Regularity of Area Minimizing Currents Iii: Blow-up

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren’s partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions. 0. Introduction In this paper we complete the proof of a slightly improved version of the celebrated Almgren’s partial regularity result for area minimizing currents in a Riemannian manifold (see [1]), namely Theorem 0.3 below. Assumption 0.1. Let ε0 ∈]0, 1[, m, n̄ ∈ N \ {0} and l ∈ N. We denote by (M) Σ ⊂ R = R an embedded (m+ n̄)-dimensional submanifold of class C0 ; (C) T an integral current of dimension m with compact support spt(T ) ⊂ Σ, area minimizing in Σ. In this paper we follow the notation of [4] concerning balls, cylinders and disks. In particular Br(x) ⊂ R will denote the Euclidean ball of radius r and center x. Definition 0.2. For T and Σ as in Assumption 0.1 we define Reg(T ) := { x ∈ spt(T ) : spt(T ) ∩Br(x) is a C0 submanifold for some r > 0 } , (0.1) Sing(T ) := spt(T ) \ ( spt(∂T ) ∪ Reg(T ) ) . (0.2) The partial regularity result proven first by Almgren [1] under the more restrictive hypothesis Σ ∈ C gives an estimate on the Hausdorff dimension dimH(Sing(T )) of Sing(T ). Theorem 0.3. dimH(Sing(T )) ≤ m− 2 for any m, n̄, l, T and Σ as in Assumption 0.1. In this note we complete the proof of Theorem 0.3, based on our previous works [3, 5, 6, 4], thus providing a new, and much shorter, account of one of the most fundamental regularity result in geometric measure theory; we refer to [6] for an extended general introduction to all these works. The proof is carried by contradiction: in the sequel we will always assume the following. Assumption 0.4 (Contradiction). There exist m ≥ 2, n̄, l, Σ and T as in Assumption 0.1 such that Hm−2+α(Sing(T )) > 0 for some α > 0. 1 2 CAMILLO DE LELLIS AND EMANUELE SPADARO The hypothesis m ≥ 2 in Assumption 0.4 is justified by the well-known fact that Sing(T ) = ∅ when m = 1 (in this case spt(T ) \ spt(∂T ) is locally the union of finitely many non-intersecting geodesic segments). Starting from Assumption 0.4, we make a careful blow-up analysis, split in the following steps. 0.1. Flat tangent planes. We first reduce to flat blow-ups around a given point, which in the sequel is assumed to be the origin. These blow-ups will also be chosen so that the size of the singular set satisfies a uniform estimate from below (cp. Section 1). 0.2. Intervals of flattening. For appropriate rescalings of the current around the origin we take advantage of the center manifold constructed in [4], which gives a good approximation of the average of the sheets of the current at some given scale. However, since it might fail to do so at different scales, in Section 2 we introduce a stopping condition for the center manifolds and define appropriate intervals of flattening Ij = [sj, tj]. For each j we construct a different center manifold Mj and approximate the (rescaled) current with a suitable multi-valued map on the normal bundle of Mj. 0.3. Finite order of contact. A major difficulty in the analysis is to prove that the minimizing current has finite order of contact with the center manifold. To this aim, in analogy with the case of harmonic multiple valued functions (cp. [3, Section 3.4]), we introduce a variant of the frequency function and prove its almost monotonicity and boundedness. This analysis, carried in Sections 3, 4 and 5, relies on the variational formulas for images of multiple valued maps as computed in [5] and on the careful estimates of [4]. Our frequency function differs from that of Almgren and allows for simpler estimates. 