exists a system Γ G s, such that the graph systems Γ i corresponding to f i converge in (G s, d) to Γ. 13

24 2. Compactness of immersions with bounded second fundamental form Then the following statement is true (Theorem 3.2 in [4]): Theorem 2.7 Every sequence in F admits a subsequence that converges in the sense of graph systems. The limit manifold and immersion We like to give a short sketch of the construction of the limit manifold and limit immersion. We start with a sequence of (r, α)-immersions, convergent in the sense of graph systems, with α 2 < 1 3, δ = r 16, δ 10 -nets Qi = {q1 i,..., qi s} with s elements, intersection sets Z(j) = {1 k s : U i U i } which are independent of i, limit isometries A δ,qj i δ,qk i j and limit functions u j : B r R k. To simplify the notation, for 0 < ϱ r we set Uϱ,j i := U i ϱ,qj. i For the open ball B δ R m we set B j δ = B δ {j}. This makes s j=1 Bj δ endow s j=1 Bj δ a disjoint union. We with the topology of the disjoint union, which is defined as follows: A subset U s j=1 Bj δ is open if and only if U Bj δ B δ is open for every j. We define a relation on s j=1 Bj δ. For (x, j), (y, k) s l=1 Bl δ we set (x, j) (y, k) [k Z(j) and A j (x, u j (x)) = A k (y, u k (y))]. (2.7) Then we have (Lemma 4.1 in [4]): Lemma 2.8 The relation is an equivalence relation. This enables us to define the limit manifold. As set, M is defined to be the quotient space M = ( s j=1 B j δ ) /, (2.8) resulting from the equivalence relation of above. Let M be endowed with the quotient topology. For (x, k) s j=1 Bj δ, let [(x, k)] denote the corresponding equivalence class. Let P denote the canonical projection from s j=1 Bj δ onto M, and P j the restriction P B j δ :Bj δ P (Bj δ ). We can consider P j as a mapping defined on B δ. We note that P is injective on B j δ, and in particular P j invertible. For V B δ we set V j = V {j}. For V B δ open we define φ j V : P (V j ) V, [(x, j)] P 1 j ([(x, j)]) V, (2.9) which yields a well-defined mapping. Finally we denote the set of all such mappings by A, that is A = {φ k W : 1 k s, W B δ open}. To simplify the notation, we will often identify sets V j with V, and elements (x, j) with x. 14

25 2.2. Existence of intersection points by a fixed point argument Now the following result holds (Theorem 4.6 in [4]): Theorem 2.9 The topological space M is Hausdorff with countable basis and A is a differentiable atlas on M. Hence (M, A) induces uniquely the structure of a differentiable manifold. Furthermore it is shown in [4] that M is compact. Now we are able to give the definition of the limit immersion. We set f : M R n, (2.10) [(x, j)] A j (x, u j (x)). By the definition of, the function f is well-defined and easily seen to be an immersion Existence of intersection points by a fixed point argument We like to construct the reparametrizations ϕ i : M M i. This is done by a kind of projection from the limit surface onto each of the surfaces f i. Our starting point is a sequence of (r, α)-immersions f i : M i R n in F, which converges in the sense of graph systems to a limit immersion f : M R n. Here we require α We shall define the projection locally on charts φ j : P (B j δ ) B δ. Using such a chart, we shall often identify the set P (B j δ ) with the ball B δ. Let A j and A i j denote the isometries of the previous sections corresponding to f and f i respectively. As the following constructions are invariant under translations and rotations, we may assume A j = Id R n and replace A i j by A 1 j A i j. Then f(p (B j δ )) is the graph of a function u j : B δ R k with u j (0) = 0, Du j (0) = 0. The set f i (Ur,j i ) is the graph of a function ui j : B r R k, however translated and rotated relatively to the limit immersion by A 1 j A i j. But actually Ai j A j as i in the sense of the metric (2.6). Hence the translation and rotation A 1 j A i j gets arbitrarily small relative to f as i. Hence we may assume that also f i (Ur,j i ) is the graph of a function on a subset of R m R m R k, which shall be denoted in the following by ũ i j. Furthermore for all ϱ with 0 < ϱ < r there is an N N, such that for all i > N {(x, ũ i j(x)) : x B r ϱ } f i (U i r,j). (2.11) This is the situation represented in Figure

