Rectifiability Properties of Brakke Motions in the Sense of Varifolds

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 25, Rectifiability Properties of Brakke Motions in the Sense of Varifolds Peibiao Zhao 1 and Xiaoping Yang 2 Department of Applied Mathematics Nanjing University of Science and Technology Nanjing , Jiangsu Province, P.R. China pbzhao@mail.njust.edu.cn, yangxp@mail.njust.edu.cn Abstract This paper investigates the Brakke motions and obtains some interesting results of Brakke motions in the sense of varifolds. Mathematics Subject Classification: 28A25, 53C10, 49Q15 Keywords: Brakke Motion, Brakke Flows, Varifolds, Radon Measures 1 Introduction It is well known that Geometric Measure Theory (abridged as GMT) is an area of analysis concerned with solving geometric problems via measure theoretic techniques. The canonical motivating physical problem is probably the so-called Plateau problem investigated experimentally by Plateau in the nineteenth century [3]: given a boundary wire, how does one find the minimal soap film which spanned it? Up to now, the techniques and ideas from GMT have been found useful in the study of PDE, the calculus of variations, harmonic analysis and fractals. We know that many variational problems are solved by enlarging the allowed class of solutions, showing that in this enlarged class a solution exists, and then showing that the solution possesses more regularity than an arbitrary element of the enlarged class. Much of works in GMT has directed towards placing this informal description on a formal footing appropriate for the study of surfaces. The key concept of underlying the whole theory is of rectifiability. It is a measure theoretic notion of smoothness. As we know that it is common 1 Supported by a Grant-in-Aid for Science Research from NJUST (BK , KN11008) and Partly by NNSF ( ) 2 Supported by NNSF( )

2 1202 Peibiao Zhao and Xiaoping Yang in analysis to construct measures as a solution to equations, for instance, the regularization and partial regularity for motion by mean curvature, and we would like to be able to deduce something about the structure of these measures. To solve a minimal surface problem, a natural approach would be to take a sequence of approximating sets whose areas are decreasing and finally extract a convergent subsequence with the hope that the limit would possess the required properties. Unfortunately, the usual notion of convergence for surface sequence sets in Euclidean spaces are not suited to this. Thus, the current theory arises at the historic moment. The current theory, introduced by de Rham and extensively developed by H.Federer and in [2, 4, 5, 6, 9, 10, 11, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25], was developed as a way around this obstacle for oriented surface. In essence, currents are generalized surfaces which will be studied in detail in the next paper. The theory of currents is ideally suited for investigating oriented surfaces but for un-oriented surface, problems arise. The theory of varifolds was emerging as the times require. The theory of varifolds was initiated by Almgren and extensively developed by Allard [1], [6, 8, 15] as an alternative notion of surfaces which didn t require an orientation. In order to consider the varifolds tangent, we, in particular, focus our attention on rectifiable varifolds. Now we single out the arrangement of this paper as follows. The first two sections of this paper are devoted to introducing some necessary notations and terminologies and studying some simple properties for general varifolds. In the third section, we will focus our attention on the moving varifolds in the sense of Brakke. In this setting, we obtain some interesting results for Brakke motion based on Varifolds. 2 Fundamental Definitions of Varifolds In order to model the varifolds, F. Morgan [16] posed a description in images for varifolds. Roughly speaking, A varifold is a measure concentrated on a set in space and certain tangent planes. The notion of varifolds allows multiplicity, but neither cancellation nor obvious definition of orientation. Also tangent planes need not be associated with the set. More precisely, an m dimensional varifold is a Radon measure on R n G(n, m), where G(n, m) is the Grassmannian of un-oriented unit m planes through the origin in R n. Let us describe the precise definition as follows. All other related notations and terminologies, one can refer to [18] for details. Let G(n, m) denote the collection of all m dimensional subspaces of R n, equipped with the metric ρ(s, T )= P S P T = ( n (P ij S P ij T )2) 1/2, where i,j=1 P S,P T denote the orthogonal projections of R n onto S, T G(n, m) respectively, and P ij S = e i P S (e j ),P ij T = e i P T (e j ) are corresponding metrics w.r.t.