0.4. Convergence to Dir-minimizer and contradiction. Based on the previous steps, we can blow-up the Lipschitz approximations from the center manifoldMj in order to get a limiting Dir-minimizing function on a flat m-dimensional domain. We then show that the singularities of the rescaled currents converge to singularities of that limiting Dir-minimizer, contradicting the partial regularity of [3, Section 3.6] and, hence, proving Theorem 0.3. Acknowledgments. The research of Camillo De Lellis has been supported by the ERC grant agreement RAM (Regularity for Area Minimizing currents), ERC 306247. The authors are warmly thankful to Bill Allard for several enlightening conversations and his constant support. 1. Flat tangent cones Definition 1.1 (Q-points). For Q ∈ N, we denote by DQ(T ) the points of density Q of the current T , and set RegQ(T ) := Reg(T ) ∩DQ(T ) and SingQ(T ) := Sing(T ) ∩DQ(T ). Definition 1.2 (Tangent cones). For any r > 0 and x ∈ R, ιx,r : R → R is the map y 7→ y−x r and Tx,r := (ιx,r)]T . The classical monotonicity formula (see [10] and [6, Lemma A.1]) implies that, for every rk ↓ 0 and x ∈ spt(T ) \ spt(∂T ), there is a subsequence (not relabeled) for which Tx,rk converges to an integral cycle S which is a cone REGULARITY OF AREA MINIMIZING CURRENTS III: BLOW-UP 3 (i.e., S0,r = S for all r > 0 and ∂S = 0) and is (locally) area-minimizing in R. Such a cone will be called, as usual, a tangent cone to T at x. Fix α > 0. By Almgren’s stratification theorem (see [10, Theorem 35.3]), for Hm−2+αa.e. x ∈ spt(T )\spt(∂T ), there exists a subsequence of radii rk ↓ 0 such that Tx,rk converge to an integer multiplicity flat plane. Similarly, for measure-theoretic reasons, if T is as in Assumption 0.4, then for Hm−2+α-a.e. x ∈ spt(T ) \ spt(∂T ) there is a subsequence sk ↓ 0 such that lim infkH ∞ (DQ(Tx,sk)∩B1) > 0 (see again [10]). Obviously there would then be Q ∈ N and x ∈ SingQ(T ) where both subsequences exist. The two subsequences might, however, differ: in the next proposition we show the existence of one point and a single subsequence along which both conclusions hold. For the relevant notation (concerning, for instance, excess and height of currents) we refer to [6, 4]. Proposition 1.3 (Contradiction sequence). Under Assumption 0.4, there are m,n,Q ≥ 2, Σ and T as in Assumption 0.1, reals α, η > 0, and a sequence rk ↓ 0 such that 0 ∈ DQ(T ) and the following holds: lim k→+∞ E(T0,rk ,B6 √ m) = 0, (1.1) lim k→+∞ Hm−2+α ∞ (DQ(T0,rk) ∩B1) > η, (1.2) H ( (B1 ∩ spt(T0,rk)) \DQ(T0,rk) ) > 0 ∀ k ∈ N. (1.3) The proof is based on the following lemma. Lemma 1.4. Let S be an m-dimensional area minimizing integral cone in R such that ∂S = 0, Q = Θ(S, 0) ∈ N, H ( DQ(S)) > 0 and H(SingQ(S)) = 0. Then, S is an m-dimensional plane with multiplicity Q. Proof. For each x ∈ RegQ(S), let rx be such that S B2rx(x) = Q JΓK for some regular submanifold Γ and set U := ⋃ x∈RegQ(S) Brx(x). Obviously, RegQ(S) ⊂ U ; hence, by assumption, it is not empty. Fix x ∈ spt(S) ∩ ∂U . Let next (xk)k∈N ⊂ RegQ(S) be such that dist (x,Brxk (xk)) → 0. We necessarily have that rxk → 0: otherwise we would have x ∈ B2rxk (xk) for some k, which would imply x ∈ RegQ(S) ⊂ U , i.e. a contradiction. Therefore, xk → x and, by [10, Theorem 35.1], Q = lim sup k→+∞ Θ(S, xk) ≤ Θ(S, x) = lim λ↓0 Θ(S, λx) ≤ Θ(S, 0) = Q. This implies x ∈ DQ(S). Since x ∈ ∂U , we must then have x ∈ SingQ(S). Thus, we conclude that Hm−1(spt(S) ∩ ∂U) = 0. It follows from the standard theory of rectifiable currents (cp. Lemma A.2) that