26 2. Compactness of immersions with bounded second fundamental form f 1 (Ur,j 1 ) f 2 (Ur,j 2 ) f 3 (Ur,j 3 ). f (limit immersion) ( ( ( ) ) ) 0 B δ B r ϱ B r Figure 2.1 Position of the immersions f i relative to the limit immersion. Note that the figure is not true to scale, because in the proof we have r = 16δ. As Du i j C 0 (B r ) α, we surely may assume Dũ i j C 0 (B r ϱ ) 2α for i sufficiently large. Moreover, by the graph convergence, for any ε > 0 we have ũ i j (0) < ε for i large. Finally we like to simplify notation. All the following considerations are performed locally on P (B j δ ). We will fix the index j and suppress it in the notation. Hence we shall write e.g. u instead of u j and ũ i instead of ũ i j. For technical reasons which are described in [4], p. 29, in general one cannot project in normal direction if the limit immersion is not C 2. However, there are several ways for solving this problem. First one could smoothen the limit immersion f in order to obtain an immersion g which is at least C 2 (or even C ) and which is C 1 -close to f. This technique was used in the diploma thesis [4]. Similarly one could use one of the approximation theorems for immersions in [13]. Slightly different is the approach using an averaged normal projection. It is described in [22] for codimension 1. A generalization to arbitrary codimension using the Riemannian center of mass is presented in Chapter 3.5 of the present paper. For this chapter we like to assume that we have already found (by one of the preceding methods) a smooth mapping ν : M G n,k, which is close to the normal of f. Let us explain what that means: As explained above f(p (B j δ )) is the graph of a function u on B δ R m {0} R m R k with Du C 0 (B δ ) α. For q P (B j δ ) consider the subspace ν f (q), where ν f : M G n,k is the normal of f. Then ν f (q) is a graph over {0} R k ; more precisely there is a linear map Ñq : R k R m such that ν f (q) = {(Ñq(z), z) : z R k } R m R k = R n. 16

27 2.2. Existence of intersection points by a fixed point argument Moreover, as Du C 0 (B δ ) α, for the operator norm we have Ñq α. The property of ν being close to ν f (which can be reached by any of the described methods) shall mean, that for all q P (B j δ ) also the subspace ν(q) is the graph of a linear map N q : R k R m over {0} R k and that N q 2α. (2.12) Identifying P (B j δ ) with the ball B δ as described above, we may similarly assign to each x B δ a linear map N x : R k R m. We like to show, that for q P (B j δ ) the affine subspace f(q) + ν(q) has exactly one point of intersection with the set f i (U4δ,j i ). This shall be done by using a fixed point argument, replacing the original argument in [4] which was based on Brouwer s invariance of domain. For that, in addition to (2.11), assume {(x, ũ i (x)) : x B 4δ ϱ } f i (U4δ,j i ), (2.13) where ϱ is small, say ϱ = δ 2 (suppose (2.11) is satisfied with the same ϱ). The mapping F : For x B δ we denote by F (x) the unique intersection point of the affine subspace h(x) := (x, u(x)) + ν(x) with R m {0}. In that way, we obtain the mapping F : B δ R m, x x N x (u(x)). (2.14) The mappings G i x: For x B δ and y B r ϱ we denote by G i x(y) the unique intersection point of the affine subspace (y, ũ i (y)) + ν(x) with R m {0}. In that way we obtain for each fixed x B δ a mapping G i x : B r ϱ R m, y y N x (ũ i (y)). (2.15) 17

28 2. Compactness of immersions with bounded second fundamental form h(x) = (x, u(x)) + ν(x) (y, ũ i (y)) + ν(x) (y, ũ i (y)) ) f i ( f (x, u(x)) ( ( ( ) ) ) 0 B δ B 4δ ϱ B r ϱ F (x) x B δ G i x(y) y B r ϱ Figure 2.2 The mappings F and G i x. The part between the parentheses on the immersion f i represents the set f i (U i 4δ,j ). The mappings H i x: For y B 4δ 2ϱ and ε sufficiently small we have ũ i (y) 2α(4δ 2ϱ) + ε 8αδ, hence y G i x(y) = N x (ũ i (y)) 16α 2 δ. For x B δ we have u(x) αδ, hence F (x) = x N x (u(x)) δ + 2α 2 δ. Using α , ϱ = δ 2, this yields y G i x(y) + F (x) (1 + 18α 2 )δ 3δ = 4δ 2ϱ. With that we define for each fixed x B δ a mapping H i x : B 4δ 2ϱ B 4δ 2ϱ, y y G i x(y) + F (x). (2.16) Lemma 2.10 For q P (B j δ ) the affine subspace f(q) + ν(q) has exactly one point of intersection with the set f i (Ur,j i ). This point lies in f i (U4δ,j i ). Proof: We pass to the local representation and consider h(x) = (x, u(x)) + ν(x) for x B δ. Using 18