3 Rectifiability properties of Brakke motions 1203 the standard orthonormal basis e 1,,e n for R n. Let A be a subset of R n,we define the Grassmannian bundle as G m (A) =A G(n, m) equipped with the product metric. For the sake of convenience, we denote by M 1 (D) =M(D) the set of all Radon measures on D. Definition 2.1 V is said to be a m varifold if V is a Radon measure on G m (R n ). If U is an open subset of R n, we mean any Radon measure V on G m (U). Given such a m varifold V on U, there corresponds a Radon measure μ = μ V on U(called the weight of V ) defined by μ(a) =V (π 1 (A)),A U, where, here and subsequently, π is the projection (x, S) x of G m (U) onto U. Definition 2.2 Let M be a countably m rectifiable, H m measurable subset of R n, and let θ be a positive locally H m integrable function on M. Corresponding to such a pair (M, θ), we define the rectifiable m varifold (M, θ) V to be simply the equivalence class of all pairs ( M, θ), where M is the countably m rectifiable with H m (MΔ M) =0and θ = θ H m a.e. on M M, and MΔ M =(M \ M) ( M \ M), θ is called the multiplicity function of (M, θ). (M, θ) is called an integer multiplicity rectifiable m varifold if the V multiplicity V function is integer-valued H m a.e. Associated to a rectifiable m varifold (M, θ), there is a Radon measure μ V defined by μ V = H V m θ, where we adopt the convention that θ 0onR n \ M. In other words, for H m measurable A, there holds μ V (A) = A M θdhm. Remark 2.1 It is well known that a general m varifold V is a Radon measure on G m (M), by Riesz representation theorem, we write V (ϕ) = ϕ(x, S)dV (x, S), where ϕ Cc 0(G m(m), R) is a function on the total space of G m (M), and for each x M, S runs over G x,m (M), the space G x,m (M) is the point x corresponding to G m (M). We give the set V m (M) of all general m varifolds the topology of convergence of Radon measures. Associated to V is a Radon measure μ V on M, defined by μ V = π # V, where π : G m (M) M is the projection map. Inversely, if μ is an m rectifiable Radon measure, then for H m a.e.x M, T x M exists, Θ m (μ, x) > 0. Take M 0 (μ) ={x M :Θ m (μ, x) exists and is positive}, then, if H m M 0 is locally finite, then for H m M 0 a.e.x M, T x μ = T x M 0, where T x μ is the m dimensional approximate tangent space of μ. It is obvious that T x μ is a μ measurable section of G m (M) defined μ a.e.. Therefore it works sense to define a varifold V μ ˆ=V (μ) by the formula V μ (ϕ) = ϕ(x, S)dV μ (x, S) = ϕ(x, T x μ)dμ(x) for ϕ Cc 0 (G m (M), R). That is to say, we give again in another way the definition of m rectifiable varifolds, i.e., integer rectifiable m varifolds.

4 1204 Peibiao Zhao and Xiaoping Yang We write IV m (M) for the set of integer rectifiable m varifolds. If we denote by IM m the space of integer m rectifiable Radon measures. It is not hard to know that there holds Remark 2.2 The function V : IM m IV m, μ : V m Mset up a one-to-one correspondence between IM m and IV m. By the definition of varifolds, one gets V V m (M) isanm dimensional rectifiable varifold in M if there exist real numbers l 1,l 2, and H m measurable subsets U 1,U 2, of M which meet every compact subset of the open set U s.t. V = l i V (U i ). If l i are positive integers, we say that V is an m dimensional i=1 integral varifold. Throughout this paper we adopt the notations RV m (M) and IV m (M) by the spaces of m dimensional rectifiable varifolds in M and m dimensional integral varifolds in M. In general, the mass M(V )ofv is defined by M(V )=μ V (U) =V (G m (U)), for U R n. It is obvious that for any m rectifiable varifold V(M, θ) on U (simply V(M) in case θ 1onM), μ = H m θ and TM = {(x, T x M):x M }, where M is the set of x M s.t. M has an approximate tangent space T x M w.r.t. θ at x. In other words, μ V = H m θ ˆ=μ. In fact, M = {x U : T x,θ(x) exist} is H m measurable, countably m rectifiable, θ is locally H m integrable on M, and V = (M, θ). V Definition 2.3 V = V (M, θ) is stationary in U if div M MXdμ V C 1 vector fields X on U having compact support in U. =0for any In other words, V is stationary in U if δv (x) = 0 for any C 1 vector fields X on U having compact support in U. Furthermore, it is well known that K.Brakke introduced varifolds moving by mean curvature in his 1978 book [7], while his method has some drawbacks, it is very general and has an impressive regularity theory. Our definition involves Radon measures on M in place of the varifolds of [7]. For this purposes, V can be reconstructed from μ V whenever it is m rectifiable. Let M be a manifold and assume that μ IM m (M), Y C 1 c (TM), and {Φ s } s ( δ,δ) is a smooth family of diffeomorphisms with s s=0φ s = Y, Φ 0 = id, Φ s M\U = id for some U M. We consider the following functional