S ′ := S U has 0 boundary in R. Moreover, since S is an area minimizing cone, S ′ is also an area-minimizing cone. By definition of U we have Θ(S ′, x) = Q for ‖S ′‖-a.e. x and, by semicontinuity, Q ≤ Θ(S ′, 0) ≤ Θ(S, 0) = Q. 4 CAMILLO DE LELLIS AND EMANUELE SPADARO We apply Allard’s theorem and deduce that S ′ is regular, i.e. S ′ is an m-plane with multiplicity Q. Finally, from Θ(S ′, 0) = Θ(S, 0), we infer M(S B1) = M(S ′ B1) and then S ′ = S. Proof of Proposition 1.3. Let m > 1 be the smallest integer for which Theorem 0.3 fails. By Almgren’s stratification theorem (cp. Theorem A.3), there must be an integer rectifiable area minimizing current R of dimension m and a positive integer Q such that the Hausdorff dimension of SingQ(R) is larger than m−2. We fix the smallest Q for which such a current R exists. Recall that, by the upper semicontinuity of the density and a straightforward application of Allard’s regularity theorem (see Theorem A.1), Sing1(R) = ∅, i.e. Q > 1. Let α ∈]0, 1] be such that H(SingQ(R)) > 0. By [10, Theorem 3.6] there exists a point x ∈ SingQ(R) such that SingQ(R) has positive Hm−2+α ∞ -upper density: i.e., assuming without loss of generality x = 0 and ∂R B1 = 0, there exists rk ↓ 0 such that lim k→+∞ Hm−2+α ∞ ( SingQ(R0,rk) ∩B1 ) = lim k→+∞ Hm−2+α ∞ ( SingQ(R) ∩Brk ) rm−2+α k > 0 . Up to a subsequence (not relabeled) we can assume that R0,rk → S, with S a tangent cone. If S is a multiplicity Q flat plane, then we set T := R and we are done: indeed, (1.3) is satisfied by Theorem A.1, because 0 ∈ Sing(R) and ‖R‖ ≥ H spt(R). Assume therefore that S is not an m-dimensional plane with multiplicity Q. Taking into account the convergence of the total variations for minimizing currents [10, Theorem 34.5] and the upper semicontinuity of Hm−2+α ∞ under the Hausdorff convergence of compact sets, we get Hm−2+α ∞ ( DQ(S) ∩ B̄1 ) ≥ lim inf k→+∞ Hm−2+α ∞ ( DQ(R0,rk) ∩ B̄1 ) > 0. (1.4) We claim that (1.4) implies Hm−2+α ∞ (SingQ(S)) > 0. (1.5) Indeed, if all points of DQ(S) are singular, then this follows from (1.4) directly. Otherwise, RegQ(S) is not empty and, hence, H(DQ(S) ∩ B1) > 0. In this case we can apply Lemma 1.4 and infer that, since S is not regular, then H(SingQ(S)) > 0 and (1.5) holds. We can, hence, find x ∈ SingQ(S) \ {0} and rk ↓ 0 such that lim k→+∞ Hm−2+α ∞ ( SingQ(Sx,rk) ∩B1 ) = lim k→+∞ Hm−2+α ∞ ( SingQ(S) ∩Brk(x)) rm−2+α k > 0. Up to a subsequence (not relabeled), we can assume that Sx,rk converges to S1. Since S1 is a tangent cone to the cone S at x 6= 0, S1 splits off a line, i.e. S1 = S2× JRvK, for some area minimizing cone S2 in Rm−1+n and some v ∈ R (cp. the arguments in [10, Lemma 35.5]). Since m is, by assumption, the smallest integer for which Theorem 0.3 fails, H(Sing(S2)) = 0 and, hence, H(SingQ(S1)) = 0. On the other hand, arguing as we did for (1.4), we have Hm−2+α ∞ (DQ(S1) ∩ B̄1) ≥ lim sup k→+∞ Hm−2+α ∞ (DQ(Sx,rk) ∩ B̄1) > 0. REGULARITY OF AREA MINIMIZING CURRENTS III: BLOW-UP 5 Thus RegQ(S1) 6= ∅ and, hence, H(DQ(S1)) > 0. We can apply Lemma 1.4 again and conclude that S1 is an m-dimensional plane with multiplicity Q. Therefore, the proposition follows taking T := τ]S, with τ the translation map y 7→ y − x, and Σ the tangent plane at 0 to the original Riemannian manifold. 