Center Manifold: a Case Study

Following Almgren’s construction of the center manifold in his Big regularity paper, we show the C regularity of area-minimizing currents in the neighborhood of points of density one, without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren’s construction in its full generality. 0. Introduction In this note we consider area-minimizing integral currents T of dimension m in R. The following theorem is the cornerstone of the regularity theory. It was proved for the first time by De Giorgi [2] for n = 1 and then extended later by several authors (the constant ωm denotes, as usual, the Lebesgue measure of the m-dimensional unit ball). Theorem 0.1. There exist constants ε, β > 0 such that, if T is an area-minimizing integral current and p is a point in its support such that θ(T, p) = 1, supp (∂T ) ∩ Br(p) = ∅ and ‖T‖(Br(p)) ≤ (ωm + ε) r, then supp (T ) ∩Br/2(p) is the graph of a C function f . Once established this ε-regularity result, the regularity theory proceeds further by deriving the usual Euler–Lagrange equations for the function f . Indeed, it turns out that f solves a system of elliptic partial differential equations and the Schauder theory then implies that f is smooth (in fact analytic, using the classical result by Hopf [5]). In his Big regularity paper [1], Almgren observes that an intermediate regularity result can be derived as a consequence of a more complicated construction without using the nonparametric PDE theory of minimal surfaces (i.e. without deriving the Euler-Lagrange equation for the graph of f). Indeed, given a minimizing current T and a point p with θ(T, p) = Q ∈ N, under the hypothesis that the excess is sufficiently small, Almgren succeeds in constructing a C regular surface (called center manifold) which, roughly speaking, approximates the “average of the sheets of the current” (we refer to [1] for further details). In the introduction of [1] he observes that, in the case Q = 1, the center manifold coincides with the current itself, thus implying directly the C regularity. The aim of the present note is to give a simple direct proof of this remark, essentially following Almgren’s strategy for the construction of the center manifold in the simplified setting Q = 1. At this point the following comment is in order: the excess-decay leading to Theorem 0.1 remains anyway a fundamental step in the proof of this paper (see Proposition 1.2 below) and, as far as we understand, of Almgren’s approach as well. One can take advantage of the information contained in Theorem 0.1 at several levels but we have decided to keep its use to the minimum. 1 2 C. DE LELLIS AND E. N. SPADARO 1. Preliminaries 1.1. Some notation. From now on we assume, without loss of generality, that T is an area-minimizing integer rectifiable current in R satisfying the following assumptions: ∂T = 0 in B1(0), θ(T, 0) = 1 and ‖T‖(B1) ≤ ωm + ε (H) The choice of the small constant ε will be specified later. In what follows, B r (q), B n r (u) and B m+n r (p) denote the open balls contained, respectively, in the Euclidean spaces R, R and R. Given a m-dimensional plane π, Cπ r (q) denotes the cylinder B r (q) × π⊥ ⊂ π × π⊥ = R and P : π × π⊥ → π the orthogonal projection. Central points, supscripts and subscripts will be often omitted when they are clear from the context. We will consider different systems of cartesian coordinates in R. A corollary of De Giorgi’s excess decay theorem (a variant of which is precisely stated in Proposition 1.2 below) is that, when ε is sufficiently small, the current has a unique tangent plane at the origin (see Corollary 1.4). Thus, immediately after the statement of Corollary 1.4, the most important system of coordinates, denoted by x, will be fixed once and for all in such a way that π0 = {xm+1 = . . . = xm+n = 0} is the tangent plane to T at 0. Other systems of coordinates will be denoted by x′, y or y′. We will always consider