Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1<= l<= m+2 and an m-dimensional subset S of R^n we also introduce global versions of this discrete curvature, by taking supremum with respect to m+2-l points on S. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on S. Next, we prove that if S is compact and m-Ahlfors regular and if p is greater than ml, then the P. Jones' \beta-numbers of S must decay as r^t with r \to 0 for some t in (0,1). If S is an immersed C^1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C^{1,t} manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent t to the optimal one a = 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem. Moreover, we obtain a qualitative control over the local graph representations of S only in terms of the energy.


Introduction
Menger curvature is defined for three points x 0 , x 1 , x 2 in R n as follows where H l denotes the l-dimensional Hausdorff measure and △(x 0 , . . . , x l ) is the convex hull of the set {x 0 , . . . , x l }. Using the sine theorem one easily sees that c(x 0 , x 1 , x 2 ) is just the inverse of the radius of the circumcircle of △(x 0 , x 1 , x 2 ). Let γ ⊆ R 3 be a closed, Lipschitz curve with arc-length parameterization Γ, i.e. Γ : S L → R 3 is such that γ = Γ(S L ) and |Γ ′ | = 1 a.e.here S L = R/LZ denotes the circle of length L. We set Using these quantities we define where (γ) i is the Cartesian product of i copies of γ. Gonzalez and Maddocks [7] suggested that these functionals can serve as knot energies, i.e. energies which separate knot types by infinite energy barriers. Gonzalez, Maddocks, Schuricht and von der Mosel [6] showed that whenever c 0 [γ] < ∞ then γ is an embedded (without self-intersections) manifold of class C 1,1 = W 2,∞ . The functionals M 1 p , M 2 p and M 3 p poses a similar property. For i = 1, 2, 3 if M i p (γ) < ∞ for some p > i then γ is an embedded manifold of class C 1,1−i/p (see the articles by Strzelecki, Szumańska and von der Mosel [22,23] and by Strzelecki and von der Mosel [24]). Furthermore, in [24] the authors proved that M 1 p (γ) is finite if and only if γ is an image of a W 2,p function. Later Blatt [2] showed that for i = 2, 3 and p > i the energy M i p (γ) < ∞ if and only if γ belongs to the Sobolev-Slobodeckij space W 1+s,p , where s = 1 − i−1 p . Note that, W 1+s,p (R) ⊆ C 1,1−i/p (R) whenever p > i, so these results deliver geometric counterparts of the Sobolev-Morrey embedding.
For p below the critical level (i.e. p < i) one cannot expect that finiteness of M i p (γ) implies smoothness. Scholtes [20] showed that if γ is a polygon in R 2 then M i p (γ) < ∞ if and only if p < i. For a 1-dimensional Borel set E ⊆ R 2 a famous result of David and Léger [15] says that M 3 2 (E) is finite if and only if E is rectifiable. This was a crucial step in the proof of Vitushkin's conjecture characterizing removable sets E for bounded analytic functions.
There are some generalizations of these results to higher dimensions. Lerman and Whitehouse [16,17] suggested a few possible definitions of discrete curvatures of Menger-type. They used these curvatures to characterize uniformly rectifiable measures in the sense of David and Semmes [4]. In this article we use a modified version (having different scaling) of one of the quantities introduced in [16].
Our research has been motivated directly by the work of Strzelecki and von der Mosel [25], where the authors work with 2-dimensional surfaces in R 3 . They define the discrete curvature of four points x 0 , x 1 , x 2 , x 3 ∈ R 3 by the formula For Σ ⊆ R 3 a compact, closed, connected, Lipschitz surface they also define In [25] the authors prove that if M SvdM p (Σ) ≤ E < ∞ for some p > 8 = dim(Σ 4 ), then Σ has to be an embedded manifold of class C 1,1−8/p with local graph representations whose domain size is controlled solely in terms if E and p. This additional control of the graph representations allowed them to prove [25,Theorem 1.5] that any sequence (Σ j ) j∈N of compact, closed, connected, Lipschitz surfaces containing the origin and with uniformly bounded measure and energy, i.e. M SvdM p (Σ j ) ≤ E and H 2 (Σ j ) ≤ A for each j ∈ N, contains a subsequence Σ j l , which converges in C 1 topology to some C 1,1−8/p compact, closed, connected manifold. This in turn allowed them to solve some variational problems with topological constraints (see [25,Theorems 1.6 and 1.7]).
Similar regularity results were also obtained by Strzelecki and von der Mosel [26] for yet another energy and T x Σ is the tangent space to Σ at x. The quantity R tp (x, y) is called the tangent-point radius, because it measures the radius of the sphere tangent to Σ at x and passing through y.
In this paper we define energy functionals for m-dimensional subsets Σ of R n (we always assume m ≤ n) and we study regularity of sets with finite energy. For m+2 points x 0 , . . . , x m+1 in R n we set (cf. [ We prove that these functionals can be called geometric curvature energies, i.e. for sets Σ of relatively little smoothness, finiteness of the energy guarantees both embeddedness and higher regularity. Of course, the condition E l p (Σ) < ∞ cannot guarantee that Σ is a manifold (even for large p) just for any m-dimensional set Σ. The main issue is that E l p (Σ \ A) ≤ E l p (Σ) for any set A, so creating holes in Σ decreases the energy. Hence, we need to work with a restricted class of sets. We say that Σ is locally lower Ahlfors regular if Here B(x, r) denotes the n-dimensional open ball of radius r centered at x. We also need a variant of the P. Jones' beta numbers introduced in [10] and the bilateral beta numbers, which originated from Reifenberg's work [19] and his famous topological disc theorem (see [21] for a modern proof). We define is the Hausdorff distance and G(n, m) denotes the Grassmannian of m-dimensional linear subspaces of R n . The β-number measures the flatness of Σ in a given scale in a scaling invariant way. The θ-number measures additionally the size of holes in that scale. Using these notions we can formulate our first Proposition 1. Let Σ ⊆ R n be a compact set satisfying (Ahl) and let l ∈ {1, . . . , m + 2}.