29 2.2. Existence of intersection points by a fixed point argument the definition of Hx i and α , we estimate H i x(ξ) H i x(ζ) = N x (ũ i (ξ) ũ i (ζ)) 2α N x ξ ζ 4α 2 ξ ζ 1 ξ ζ. 2 Hence H i x is a contraction. By the Banach fixed point theorem there is exactly one y B 4δ 2ϱ with H i x(y) = y, that is with G i x(y) = F (x). By the definitions of F and G i x, the affine subspaces h(x)= (x, u(x))+ν(x) and (y, ũ i (y))+ν(x) intersect each other in F (x) = G i x(y) and are parallel, hence h(x) = (y, ũ i (y)) + ν(x) and (y, ũ i (y)) h(x). By (2.13), the affine subspace h(x) intersects the set f i (U i 4δ,j ) in (y, ũi (y)). Similarly, we show that there is only one point of intersection with f i (Ur,j i ): For that we assume that we have chosen r slightly smaller in the beginning, such that also the set f i (Ur+2ϱ,j i ) is the graph of a function ũ i on a subset of R m with Dũ i C 0 2α, and such that f i (U i r,j) {(x, ũ i (x)) : x B r+ϱ } f i (U i r+2ϱ,j). Now for each fixed x B δ define a function H i x : B r+ϱ B r+ϱ, y y G i x(y) + F (x), where we also extend G i x to the ball B r+ϱ. Using r = 16δ, ϱ = δ 2, α and assuming ε to be small, one shows y G i x(y) + F (x) r 2. Hence H x i is well-defined. Then also H x i is a contraction and there is exactly one y B r+ϱ with G i x(y) = F (x). By the definitions of G i x and F, this shows the statement. With the preceding lemma we are able to give a definition of the mappings ϕ i : M M i. For that let x M. Then x P (B j δ ) for some j. The set h(x) intersects f i (Ur,j i ) in exactly one point S x. Furthermore there is exactly one point σ x Ur,j i with f i (σ x ) = S x. We set ϕ i (x) := σ x. The mappings ϕ i are well-defined by the argumentation of Lemma 5.7 in [4]. It holds (Theorem 5.15 in [4]): Theorem 2.11 The mappings ϕ i : M M i are diffeomorphisms. 19

30 2. Compactness of immersions with bounded second fundamental form Convergence of the mappings f i ϕ i The next step in the proof is to show convergence of f i ϕ i in the C 1 -topology. For that one has to show that f i ϕ i converges locally in C 1 with respect to any chart of the atlas A constructed above. For the details of the proof we refer to [4]. However we like to point out how to obtain higher convergence, that is convergence in C k 1 for k 2 (here k = degree of differentiability, k = codimension). For that assume in addition to (2.1) the bounds l A(f) L (M) A l for any l with 0 l k 2, (2.17) where k 2. Here we assume that each immersion is sufficiently smooth, that is at least of class C k. Additionally assume that also the mapping ν : M G n,k we used to construct the diffeomorphisms ϕ i is at least C k. For α > 0 choose r > 0 such that each immersion is an (r, α)-immersion. We need a bound for higher derivatives of the graph functions u: Lemma 2.12 For B r R m and n = m+k let f C k (B r, R n ) a mapping of the type f(x) = (x, u(x)) R m R k with u(0) = 0. Suppose Du C 0 (B r) α < and l A(f) L (B r ) A l < for any l with 0 l k 2. Then u C k (B r ) C(r, α, A 0,..., A k 2 ) for a universal constant C(r, α, A 0,..., A k 2 ) <. Proof: We have Du C 0 (B r) α, hence u C 0 (B r) αr. estimated by induction, see [28], Lemma 8.2. The higher derivatives of u are easily Starting from this situation the calculations in Chapter 6 in [4] show that a subsequence of f i ϕ i converges locally in C k 1 to f. We will make use of this when proving higher order convergence in Theorem The L p -bound for the second fundamental form of the limit The final step in the proof of Theorem 2.1 consists of showing the bound A(f) L p (M) A for the limit immersion f. In particular we like to show this bound for the case m = 1 and 1 < p < 2. This case was omitted in [4] (see Remark 7.3 there). Although the argumentation is trivial, it shall be included here for the sake of completeness. First we need the following statement by C.B. Morrey: Lemma 2.13 Let Ω R m be a bounded domain and F C 2, F = F (x, ζ, ξ) a nonnegative function, which is convex in ξ for any fixed (x, ζ) and which assigns to each x Ω and v W 2,p (Ω, R n ) a value F (x, Dv, D 2 v) 0. Then the functional F(v) = Ω F (x, Dv, D2 v)dl m is lower semi-continuous with respect to weak convergence in W 2,1 (Ω, R n ). 20