5 Rectifiability properties of Brakke motions 1205 I[V μ ]= f(x)dμ(x), f>0 smooth. The first variation of I is M s s=0i[φ s #(V μ U)] = s s=0 f(φ s (x)) J μ Φ s (x) dμ(x) = x s=0f(φ s (x)) + f(x) x s=0 J μ Φ s (x) dμ = f Y + fs : Ydμ = f S Y + S : (fy)dμ = δv μ (fy )+ f S Ydμ Then, we arrive at the weighted first variation formula s s=0i[φ s #(V μ U)] = ( f H + f S ) Y dμ, if δv is absolutely continuous w.r.t. μ. Let M be complete, μ a Radon measure on M, and φ(φ : M R) Cc 2(M, R+ ). Definition 2.4(Brakke Motion) Let {μ t } t 0 be a family of Radon measures on M. {μ t } t 0 is called a Brakke Motion provided that for all t 0 and all φ Cc 2 (M, R + ), D t μ t (φ) B(μ t,φ). Here D t0 f(t) = lim sup f(t) f (t 0) t t0 t t 0, B(μ, φ) is defined as follows: {, singular case B(μ, φ) = φh 2 + φ S H dμ, nonsingular case where = M and S = S(x) T x μ for H m a.e.x {φ>0}, and the singular case is as the following (i) μ {φ >0} is not an m rectifiable Radon measure; (ii) δv {φ >0} is not an Radon m measure on {φ >0}, where V = V μ {φ >0}; (iii) δv {φ >0} is singular w.r.t. μ {φ >0}; (iv) φh 2 dμ =, where H = d(δv ) {φ >0}. dμ Take f = φ, x = Y = H, then weighted first variation formula becomes t d φdh m = φh 2 + φ S HdH m dt Γ t Γ t where {Γ t } t 0 is a smooth family of submanifolds evolving by mean curvature.

6 1206 Peibiao Zhao and Xiaoping Yang Remark 2.3 If μ 0 is stationary i.e., δv μ0 =0, then μ t = μ 0 for all t 0 by Definition 2.4. For comparing Definition 2.4 with that of [7], we give the so-called Brakke flow as follows. Definition 2.5 Let {μ t } t 0 be Radon measures on R n. We say that {μ t } t 0 is a Brakke flow (In general, Brakke motion) if lim sup μs(φ) μt(φ) B(μ s t s t t,φ) for any t 0 and φ Cc 2 (R n [0, ))). In particular, if μ t IM m (M), then we say that it is an integral Brakke flow. 3 Main Theorems and Proofs By virtue of the statements posed in section 2 and notice that μ is a continuous map but V is not. From Definitions of varifolds, rectifiable m varifolfds, etc, it is easy to derive the following properties. Proposition 3.1 Let U be an H m measurable subset of R n, and let θ : R n Z + be locally H m -integrable, with U = {θ >0} H m a.e. we define μ(u, θ) = H m θ, then (1) Θ m (μ, x) exists and equals θ(x) for H m a.e.x R n ; (2) U = M 0 H m a.e; (3) Suppose that U C 0 ( C i ) where H m (C 0 )=0and each C i is an i 1 embedded into C 1 m dimensional submanifold. Then for H m a.e. x U, T x μ = T x C i for every x C i ; (4) Θ m (μ, x) =0for H m a.e. x sptμ U ; (5) μ(sptμ U )=0. Proof Since the simplicity of the proof of Proposition 3.1, we only to prove (1). Indeed, for H m a.e. x C i Θ m (μ, x) =Θ m (μ i,x) where μ i = μ(c i,θ C i ), then we get Θ m (μ, x) = μ i (B ρ (x)) lim ρ 0 α(m)ρ α(m)ρ m m H m C i (B ρ (x)) Hm C i (B ρ (x)) α(m)ρ m = dμ i d(h m C i ) Θm (H m C i,x) = θ(x) By the Lebesgue differentiation theorem and the fact that μ i = μ(c i,θ C i ). By a direct checking, the other properties can be derived. This completes the proof of Proposition 3.1. Let S be a m dimensional subspace of R n, we will denote by S v the projection of vector v onto S, w v the inner product of vectors w, v, A : B