Applying the result of David, Kenig and Toro [3, Proposition 9.1] (cf. Proposition 1.4) we then obtain Theorem 1. Let Σ ⊆ R n be a compact set satisfying (Ahl) and such that If E l p (Σ) < ∞ for some p > ml, then Σ is a closed, embedded manifold of class C 1,λ/κ . This motivates the following Definition 1. We say that a set Σ ⊆ R n is an m-fine set if it is m-dimensional, compact and satisfies (Ahl) and (θ β).
Examples of m-fine sets include closed m-dimensional Lipschitz submanifolds of R n and also images of maps ϕ : M → R n , where M is an abstract, closed C 1 manifold and ϕ is an immersion. Other examples are described in Section 2.2.
The condition (θ β) is purely geometric but it is hard to understand what kind of behavior it implies. It gives control over the size of holes in Σ but it does not imply that the topological boundary of Σ is empty. In [26, Definition 2.9] (cf. Definition 3.2) the authors considered a class of admissible sets satisfying a different set of conditions. Their idea was to use the topological linking number to prevent holes in Σ. Any admissible set in the sense of [26] with finite E l p -energy for some p > ml, satisfies the (θ β) condition (see [13,Theorem 4.15] for the case l = m + 2), hence, by Theorem 1, it is a closed C 1,λ/κ -manifold.
Once we have estimates on the β-numbers (Proposition 1), the regularity result (Theorem 1) follows quite easily but the key point is that one can get a uniform (not depending on Σ) control over the local graph representations of Σ only in terms of the energy bound E and the parameters m, l and p. To show that this is true we first prove the following uniform, with respect to Σ, estimate on the local lower Ahlfors regularity of Σ.
Theorem 2. Let Σ ⊆ R n be an m-fine set. If E l p (Σ) ≤ E < ∞ for some p > ml, then where ω m = H m (B(0, 1) ∩ R m ) is the measure of the unit ball in R m .
Theorem 1 together with Theorem 2 give us estimates on the β-numbers independent of Σ. Knowing that Σ is a compact, closed, C 1,λ/κ -submanifold of R n , we prove that also the constant M θβ from the (θ β) condition can be replaced by an absolute constant. Then we obtain estimates on the oscillation of tangent planes of Σ solely in terms of E, m, l and p. This allows to prove that the size of a single patch of Σ representable as a graph of some function is controlled solely in terms of E, m, l and p. Next we bootstrap the exponent λ κ to the optimal one α = 1 − ml p (see [14] and [1] for the proof that this is indeed optimal). Theorem 3. Let Σ ⊆ R n be an m fine set. If E l p (Σ) ≤ E < ∞ for some p > ml, then Σ is a closed C 1,α -manifold. Moreover, there exist two constants R g = R g (E, m, l, p) > 0 and C g = C g (E, m, l, p) > 0 such that This work already lead to a few other results. In our joint work with Szumańska [14] we have constructed an example of a function f ∈ C 1,α 0 ([0, 1] m ), where α 0 = 1 − m(m+1) p , whose graph has infinite E m+2 p -energy and we proved that for any α 1 > α 0 the graphs of C 1,α 1 functions always have finite energy. Later this result was complemented by our joint work with Blatt [1], where we have shown that a C 1 -submanifold of R n has finite E l p -energy for some p > m(l − 1) and l ∈ {2, . . . , m+2} if and only if it is locally a graph of a function in the Sobolev-Slobodeckij space W 1+s,p , where s = 1 − m(l−1) p . In another article [12] written jointly with Strzelecki and von der Mosel, we have shown that an m-fine set Σ ⊆ R n is a W 2,p -manifold if and only if it satisfies the condition E 1 p (Σ) < ∞. The paper [12] includes Theorem 3 for the E 1 p -energy and a counterpart of Theorem 3 for a modified version of the E tp p -energy, where one integration was replaced by taking the supremum. In a forthcoming joint article with Strzelecki and von der Mosel [11] we also prove a compactness result similar to [25,Theorem 1.5] for the E l p and E tp p energies.
Organization of the paper. In Section 1 we describe the notation, we state precisely the result of [3] about Reifenberg flat sets with vanishing constant and we prove some auxiliary propositions about roughly regular simplices and about the metric on the Grassmannian. In 1.4 we also show that C 2 -manifolds have finite E l p -energy for any p > 0. In Section 2 we prove Proposition 1 and Theorem 1 and we give some examples of m-fine sets. In Section 3 we establish Theorem 2. For this we need to define another class of admissible sets and prove some more auxiliary results about cones and homotopies inside cones. In Section 4 we prove a counterpart of Theorem 3, where α is replaced with λ/κ. In Section 5 we bootstrap the exponent λ/κ to the optimal α = 1 − ml p and consequently establish Theorem 3.

Preliminaries
1.1. Notation. We write S for the unit (n − 1)-dimensional sphere centered at the origin and we write B for the unit n-dimensional open ball centered at the origin. We also use the symbols S r = rS, B r = rB, S(x, r) = x + rS and B(x, r) = x + rB.
If v = (v 1 , . . . , v n ) is a vector in R n , we write |v| = |v i | 2 = v, v for the standard Euclidean norm of v. If A : R k → R l is a linear operator, we write A = sup |v|=1 |Av| for the operator norm of A.
The symbol G(n, m) denotes the Grassmann manifold of m-dimensional linear subspaces of R n . Whenever we write U ∈ G(n, m) we identify the point U of the space G(n, m) with the appropriate m-dimensional subspace of R n . In particular any vector u ∈ U is treated as an n-dimensional vector in the ambient space R n which happens to lie in U ⊆ R n .