2. Intervals of flattening For the sequel we fix the constant cs := 1 64 √ m and notice that 2−N0 < cs, where N0 is the parameter introduced in [4, Assumption 1.8]. It is always understood that the parameters β2, δ2, γ2, ε2, κ, Ce, Ch,M0, N0 in [4] are fixed in such a way that all the theorems and propositions therein are applicable, cf. [4, Section 1.2]. In particular, all constants which will depend upon these parameters will be called geometric and denoted by C0. On the contrary, we will highlight the dependence of the constants upon the parameters introduced in this paper p1, p2, . . . by writing C = C(p1, p2, . . .). We recall also the notation introduced in [4, Assumption 1.3]. If Σ ∩ B7m has no boundary in B7m and for any p ∈ Σ ∩ B7m there is a map Ψp : TpΣ ⊃ Ω → (TpΣ) parametrizing it, then c(Σ ∩B7m) := supp∈Σ∩B7√m ‖DΨp‖C2,ε0 . Obviously these assumptions might fail for a general Σ (in fact c(Σ ∩B7m) need not be well-defined). However, having fixed a point q ∈ Σ, given its C0 regularity, c(ιq,r(Σ)∩B7m) is well-defined whenever r is sufficiently small and converges to 0 as r ↓ 0. In particular, by Proposition 1.3 and simple rescaling arguments, we assume in the sequel the following. Assumption 2.1. Let ε3 ∈]0, ε2[. Under Assumption 0.4, there exist m,n,Q ≥ 2, α, η > 0, T and Σ for which: (a) there is a sequence of radii rk ↓ 0 as in Proposition 1.3; (b) the following holds: T0Σ = R × {0}, spt(∂T ) ∩B6m = ∅, 0 ∈ DQ(T ), (2.1) ‖T‖(B6mr) ≤ r ( Qωm(6 √ m) + ε3 ) for all r ∈ (0, 1), (2.2) c(Σ ∩B7m) ≤ ε3 . (2.3) 2.1. Defining procedure. We set R := { r ∈]0, 1] : E(T,B6mr) ≤ ε3 } . (2.4) Observe that, if {sk} ⊂ R and sk ↑ s, then s ∈ R. We cover R with a collection F = {Ij}j of intervals Ij =]sj, tj] defined as follows. t0 := max{t : t ∈ R}. Next assume, by induction, to have defined tj (and hence also t0 > s0 ≥ t1 > s1 ≥ . . . > sj−1 ≥ tj) and consider the following objects: Tj := ((ι0,tj)]T ) B6 √ m, Σj := ι0,tj(Σ) ∩ B7m; moreover, consider for each j an orthonormal system of coordinates so that, if we denote by π0 the m-plane R × {0}, then E(Tj,B6m, π0) = E(Tj,B6m) (alternatively we can keep the system of coordinates fixed and rotate the currents Tj). 6 CAMILLO DE LELLIS AND EMANUELE SPADARO Let Mj be the corresponding center manifold constructed in [4, Theorem 1.17] applied to Tj and Σj with respect to the m-plane π0; the manifold Mj is then the graph of a map φj : π0 ⊃ [−4, 4] → π⊥ 0 , and we set Φj(x) := (x,φj(x)) ∈ π0×π 0 . Then, (Stop) either there is r ∈]0, 3] and a cube L of the Whitney decomposition W (j) of [−4, 4] ⊂ π0 of [4, Definition 1.10 & Proposition 1.11] (applied to Tj) s.t. `(L) ≥ cs r and L ∩ B̄r(0, π0) 6= ∅ (2.5) (in what follows Br(p, π) and B̄r(p, π) will denote the open and closed disks Br(p)∩ (p+ π), B̄r(p) ∩ (p+ π)); (Go) or there exists no radius as in (Stop). We will prove below that, if r is as in (Stop), then r < 2−5. This justifies the following: in case (Go) holds, we set sj := 0, i.e. Ij :=]0, tj], and end the procedure; in case (Stop) holds we let sj := r̄ tj, where r̄ is the maximum radius satisfying (Stop). We choose then tj+1 as the largest element in R∩]0, sj] and proceed inductively. 2.2. First consequences. The following is a list of easy consequences of the definition. Given two sets A and B, we define their separation as the number sep(A,B) := inf{|x−y| : x ∈ A, y ∈ B}. Proposition 2.2. Assuming ε3 sufficiently small, then the following holds: (i) sj < tj 25 and the family F is either countable and tj ↓ 0, or finite and Ij =]0, tj] for the largest j; (ii) the union of the intervals of F cover R, and for k large enough the radii rk in Assumption 2.1 belong to R; (iii) if r ∈] sj tj , 3[ and J ∈ W (j) n intersects B := pπ0(Br(pj)), with pj := Φj(0), then J is in the domain of influence W (j) n (H) (see [4, Definition 3.3]) of a cube H ∈ W (j) e with `(H) ≤ 