7 Rectifiability properties of Brakke motions 1207 the inner product of bilinear forms A and B, respectively. Take V V m (R n ), μ = μ V,Φ:R n R n be C 1. The push-forward Φ # (V ) defined by Φ # (V )(ϕ) = ϕ(x, S)d(Φ # V )(x, S) = ϕ(x, S) J S Φ dv (x, S) for ϕ C 0 c (G m(r n, R)), where the Jacobian is given by the following J S Φ(x) = det[(dφ(x) S) dφ(x) S] for x R n, S G x,m (R n ). The first Variation δv of V is the linear functional on section Y C 1 c (Rn ) given by δv (Y )= d dt t=0φ t # (V )(U), where spty U Rn and {Φ t } t ( ɛ,ɛ) is a family of diffeomorphisms supported in U with ( t ) t=0φ t = Y,Φ 0 = id. Let denote the covariant differentiation on M, then t t=0 J S Φ t (x) = S : Y (x) = m < ei Y (x),e i > for any orthonormal basis {e 1,,e m } of S. Thus δv (Y ) = G m(r n ) S : Y (x)dv (x, S). For U R n open, define the total first variation by i=1 δv (U) = sup{δv (Y ):Y C 1 c (R n U ), Y 1} If δv is a Radon measure, then δv is a R n valued Radon measure with δv as a total variation measure. In this setting, δv can be decomposed as δv = μ H + δv sing, where H is a locally μ integrable section of R n and δv sing is supported on a set of μ measure zero. H is also called the generalized mean curvature vector of V. For a m rectifiable Radon measure μ, we can take S = S(x) =T x μ, then the first variation formula becomes δv μ (Y )= d H dt t=0φ t #(V )(U) = S : Ydμ= Vµ Ydμ+ δv sing (Y ) If we still write, H = H Vµ for short, then the formula above becomes Now, we can state the following δv μ (ϕ) = H Ydμ+ δvsing (Y ) Theorem 3.1 (Allard) Let {μ i } i 1 IM m (M) with sup i 1 (μ i (U)+( i V μi )(U)) < for each U M, where M is a complete Riemannian manifold, assumed

8 1208 Peibiao Zhao and Xiaoping Yang to be isometrically embedded in a high-dimensional Euclidean space. Then there exists μ IM m (M) and a subsequence {μ ij } j 1 s.t. (1) μ ij μ as Radon measures on M; (2) V μij V μ as Radon measures on G m (M); (3) δv μij δv μ as TM valued Radon measures; (4) Lower semi-continuity of total first variation: δv μ lim inf δv μ ij as ij a Radon measure. At the same time by [18], we have the so-called first rectifiability theorem as follows Theorem 3.2 Suppose that V is an m varifold on U which has a tangent space T x with multiplicity θ(x) (0, ) for μ V a.e.x U. Then, V is an m rectifiable varifold. For integer varifolds, we can state the Brakke s orthogonality Theorem as follows Theorem 3.3 [7] If V is an integer varifold with locally bounded first variation, then the vector H (x) is orthogonal to the tangent space S(x) ={S ij (x)} for μ V almost all points x U. Moreover, there holds the following lim r 0 B r m 1 S(x) S(x 0 ) 2 dμ V =0for μ V a.e. x 0 U r(x 0 ) Theorem 3.3 implies that the Krakke s moving varifolds, which will be investigated in the following, are well worth doing. Proposition 3.2 If t M[μ 0 s]ds <, then M[μ t ] M[μ 0 ]. Proof Take test function φ = φ R Cc 2(M, R+ ) satisfying φ =1onB R (x 0 ), and φ =0offB 2R (x 0 ), 0 φ 1 Dφ 2 for fixed x R 0 M. Then, we get t t φ 2 dμ t φ 2 dμ 0 B(μ s,φ 2 )ds φ 3 dμ s ds 0 4 t M[μ R 2 s ]ds Letting R, we get the required result. Proposition This completes the proof of Theorem 3.4 Let M be complete, {μ i t} t 0, i =1, 2, be a sequence of integral Brakke flows in M, and sup μ i t C(U) < for each U M. Then, i,t (1) there exist a subsequence {μ i j t } t 0,j =1, 2,, and an integral Brakke flow (motion) {μ t } t 0 s.t. μ i j t μ t as Radon measures for each t 0; (2) For a.e.t 0, there is a subsequence {i j} j 1 of {i j } j 1 s.t. lim V (μ i j t )= j V (μ t ) as varifolds.