If A is any set, then we write id A : A → A for the identity mapping. Let H ∈ G(n, m). We use the symbol π H to denote the orthogonal projection onto H and π ⊥ H = I − π H to denote the orthogonal projection onto the orthogonal complement H ⊥ . We write aff{x 0 , . . . , x m } for the smallest affine subspace of R n containing points x 0 , . . . , x m ∈ R n , i.e. In the course of the proofs we will frequently use cones and "conical caps" of different sorts.
Remark 1.1. We use the notation C = C(x, y, z) to denote that C depends solely on x, y and z. The symbols C,Ĉ,C,C are used to denote general constants, whose values may change in different parts of the text. Subscripts in constants (like "C θ ") do not denote dependences but are used to name the constant and distinguish it from other constants. Subscripted constants always have global meaning and do not change.
1.2. Reifenberg flat sets. For convenience we introduce the following Definition 1.2. Let Σ ⊆ R n be any set. Let x ∈ Σ and r > 0. We say that H ∈ G(n, m) is the best approximating m-plane for Σ in B(x, r) and write H ∈ BAP m (x, r) if the following condition is satisfied . Since G(n, m) is compact, such H always exists, but it might not be unique, e.g. consider the set Σ = S ∪ {0} and take x = 0, r = 2.
Recall the definitions of β Σ m and θ Σ m given in the introduction. In [3], the authors define the β and θ numbers in a slightly different way using open balls instead of closed ones. This does not change much since both definitions lead to comparable quantities (see [ The following proposition was proved by David, Kenig and Toro. Proposition 1.4 (cf. [3], Proposition 9.1). Let τ ∈ (0, 1) be given. Suppose Σ is a Reifenbergflat set with vanishing constant of dimension m in R n and that, for each compact subset K ⊆ Σ there is a constant C K such that β Σ m (x, r) ≤ C K r τ for each x ∈ K and r ≤ 1. Then Σ is a C 1,τ -submanifold of R n .

Voluminous simplices.
Here we define the class of (η, d)-voluminous simplices, where η measures the "regularity" of a simplex. The curvature K of any such simplex is controlled in terms of η and d. A very similar notion was used by Lerman and Whitehouse in [16, § 3.1], where these kind of simplices were called 1-separated. We derive estimates of the distance by which we can move each vertex of an (η, d)-voluminous simplex without losing the lower bound on the curvature. We will use this result to obtain a lower bound on the E l p -energy in the proof of Proposition 2.1. Definition 1.5. Let T = △(x 0 , . . . , x k ) be a simplex in R n and let d ∈ (0, ∞) and η ∈ (0, 1). We say that T is (η, d)-voluminous if diam(T) ≤ d and h min (T) ≥ ηd .
Let us recall the definition of the outer product: Definition 1.7. Let w 1 , . . . , w l be vectors in R n . We define the outer product w 1 ∧ · · · ∧ w l to be the vector in R ( n l ) , whose coordinates are exactly the l-minors of the (n × l)-matrix (w 1 , . . . , w l ). Remark 1.8. A standard fact from linear algebra says that the length |w 1 ∧· · ·∧w l | of the outer product of w 1 , . . . , w l is equal to the l-dimensional volume of the parallelotope spanned by w 1 , . . . , w l . In particular |w 1 ∧ · · · ∧ w l | ≤ |w 1 | · |w 2 | · · · |w k |.
There exists a number ς k = ς k (η) ∈ (0, 1) such that for any simplex T 1 = △ T 1 = △(y 0 , . . . , y k ) satisfying |x i − y i | ≤ ς k d for each i = 1, . . . , k the following estimate Proof. Letς ∈ (0, 1) be some number and let T 1 = (y 0 , . . . , y k ) be such that Whenever we take an outer product of j vectors from the set {w 1 , . . . , w k } and (k − j) vectors from the set {v 1 , . . . , v k } we obtain a vector of length at most d k−j (ςd) j . Hence we can write 10. Let x, s ∈ R and s > 0. When |x| ≈ 0, the function (1 + x) s behaves asymptotically like 1 + sx, hence there exists a constant C ς = C ς (k) > 1 such that 1.4. The E l p -energy for smooth manifolds. Observe that K(αT ) = 1 α K(T ) for any α > 0, so our curvature behaves under scaling like the original Menger curvature c. If △ T is a regular simplex (meaning that all the side lengths are equal), then K(T ) ≃ 1 diam T ≃ R(T ) −1 , where R(T ) is the radius of a circumsphere of T . For m = 1 one easily sees that we always have We emphasis the behavior on regular simplices because small, close to regular (or voluminous) simplices are the reason why E l p (Σ) might get very big or infinite. For the class of (η, d)-voluminous simplices T the value K(T ) is comparable with yet another possible definition of discrete curvature (cf. [17, §10]) . This last factor prevents K ′ from blowing up on simplices with vertices on smooth manifolds.
It occurs that one cannot define k-dimensional Menger curvature using integrals of R −1 . This "obvious" generalization of the Menger curvature fails because of examples (see [25, Appendix B]) of very smooth embedded manifolds for which this kind of curvature would be unbounded. For the curvature K we have the following is finite for every p > 0 and every l ∈ {1, . . . , m + 2}. Lemma 1.12. Let Σ ⊆ R n be any set and let T = (x 0 , . . . , x m+1 ) ∈ Σ m+2 . We set T = △ T and d = diam(T). There exists a constant C Kβ = C Kβ (m, n) such that we have Proof. 2 Without loss of generality we can assume that x 0 = 0. If the vectors {x 1 , . . . , x m+1 } are not linearly independent, then H m+1 (T) = 0 and there is nothing to prove. Let x 1 , . . . x m+1 be linearly independent and let W denote the (m + 1)-dimensional vector space spanned be these vectors. Set Then, the set T + S is isometric with T × S and the following holds Using compactness of the Grassmannian we can find a vector space V ∈ G(n, m) such that Observe also that the mapping Q : G(n, m) → R n given by Q(V ) = P V (y) is continuous for any choice of y ∈ R n . In consequence, we get the estimate The author wishes to thank Simon Blatt for significantly simplifying this proof while we were working on [1].