3 cs r and max {sep (H,B), sep (H, J)} ≤ 3 √ m`(H) ≤ 3r 16 ; (iv) E(Tj,Br) ≤ C0ε3 r2−2δ2 for every r ∈] sj tj , 3[. (v) sup{dist(x,Mj) : x ∈ spt(Tj)∩p j (Br(pj))} ≤ C0 (m j 0) 1 2m r2 for every r ∈] sj tj , 3[, where mj0 := max{c(Σj),E(Tj,B6m)}. Proof. We start by noticing that sj ≤ tj 25 follows from the inequality 2 −N0−1 < cs (cf. [4, Assumption 1.8]) because all cubes in the Whitney decomposition have side-length at most 2−N0−6 (cf. [4, Proposition 1.11]). In particular, this implies that the inductive procedure either never stops, leading to tj ↓ 0, or it stops because sj = 0 and ]0, tj] ⊂ R, thus proving (i). The first part of (ii) follows straightforwardly from the choice of tj+1, and the last assertion holds from E(T,B6mrk)→ 0. REGULARITY OF AREA MINIMIZING CURRENTS III: BLOW-UP 7 Regarding (iii), let H ∈ W (j) e be as in [4, Definition 3.3] and choose k ∈ N\{0} such that `(H) = 2 `(J). Observe that ‖Dφj‖C2,κ ≤ C0ε3 by [4, Theorem 1.17]. If ε3 is sufficiently small, we can assume Br/2(0, π0) ⊂ B ⊂ Br(0, π0) . (2.6) Now, by [4, Corollary 3.2], sep(H, J) ≤ 2 √ m`(H) and sep (B,H) ≤ sep (H, J) + 2 √ m`(J) ≤ 3 √ m`(H). Both the inequalities claimed in (iii) are then trivial when r > 1 4 , because `(H) ≤ 2−N0−6 ≤ 2cs ≤ 2−9/ √ m. Assume therefore r ≤ 1 4 and note that H intersects B2r+3m`(H). Let ρ := 2r + 3 √ m`(H). Observe that 2r < ρ < 1. Since for such ρ (Stop) cannot hold, we have that `(H) < cs ( 2r + 3 √ m`(H) ) = 2 cs r + 3`(H) 16 . Therefore, we conclude that `(H) ≤ 3 csr and sep(H,B) ≤ 9 √ mcsr < 3r/16. We now turn to (iv). If r ≥ 2−N0 , then obviously E(Tj,Br) ≤ (4 √ m 20)2r2E(Tj,B4√m) ≤ (4 √ m 2)rε3 . Otherwise, let k ≥ N0 be the smallest natural number such that 2−k+1 > r and let L ∈ W (j),k ∪ S (j),k be a cube so that 0 ∈ L (cf. [4, Definition 1.10], `(H) = 2−k). By [4, Proposition 4.2(v)], |pL| ≤ ( √ m+ C0(m j 0) /2m) ≤ 2 √ m`(H) and so it follows easily that Br ⊂ BL. From condition (Go) we have L 6∈ W . Thus, by [4, Proposition 1.11], we get E(Tj,Br) ≤ C0E(Tj,BL) ≤ C0ε3 r2−2δ2 . Finally, (v) follows from [4, Corollary 2.2 (ii)], because by (Go), for every r ∈] sj tj , 3[, every cube L ∈ W (j) which intersects Br(0, π0) satisfies `(L) < csr. 3. Frequency function and first variations Consider the following Lipschitz (piecewise linear) function φ : [0 +∞[→ [0, 1] given by φ(r) :=  1 for r ∈ [0, 1 2 ], 2− 2r for r ∈ ] 2 , 1], 0 for r ∈ ]1,+∞[. For every interval of flattening Ij =]sj, tj], let Nj be the normal approximation of Tj on Mj in [4, Theorem 2.4]. Definition 3.1 (Frequency functions). For every r ∈]0, 3] we define: Dj(r) := ∫ Mj φ ( dj(p) r ) |DNj|(p) dp and Hj(r) := − ∫ Mj φ′ ( dj(p) r ) |Nj|(p) d(p) dp , where dj(p) is the geodesic distance on Mj between p and Φj(0). If Hj(r) > 0, we define the frequency function Ij(r) := rDj(r) Hj(r) . 8 CAMILLO DE LELLIS AND EMANUELE SPADARO The following is the main analytical estimate of the paper, which allows us to exclude infinite order of contact among the different sheets of a minimizing current. Theorem 3.2 (Main frequency estimate). If ε3 is sufficiently small, then there exists a geometric constant C0 such that, for every [a, b] ⊂ [ sj tj , 3] with Hj|[a,b] > 0, we have Ij(a) ≤ C0(1 + Ij(b)). (3.1) To simplify the notation, in this section we drop the index j and omit the measure H in the integrals over regions ofM. The proof exploits four identities collected in Proposition 3.5, which will be proved in the next sections. Definition 3.3. We let ∂r̂ denote the derivative with respect to arclength along geodesics starting at Φ(0). We set