9 Rectifiability properties of Brakke motions 1209 Proof We prove this theorem in terms of the approaches offered by L. Ambrosio and B. Kirchheim [2] or that by L. Simon [18]. To complete the proof of Theorem 3.4, we first derive that the following two facts are tenable. The first fact is the upper semicontinuity of B(μ, φ); The second fact is aplinear growth of {μ t }, and the related properties of {μ t }. That is, we first prove the following Proposition 3.3 Let {μ i } i 1 IM m (M) be a sequence with the property B(μ i,φ) k, μ i ({φ >0}) k for all i 1,φ Cc 2 (M, R + ),k >0. Then, there exists a subsequence {μ ij } j 1 and a Radon measure μ IM m ({φ >0}) s.t. (a) μ ij {φ >0} μ as Radon measures on {φ >0}; (b) V (μ ij {φ >0}) V (μ) as varifolds over {φ >0}. If sup μ i ({φ > i 1 0}) <, then, we also have the following lim sup B(μ i,φ) B(μ, φ). i Proof. By the hypotheses of Proposition 3.3, Allard s compactness theorem implies that there is a subsequence {μ ij } j 1 and a Radon measure μ IM m ({φ >0}) s.t. μ ij {φ >0} μ as radon measures on {φ >0}, and V (μ ij {φ >0}) V (μ) as varifolds over {φ >0}. On the other hand, since k φh 2 + φ S Hdμ i 1 2 φh2 + 1 φ 2 2 φ i. Indeed, we have B(μ, φ) = φh 2 φ S Hdμ = φ H 1 φ : S φ : S 2 dμ 2 φ 4 φ φ 2 By [6,12], one gets B(μ, φ) 1 dμ (max 4 φ 2 φ /2)μ({φ >0}) Then, φh 2 dμ i C(φ). By Cauchy-Schwarz inequality, for each U {φ >0}, then we have δv μi (U) C(φ, U). For μ i μ as Radon measures on {φ >0} with sup μ i ({φ >0}) <. If lim sup B(μ i,φ)=, we are done. Otherwise, i i by taking a sequence, we obtain lim B(μ i,φ) exists and is finite. Then by (a) i we know that V (μ i {φ >0}) V (μ {φ >0}) as varifolds over {φ >0}. This is the end of the proof of the first fact. We now continue to prove the second fact as follows. Proposition 3.4 Let {μ t } t 0 satisfy μ t (U) C(U) for t 0. Then, (c) μ t kt is nonincreasing for φ Cc 2 (M, R + ), where k is a constant depending on {φ >0};