The vertices of T lie in Σ ∩ B(x 0 , d) and T is convex, so we also have Let y ∈ T + S and let t ∈ T and s ∈ S be such that s + t = y. Using the triangle inequality we see that (4) and (5) we obtain the desired estimate.
Proof of Proposition 1.11. Since M is a compact C 2 -manifold, it has a tubular neighborhood of some radius ε > 0 and the nearest point projection p : M ε → M is a well-defined, continuous function (see e.g. [5] for a discussion of the properties of the nearest point projection mapping).
To find ε one proceeds as follows. Take the principal curvatures κ 1 , . . . , κ m of M . These are continuous functions M → R, because M is a C 2 manifold. Next set Such maximal value exists due to continuity of κ j for each j = 1, . . . , m and compactness of M . We will show that for all r ≤ ε and all x ∈ Σ we have Next, we apply Lemma 1.12 and get the desired result. Choose r ∈ (0, ε]. Fix some point x ∈ Σ and pick a point y ∈ T x M ⊥ with |x − y| = ε. Note that y belongs to the tubular neighborhood M ε and that p(y) = x. Hence, the point x is the only point of M in the ball B(y, ε). In other words M lies in the complement of B(y, ε). This is true for any y satisfying y ∈ T x M ⊥ and |x − y| = ε, so we have Pick another pointx ∈ Σ ∩ B(x, r). We then have Using (7) and simple trigonometry, it is ease to calculate the maximal distance ofx from the tangent space T x M . Let z be any point in the intersection ∂B(x, r) ∩ ∂B(y, ε). Note that points of M ∩ B(x, ε) must be closer to T x M than z. In other words This situation is presented on Figure 1. Let α be the angle between T x M and z and set h = dist(z, T x M ). We use the fact that the distance |z − x| is equal to r.
This shows (6) and thus finishes the proof.
Remark 1.14. Note that the only property of M , which allowed us to prove Proposition 1.11 was the existence of an appropriate tubular neighborhood M ε . One can easily see that Proposition 1.11 still holds if M is just a set of positive reach as defined in [5]. .55] for the reference. We treat G(n, m) as a metric space with the following metric Note that this metric is different from the geodesic distance on the Grassmannian. However, the topology induced by the metric d Gr agrees with the standard quotient topology which is the same as the topology induced by the geodesic distance. Remark 1.16. Let I : R n → R n denote the identity mapping. We will frequently use the following identity without reference Definition 1.17. Let V ∈ G(n, m) and let (v 1 , . . . , v m ) be the basis of V . Fix some radius ρ > 0 and a small constant ε ∈ (0, 1) We say that We proceed by induction.
Proof. Without loss of generality, we can assume that ϑ < 1. If ϑ ≥ 1 then we can set C π = 2 and there is nothing to prove. Set We calculate Proof of Proposition 1.18. Dividing each v i by ρ 0 , we get a ρε-basis with ρ = 1. Hence we can assume that ρ 0 = 1. Without loss of generality we may also assume that ϑ < 1. Indeed, we always have the trivial estimate d Gr (U, V ) ≤ 2, so if ϑ ≥ 1 we can set C ρε = 2.

Geometric Morrey-Sobolev embedding
In this section we prove Theorem 1 which is a geometric counterpart of the Morrey-Sobolev embedding W 2,p (R k ) ⊆ C 1,1−k/p for p > k. We also give some examples of m-fine sets to which Theorem 1 applies.
Proof. We shall estimate the E l p -energy of Σ. Recall that ς m+1 ≤ 1 4 was defined by (2).
sup y l ,...,y m+1 ∈Σ K p (△(y 0 , . . . , y m+1 )) dH ml (y 0 ,...,y l−1 ) . Proposition 1.9 combined with Remark 1.6 lets us estimate the integrand sup y l ,...,y m+1 ∈Σ Since Σ satisfies (Ahl), we get a lower bound on the measure of the sets over which we integrate Plugging the last two estimates into (12) and recalling (3) we obtain Proposition 2.1 is interesting in itself. It says that whenever the energy of Σ is finite, we cannot have very small and voluminous simplices with vertices on Σ. It gives a bound on the "regularity" (i.e. parameter η) of any simplex in terms of its diameter d and we see that η goes to 0 when we decrease d. Now we are ready to prove Proposition 1.
The existence of such simplex follows from the fact that the set Σ ∩ B(x, r) is compact and from the fact that the function T → H m+1 (△ T ) is continuous with respect to x 0 , . . . , x m+1 .
Rearranging the vertices of T we can assume that h min Note that the distance of any point y ∈ Σ ∩ B(x, r) from the affine plane x 0 + H has to be less then or equal to h min (T) = dist(x m+1 , x 0 + H). If we could find a point Now we only need to estimate h min (T) = h m+1 (T) from above. Of course T is (h min (T)/(2r), 2r)voluminous, so applying Proposition 2.1 we obtain Putting (13) and (14) together we get Having Proposition 1 at our disposal we can easily prove Theorem 1.
Proof of Theorem 1. We know already that β Σ m (x, r) ≤ C(m, l, p, A Ahl , E)r λ/κ for r < R Ahl . We assumed (θ β), so Σ is Reifenberg flat with vanishing constant. We finish the proof by applying Proposition 1.4.
If we choose R x small then we can make the Lipschitz constant of Φ x smaller than some ε > 0. Due to compactness of M and continuity of Df we can find a global radius R Σ = min{R x : x ∈ M }. Then we can safely set A Ahl = √ 1 − ε 2 and M Σ = 4.   where See Figure 2 for a graphical presentation. Condition θ β holds at the boundary points (−1, 0) and (1, 0) of Σ, because the β-numbers do not converge to zero with r → 0 at these points. All the other points of Σ are internal points of line segments or corner points of squares, so at these points condition (θ β) is also satisfied. Hence, Σ is 1-fine. This example shows that condition (θ β) does not exclude boundary points but at any such boundary point we have to add some oscillation, to prevent β-numbers from getting too small. The same effect can be observed in the following example

Uniform Ahlfors regularity -the proof of Theorem 2
Here we give the proof of Theorem 2. First we introduce the class of admissible sets, which is tailored for proving the existence of many voluminous simplices (cf. Proposition 3.18) with vertices on Σ. Proposition 3.18 is crucial in the proof of Theorem 2. In the end we also show how to make all the emerging constants depend solely on E, m, l and p.
Let N be an (n − m)-dimensional, compact, closed submanifold of R n . We say that Σ is linked with N and write lk 2 (Σ, N ) = 1, if there exists an i ∈ I such that the map where deg 2 is the topological degree modulo 2.
For the definition of the degree of a map we refer the reader to [9, Chapter 5, § 1].
Definition 3.2 (cf. [26] Definition 2.9). Let δ ∈ (0, 1) and let I be a countable set of indices. Let Σ be a compact subset of R n satisfying (Ahl). We say that Σ is (δ, m)-admissible and write Σ ∈ A(δ, m) if the following conditions are satisfied A1 Mock tangent planes and flatness. There exists a dense subset Σ * ⊆ Σ of full measure in Σ (i.e. H m (Σ \ Σ * ) = 0) such that for each x ∈ Σ * there is an m-plane H = H x ∈ G(n, m) and a radius r 0 = r 0 (x) > 0 such that . Condition A1 ensures that at every point x ∈ Σ * one can touch Σ with an apropriate cone. Condition A2 says that at each point of Σ there is a sphere S x which is linked with Σ. This means intuitively, that we cannot move S x far away from Σ without tearing one of these sets. Example 3.10 shows that this condition is unavoidable for the theorems stated in this paper to be true.
There are three especially useful properties of lk 2 that we want to use.
Then the disk B(y, r) ∩ (y + V ) contains at least one point of Σ.
It is easy to verify that Σ ∈ A(δ, m). Take M 1 = Σ and f 1 = id M 1 . The set Z will be empty, so Σ * = Σ. At each point x ∈ Σ we set H x to be the tangent space T x Σ. Small spheres centered at x ∈ Σ and contained in x + H ⊥ x are linked with Σ; for the proof see e.g. [18, pp. 194-195]. Note that we do not assume orientability; that is why we used degree modulo 2.
where Σ i are closed, compact, m-dimensional submanifolds of R n of class C 1 . Moreover assume that these manifolds intersect only on sets of zero mdimensional Hausdorff measure, i.e.
Then Σ ∈ A(δ, m) for any δ ∈ (0, 1). Remark 3.8. Any C 1 -manifold is (δ, m)-admissible (cf. Example 3.6) for any δ ∈ (0, 1), hence any m-fine set with finite E l p -energy for some p > ml is also (δ, m)-admissible. It turns out that any (δ, m)-admissible set with finite E l p -energy for some p > ml is also m-fine. We will not use this fact in this article. The proof for the E m+2 p -energy can be found in [13,Theorem 2.13].
If we do not assume finiteness of the E l p -energy then these two classes of sets are different and none of them is contained in the other.

Now we give some negative examples showing the role of condition A2.
Example 3.10. Let H ∈ G(n, m) and let Σ = π H (S) = B ∩ H. Then Σ satisfies (Ahl) and condition A1 but it does not satisfy A2. Hence, it is not admissible. Although Σ is a compact, m-dimensional submanifold of R n of class C 1 , it is not closed.  (15) is not satisfied at any point.

Homotopies inside cones. In this section we prove a few useful facts about cones.
In the proof of Proposition 3.18 we construct a set F by glueing conical caps together. Then we need to know that we can deform one sphere lying in F to some other sphere lying in F without leaving F . To be able to do this easily we need Propositions 3.16 and 3.17.   The proofs can be found in [13, Section 4.1.1] Corollary 3.15. Let H and δ be as in Proposition 3.13. Let S 1 and S 2 be two round spheres centered at the origin, contained in the conical cap C(δ, H, ρ 1 , ρ 2 ) and of the same dimension (n − m − 1). Moreover assume that 0 ≤ ρ 1 < ρ 2 . There exists an isotopy Proof. Let r 1 and r 2 be the radii of S 1 and S 2 respectively. We have ρ 1 < r 1 , r 2 < ρ 2 . Let V 1 , V 2 ∈ G(n, n − m) be the two subspaces of R n such that S 1 ⊆ V 1 and S 2 ⊆ V 2 . In other words S 1 = S r 1 ∩ V 1 and S 2 = S r 2 ∩ V 2 . Because S 1 and S 2 are subsets of C(δ, H), we know that V 1 and V 2 are elements of G (δ, H). From Proposition 3.13 we get a continuous path γ joining V 1 with V 2 . By Corollary 3.14, this path lifts to a pathγ in the orthogonal group O(n). For z ∈ S 1 and t ∈ [0, 1] we set This gives a continuous deformation of S 1 = S r 1 ∩V 1 into S r 1 ∩V 2 . Now, we only need to adjust the radius but this can be easily done inside V 2 ∩ A(ρ 1 , ρ 2 ) so the corollary is proven.
Proposition 3. 16. Let H ∈ G(n, m). Let S be a sphere perpendicular to H, meaning that S = S(x, r) ∩ (x + H ⊥ ) for some x ∈ H and r > 0. Assume that S is contained in the conical cap C(δ, H, ρ 1 , ρ 2 ), where ρ 2 > 0. Fix some ρ ∈ (ρ 1 , ρ 2 ). There exists an isotopy Figure 3. When we move the center of a sphere to the origin, we need to control the radius so that the deformation is performed inside the conical cap.