10 1210 Peibiao Zhao and Xiaoping Yang (d) There is a co-countable set B [0, ) with μ t continuous at all t B. Proof. Since D t μ t (φ) B(μ t,φ) k 0 (φ). Taking k 0 = k, then (c) is derived by virtue of the definition of derivative. Next let A be a countable dense subset of Cc 2 (M, R + ). Then, by (c), for each φ A, there is a co-countable set B φ [0, ) s.t. μ t is continuous on B φ. Take B = φ A B φ,soμ t (φ) is continuous at each t B for each φ A. By the uniform approximation the required results follows. From these statements, we go back to continue to prove Theorem 3.4. Let B 0 be a countable dense subset of times. For t B 0, the mass bound by the hypotheses and by the compactness of Radon measures allows us to choose a subsequence {μ i j t } j 1 and a Radon measure μ t on M s.t. μ i j t μ t as Radon measures. By choosing successive subsequences and diagonalizing, we can find a single subsequence, s.t. μ i j t μ t for all t B 0. Here we also denote the single subsequence by {μ i j t } j 1. For any φ Cc 2 (M, R + ), by Proposition 3.4, we know that μ i j t (φ) k(φ)t is nonincreasing in t. Then we find μ t (φ) k(φ)t is nonincreasing for t B 0 by passing to limits as j. On the other hand, when t [0, ) \ B 0, we take any convergent subsequence {μ i k t } k 1. Let μ t = lim μ i k t,soμ t is defined for all t [0, ). By k Proposition 3.4, there is a co-countable set B 1 s.t. μ t is continuous at all t B 1. For that t, μ t is uniquely determined, independent of the subsequence {i k } k 1 chosen for that t, and hence the full sequence converges for t B 1. Since [0, ) \ B 1 is countable, repeating the procedure above, we can construct a family {μ t } t 0 of Radon measures s.t. μ i j t μ t for all t 0, and μ t (φ) k(φ)t is nonincreasing for all φ Cc 2 (M, R + ). In the following section, we prove {μ t } is an integral Brakke motion. In other words, we wish to prove D t0 μ t (φ) = max(dt 0 μ t (φ),d t + μ t (φ)). We want to prove that D t + 0 μ t (φ) B(μ t0,φ). For sequence t l t 0,ɛ l 0 with D t + 0 μ t (φ) ɛ l μt l (φ) μt 0 (φ) t l t 0. By the convergence μ i j t μ t, there is for each l a number r l, selected from the sequence {i j } j 1 s.t. D t + 0 μ t (φ) 2ɛ l μr l t l (φ) μ r l t 0 (φ) t l t 0 1 tl D t μ r l t (φ)dt. t l t 0 t 0 Let l, there exists at least one point s l [t 0,t l ] s.t. μ r l s l IM m ({φ >0}) and D t + 0 μ t (φ) 2ɛ l D t μ r l s l (φ) B(μ r l s l,φ). By Proposition 3.3, we know that there is a subsequence {μ r l s l} l 1 and a Radon measure μ IM m ({φ >0}) s.t. μ r l s μ and D t + l 0 μ t (φ) lim sup B(μ r l l s B(μ, φ). We can prove by using l,φ) the similar method for Dt 0 μ t (φ). By a direct checking and D t0 μ t (φ) B(μ t,φ), then we get the required results.

11 Rectifiability properties of Brakke motions 1211 Next, let A be a countable subset of Cc 2 (M, R + ) s.t. the open set {φ >0} for φ A cover M. Fixed T<, then for φ A, we obtain T k μ i j T (φ) μi j 0 D t μ i j t (φ)dt B(μ i j t,φ)dt 0 then for a.e.t 0, lim sup B(μ i j t,φ) >. By Proposition 3.4, we know that j there is a subsequence {i j } of {i j} such that μ i j t μ t IM m (M),V(μ i j t ) V (μ t ). Then, we completes the proof of Theorem 3.4. Acknowledgments This work was supported by the Foundation of Nanjing University of Science and Technology and the Natural Science Foundations of Province, China. References [1] Allard W.K., First variation of a Varifold, Annals of Math., 95(1972): [2] Ambrosio L. and Kirchheim B., Currents in metric spaces, Acta Math., 185:1(2000): 1-80 [3] Almgern F., Plateau s problem. An invitation to varifold geometry, W.A. Benjamin, Inc, New York-Amsterrrdam, 1966, Xii+74 pp [4] Ambrosio L. and Kirchheim B., Rectifiable sets in metric and Banach spaces, Math. Ann., 318:3(2000): [5] Anzellotti G., Serapioni R. and Tamanini I., Curvatures, functionals, currents, Indiana Univ. Math. J., 39(1990): [6] Bouchitte G., Buttazzo G. and Fragala I., Mean curvature of a measure and related variational problems, Ann. Scuola Norm. Sup. Pisa CI. Sci., XXV(4)(1997): [7] Brakke K.A., The motion of a surface by its mean curvature, No. 20, Math. Notes, Princeton Univ. Press, Preceton, NJ, 1978 [8] Delladio S. and Scianna G., Oriented and nonoriented curvature varifolds, Proc. Roy. Soc., Edinberg Sect. A. 125(1995): [9] Federer H., Geometric Measure Theory, Springer, 1969 [10] Federer H. and Fleming W.H., Normal and integral currents, Ann. of Math., 72:2 (1960):

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