Proof. Any point z ∈ S can be uniquely decomposed into a sum z = x + ry, where y ∈ S ∩ H ⊥ is a point in the unit sphere in H ⊥ . We define This gives an isotopy which deforms S to a sphere perpendicular to H and centered at the origin (see Figure 3). Fix some z = x + ry ∈ S. The sphere S is contained in C(δ, H), so it follows that This shows that the whole deformation is performed inside C(δ, H). Next, we need to continuously change the radius to the value ρ but this can be easily done inside H ⊥ ∩ (B ρ 2 \ B ρ 1 ).

The construction of voluminous simplices.
For any x 0 ∈ Σ * Proposition 3.18 stated below, ensures the existence of d = d(x 0 ) > 0 and an (η, d)-voluminous simplex with vertices on Σ ∩ B(x 0 , d) and also that at any scale below d our set Σ has big projection onto some affine m-plane. The reasoning used here mimics [25,Proposition 3.5]. Note that, finiteness of the E l p -energy is not used in the proof.

Corollary 3.19.
For any x 0 ∈ Σ * and any ρ ≤ 1 Proof. The orthogonal projection π x 0 +H is Lipschitz with constant 1 so it cannot increase the H m -measure. From (18) we know that the image of Σ ∩ B(x 0 , ρ) under π x 0 +H contains the ball Proof of Proposition 3.18. Without loss of generality we can assume that x 0 = 0 is the origin.
To prove the proposition we will construct finite sequences of • compact, connected, centrally symmetric sets • and of radii ρ 0 < ρ 1 < · · · < ρ N . For brevity, we define The above sequences will satisfy the following conditions • the interior of F i is disjoint with Σ, i.e.
• for each i ≥ 0 the set F i+1 contains a large conical cap, i.e.
Let us define the first elements of these sequences. We set H 0 = H x 0 , ρ 0 = 0 and F 0 = ∅. Next, we set Directly from the definition of an admissible set, we know that ρ 1 > 0, so the condition (20) is satisfied for i = 0. Conditions (19) and (21) are immediate for i = 0. Using Proposition 3.16 one can deform any sphere S from condition (22) to the sphere S x defined in A2 of the definition of A(δ, m). This shows that (22) is satisfied for i = 0.
We proceed by induction. Assume we have already defined the sets F i , subspaces H i and radii ρ i for i = 0, 1, . . . , I. Now, we will show how to continue the construction.
Let (e 1 , e 2 , . . . , e m ) be an orthonormal basis of H I . We choose m points lying on Σ such that Recall that x 0 = 0 and set P = span{x 1 , x 2 , . . . , x m }. It suffices to find one more point x m+1 ∈ Σ such that the distance dist(x m+1 , P ) ≥ηρ I for some positiveη. Indeed, if we set T = △(x 0 , . . . , x m+1 ), we have (24) h min (T) = Choose a small positive number h 0 = h 0 (δ) ≤ 1 2 such that (25) δ This is always possible because when we decrease h 0 to 0 the left-hand side of (25) converges to δ < 1 and the right-hand side converges to 1. We need this condition to be able to apply Proposition 3.17 later on.  If case (B) occurs, then our set Σ is almost flat in A( 1 2 ρ I , 2ρ I ) so there is no chance of finding a voluminous simplex in this scale and we have to continue our construction. Let • H I+1 = P , • ρ I+1 = inf{s > ρ I : C(δ, P, ρ I , s) ∩ Σ = ∅} and • F I+1 = F I ∪ C(δ, P, 1 2 ρ I , ρ I+1 ). We assumed (B), so it follows that This means that C(δ, P, 1 2 ρ I , 2ρ I ) does not intersect Σ and we can safely set H I+1 = P . It is immediate that ρ I+1 ≥ 2ρ I so conditions (19), (20) and (21) are satisfied. Now, the only thing left is to verify condition (22).
We are going to show that all spheres S contained in F I+1 of the form are linked with Σ. By the inductive assumption, we already know that spheres centered at H I ∩ B r I , perpendicular to H I and contained in F I are linked with Σ. Therefore, all we need to do is to continuously deform S to an appropriate sphere centered at H I and contained in F I in such a way that we never leave the set F I+1 (see Figure 5). H I P S Figure 5. First we move the center of S to x0. Then we rotate S so that it is perpendicular to HI . Finally we change the radius so that it is between 1 2 ρI−1 and ρI .
We know that F I+1 contains the conical cap C = C(δ, P, 1 2 ρ I , ρ I+1 ), so we can use Proposition 3.16 to move S inside C, so that it is centered at the origin.
From (27) we get . Using this and our inductive assumption we obtain . We have two cones that have nonempty intersection and we chose h 0 such that (25) holds, so we can apply Proposition 3.17 with α = δ and β = 2h 0 δ. Hence the intersection C(δ, H I ) ∩ C(δ, P ) contains the space H ⊥ I . Therefore H ⊥ I ∩ A( 1 2 ρ I , ρ I+1 ) ⊆ C(δ, P, 1 2 ρ I , ρ I+1 ) ∩ F I . Using Corollary 3.15 we can rotate S inside C, so that it lies in H ⊥ . Then we decrease the radius of S to the value e.g. 3 4 ρ I ∈ ( 1 2 ρ I−1 , ρ I ). Applying the inductive assumption we obtain condition (22) for i = I + 1.
The set Σ is compact and ρ i grows geometrically, so our construction has to end eventually. Otherwise we would find arbitrary large spheres, which are linked with Σ but this contradicts compactness.
From Proposition 2.1 we know that d(Σ) must satisfy (11) with η = η 0 defined by (26). Hence, we already have a positive lower bound on d(Σ). We only need to show that it does not depend on A Ahl .

3.5.
Removing the dependence on M θβ and R θβ . In this section we show that if Σ is m-fine with finite E l p -energy, then the constants M θβ and R θβ from Theorem 1 can be chosen depending solely on E, m, l and p. Proposition 3.21. Let Σ ⊆ R n be an m-fine set such that E l p (Σ) ≤ E < ∞ for some p > ml. Then there exists R 1 = R 1 (E, m, l, p) such that Σ satisfies (Ahl) and (θ β) with constants M θβ = 5, R θβ = R Ahl = R 1 and A Ahl = Fix a point x 0 ∈ Σ and a radius r ≤ R 0 . Choose some m-plane P ∈ G(n, m) such that . For brevity we set β = 2β Σ m (x 0 , r) and γ = √ 15 4 . Inspecting the proof of Proposition 3.18 we can find i ∈ N such that ρ i ≤ r < ρ i+1 . We set H = H i . Let y ∈ R n be any point such that y − x 0 ∈ H and |y − x 0 | = γr. We see that S(y, 1 4 r) ∩ (y + H ⊥ ) is linked with Σ, hence (cf. Proposition 3.5) there exists z ∈ Σ ∩ B(y, 1 4 To apply Proposition 3.17 we need to ensure the condition Substituting Ψ = β γ in (30) and recalling that γ = √ 15 4 we obtain the following inequality Note that if Ψ → 0 then the right-hand side converges to 3 4 . Let Ψ 0 be the smallest, positive root of the equation Then any Ψ ∈ (0, Ψ 0 ) satisfies (31). Recall that 1 2 β = β Σ m (x, r) ≤ C(m, l, p)E 1/κ r λ/κ , so to ensure condition (30) it suffices to impose the following constraint Now, for such r we can use Proposition 3.17 to obtain , P , γ r , r ) x 0 + C( 1 4 , H, 1 2 ρ i , r) β r Figure 6. If β is small enough, then the cone C( 8β 7γ , P ) contains H ⊥ and we can continuously transform S1 into S3 inside the conical cap C( 8β 7γ , P, 7 8 rγ, 7 8 r).
We can translate S 3 along any vector v ∈ P with |v| ≤ 1 − β 2 r without changing the linking number. This way we see that for any point w ∈ (x 0 + P ) ∩ B(x 0 , 1 − β 2 r) there exists a point z ∈ Σ such that |z − w| ≤ βr.
For any other point w ∈ (x 0 + P ) with 1 − β 2 r ≤ |w − x 0 | ≤ r we set Then we find z ∈ Σ such that |w − z| ≤ βr and we obtain the estimate Therefore the infimum over all H ∈ G(n, m) must be even smaller, so θ Σ m (x 0 , r) ≤ 5β Σ m (x 0 , r) for any r ≤ R θβ = R 1 and we can safely set M θβ = 5.

Uniform estimates on the local graph representations
For the sake of brevity we introduce the following notation π x = π TxΣ and π ⊥ x = π ⊥ TxΣ , where x ∈ Σ. The main result of this section is Theorem 4.1. Let Σ ⊆ R n be an m-fine set. If E l p (Σ) ≤ E < ∞ for some p > ml, then Σ is a closed C 1,λ/κ -manifold (by Theorem 1) and there exist constants R λκ = R λκ (E, m, l, p) and C λκ = C λκ (E, m, l, p) such that for all x ∈ Σ there exists a function F x : T x Σ → (T x Σ) ⊥ of class C 1,λ/κ such that To prove this theorem we fix a point x ∈ Σ and for each radii r > 0 we choose an m-plane P (x, r). Then we use the fact that θ Σ m (x, r) ≤ M θβ β Σ m (x, r) together with Proposition 1 to show that P (x, r) converge to the tangent plane T x Σ, when r → 0. This also gives a bound on the oscillation of T x Σ. Then we derive Lemma 4.8, which says that at some small scale we cannot have two distinct points y and z of Σ such that the vector v = (y − z) is orthogonal to T x Σ. Any such vector v would be close to the tangent plane T z Σ and this would violate the bound on the oscillation of tangent planes proved earlier. From here, it follows that there exists a small radius R λκ such that Σ ∩ B(x, R λκ ) is a graph of some function F x , which is of class C 1,λ/κ by Theorem 1.
In the sequel of this section we always assume that Σ satisfies hypothesis of Theorem 4.1.
There exists a limit lim r→0 P (r) = T x Σ ∈ G(n, m) and it does not depend on the choice of P (r) ∈ BAP m (x, r).
Applying Lemma 4.2 with x = y, r 0 = ρ j and r 1 = 1 2 r 0 = ρ j+1 we obtain Setting r 0 = ρ b and r 1 = s or r 0 = ρ a and r 1 = t we also get and d Gr (P (t), P a ) ≤ CE 1/κ ρ λ/κ a . Using these estimates we can write d Gr (P (r), P (s)) ≤ d Gr (P (r), P a ) + b−1 j=a d Gr (P j , P j+1 ) + d Gr (P b , P (s)) which shows that the Cauchy condition is satisfied, so P (r) converges in G(n, m) to some m-plane, which must be the tangent plane T x Σ.
Corollary 4.4. There exists a constant C th = C th (m, l, p) such that for all x ∈ Σ, all r ≤R 1 and all H ∈ BAP m (x, r) we have There exists a constant C tp = C tp (m, l, p) such that for all x ∈ Σ and all y ∈ Σ ∩ B(x,R 1 ) we have Choose an m-plane H ∈ BAP m (x, |y − x|). Using (33) and Corollary 4.4 we get m (x, |y − x|) + |y − x|C th E 1/κ |y − x| λ/κ ≤ C tp E 1/κ |y − x| 1+λ/κ . Lemma 4.6. There exists a constant C tt = C tt (m, l, p) such that for all x ∈ Σ and for all Proof. Let y ∈ Σ∩B(x, 1 2R 1 ). Set r 0 = 2|x−y| and r 1 = |x−y|. Choose any H 0 ∈ BAP m (x, r 0 ) and any H 1 ∈ BAP m (y, r 1 ). From Lemma 4.2 we have On the other hand Corollary 4.4 says that Putting these estimates together we obtain

4.2.
Uniform estimates on the size of maps. Combining Corollary 4.5 and Lemma 4.6 one can see that if we have two distinct points y, z ∈ Σ such that y − z ⊥ T x Σ and |y − z| |x − y| then the tangent plane T y Σ must form a large angle with the plane T x Σ. Such situation can only happen far away from x because of the bound on the oscillation of tangent planes.
Hence we can use Corollary 4.5 once again to estimate the distance between z and T y Σ. Using the definition of d Gr we may write On the other hand Lemma 4.6 gives us Putting these two estimates together we have We set R 2 = 1 2 min{R 2 (C 0 C tp ) −1 ,Ĉ(m, l, p)E −1/λ }. Corollary 4.9. For each x ∈ Σ and each y ∈ Σ ∩ B(x, R 2 ) the point y is the only point in the Recall our convention, that when we write T o Σ we always mean the appropriate subspace of R n . For x ∈ D(o) we set Observe that these mappings are well defined since R 2 is not greater thanR 2 defined in Corollary 4.7, which ensures that d Gr Proof. We want to estimate Let h ∈ S and set u = L x (h) and v = L y (h). Note that u − v ∈ T o Σ ⊥ so we can write To prove the second part of Lemma 4.12 we will use Proposition 1.18. Let (e 1 , . . . , e m ) be some orthonormal basis of T o Σ. For each i = 1, . . . , m set u i = Dϕ(x)e i and v i = Dϕ(y)e i . Then (u 1 , . . . , u m ) is a basis of T ϕ(x) Σ and (v 1 , . . . , v m ) is a basis of T ϕ(y) Σ. By Remark 4.10 for i, j = 1, . . . , m and i = j we have These estimates show that (u 1 , . . . , u m ) is a ρε-basis of T ϕ(x) Σ with ρ = 1 and ε = 3ι. Moreover Since 3ι = 3ǫρε 100 ≤ ǫ ρε we can use Proposition 1.18 to obtain Proof of Theorem 4.1. Combining Lemma 4.6 with Lemma 4.12 we get increasing its Hölder norm and in such a way that {(y, F o (y)) : Hence we may set

Optimal Hölder regularity
In the previous paragraph we showed that Σ is a closed manifold of class C 1,λ/κ but λ/κ was not an optimal exponent. Now we shall prove that for any o ∈ Σ the map F o is of class C 1,α , where α = 1 − ml p . For this purpose we employ a technique developed by Strzelecki, Szumańska and von der Mosel in [23].
The key to the proof of Theorem 3 is Lemma 5.1. It says that the oscillation of Dϕ on a ball of radius r can be bounded above by the oscillation of Dϕ on a ball of radius r/N , where N is some big number, plus a term of order r α . If we choose N big enough, then, upon iteration, the first term disappears and the sum of the second terms is still of order r α .
To prove Lemma 5.1 we choose two points x, y ∈ D(o) and we set r = |x − y|. From Lemma 4.12 we know that the oscillation of Dϕ is comparable with the oscillation of T ϕ(·) Σ. We choose points x 0 , . . . , x m and y 0 , . . . , y m near x and y respectively, such that {x i − x 0 } m i=1 and {y i − y 0 } m i=1 form a roughly (up to an error of order 1 k , where k is some big number) orthogonal bases of T o Σ. Moreover |x i − x 0 | ≈ r/N and |y i − y 0 | ≈ r/N . In the scale we are working in, we always have are also roughly (up to an error of order 1 k + ι) orthogonal and span some m-dimensional secant spaces X and Y respectively. If we choose the points y 0 , . . . , y m appropriately, then the "angle" d Gr (X, Y ) can be estimated by r α . The error we make when we pass from d Gr (T ϕ(x , T ϕ(y) ) to d Gr (X, Y ) is comparable with the oscillation of Dϕ on balls of radius r/N .
To choose "good" points y 0 , . . . , y m we first define the set of "bad parameters" B(x 0 , . . . , x l−2 ), i.e. such z ∈ D(o) that the integrand is big. From finiteness of E l p (Σ), we derive the conclusion that the measure of B(x 0 , . . . , x l−2 ) has to be smaller than the measure of a ball of radius r/(kN ), hence close to eachỹ there exists y which does not belong to B(x 0 , . . . , x l−2 ). From the fact that K l,ϕ (x 0 , . . . , x l−2 , y) is small, we derive an estimate on the distance of ϕ(y) from ϕ(x 0 ) + X, which in turn gives the estimate d Gr (X, Y ) r α .
Lemma 5.1. For all k ≥ k 0 = 100/ǫ ρε and N ≥ N 0 = 8 there exist constants C 1 = C 1 (m) and C 2 = C 2 (m, l, p, k, N ) such that for all x, y ∈ D 1 6 R λκ Using this lemma we can prove Theorem 3.
Proof of Theorem 3. Fix some a ∈ D 1 12 R λκ and a radius R ∈ (0, 1 36 R λκ ]. Taking the supremum on both sides of (40) over all x, y ∈ D(a, R) satisfying |x − y| ≤ r ≤ R we obtain the estimate Φ(r, D(a, R)) ≤ C 1 Φ 2r N , D(a, R + r) + C 2 M l p (a, R + r)r α . Choose any j ∈ N. Iterating the above inequality j times we get Φ(r, D(a, R)) ≤ C j 1 Φ 2 j N −j r, D(R + r j ) + C 2 M p (a, R + r j )r α where r j = r j−1 l=0 2 l N −l ≤ 2r. Recall that we know a priori that ϕ is C 1,λ/κ -smooth, so we can estimate the first term on the right-hand side by Φ 2 j N −j r, D(a, R + r j ) ≤ C λκ 2 jλ/κ N −jλ/κ r λ/κ , which gives Φ(r, D(a, R)) ≤ C λκ (C 1 N −λ/κ ) j r λ/κ + C 2 M l p (a, 3R)r α j−1 l=0 (C 1 N −α ) l for each j ∈ N. To ensure that the first term disappears and that the second term converges when j → ∞ we need to know the following (41) C 1 2 λ/κ N −λ/κ < 1 and C 1 N −α < 1 .
Hence, we only need to estimate d Gr (X, Y ).