positively oriented systems x′, i.e. such that there is a unique element A ∈ SO(m + n) with x′(p) = A · x(p) for every point p. An important role in each system of coordinates will be played by the oriented m-dimensional plane π where the last n coordinates vanish (and by its orthogonal complement π⊥). Obviously, given π there are several systems of coordinates y for which π = {ym+1 = . . . = ym+n = 0}. However, when we want to stress the relation between y and π we will use the notation yπ. 1.2. Lipschitz approximation of minimal currents. The following approximation theorem can be found in several accounts of the regularity theory for area-minimizing currents. It can also be seen as a special case of a much more general result due to Almgren (see the third chapter of [1]) and reproved in a simpler way in [3]. As it is customary the (rescaled) cylindrical excess is given by the formula Ex(T, C r ) := ‖T‖(Cπ r ) − ωmr ωm rm = 1 2ωm rm ˆ Cπ r |~ T − ~π| d‖T‖, (1.1) (where ~π is the unit simple vector orienting π and the last equality in (1.1) holds when we assuming ∂T = 0 in Cr and P♯(T Cr) = JBr(pK). Proposition 1.1. There are constants C > 0 and 0 < η, ε1 < 1 with the following property. Let r > 0 and T be an area-minimizing integer rectifiable m-current in Cπ r such that ∂ T = 0, P #(T ) = JB r K and E := Ex(T, C r ) ≤ ε1. CENTER MANIFOLD: A CASE STUDY 3 Then, for s = r(1 − CE), there exists a Lipschitz function f : Bs → R and a closed set K ⊂ Bs such that: Lip(f) ≤ CE; (1.2a) |Bs \K| ≤ C r E and graph(f |K) = T (K × R); (1.2b) ∣ ∣ ∣ ∣ ‖T‖(Cs) − ωm s − 1 2 ˆ Bs |Df | ∣ ∣ ∣ ∣ ≤ C r E. (1.2c) This proposition is a key step in the derivation of Theorem 0.1. In the appendix we include a short proof in the spirit of [3]. Clearly, Theorem 0.1 can be thought as a much finer version of this approximation. However, an aspect which is crucial for further developments is that several important estimates can be derived directly from Proposition 1.1. 1.3. De Giorgi’s excess decay. The fundamental step in De Giorgi’s proof of Theorem 0.1 is the decay of the quantity usually called “spherical excess” (where the minimum is taken over all oriented m–planes π) Ex(T,Br(p)) := min π Ex(T,Br(p), π), with Ex(T,Br(p), π) := 1 2 − ˆ Br(p) |~ T − ~π|d‖T‖. Proposition 1.2. There is a dimensional constant C with the following property. For every δ, ε0 > 0, there is ε > 0 such that, if (H) holds, then Ex(T,B1) ≤ ε0 and Ex(T,Br(p)) ≤ C ε0 r2−2δ for every r ≤ 1/2 and every p ∈ B1/2 ∩ supp (T ). (1.3) From now on we will consider the constant δ fixed. Its choice will specified much later. Definition 1.3. For later reference, we say that a plane π is admissible in p at scale ρ (or simply that (p, ρ, π) is admissible) if Ex(T,Bρ(p), π) ≤ Cm,nε0 ρ2−2δ, (1.4) for some fixed (possibly large) dimensional constant Cm,n. Proposition 1.2 guarantees that, for every p and r as in the statement, there exists always an admissible plane πp,r. The following is a straightforward consequence of Proposition 1.2 which will be extensively used. Corollary 1.4. There are dimensional constants C, C ′ and C ′′ with the following property. For every δ, ε0 > 0, there is ε > 0 such that, under the assumption (H): (a) if (p, ρ, π) and (p′, ρ′, π′) are admissible (according to Definition 1.3), then |~π − ~π′| ≤ C ε0 ( max{ρ, ρ′, |q − q′|} )1−δ ; (b) there exists a unique tangent plane πp to T at every p ∈ supp (T )∩B1/2; moreover, if (p, ρ, π) is admissible then |π − πp| ≤ C ′ ε0 ρ1−δ and, vice versa, if |π − πp| ≤ C ′′ ε0 ρ 1−δ, then (p, ρ, π) is admissible; (c) for every q ∈ B 1/4, there exists a unique u ∈ R such that (q, u) ∈ supp (T ) ∩B1/2. 4 C. DE LELLIS AND E. N. SPADARO Remark 1.5. An important point in the previous corollary is that the constant C ′′ can be chosen arbitrarily large, provided the constant Cm,n in Definition 1.3 is chosen accordingly. This fact is an easy consequence of the proof given in the appendix. Theorem 0.1 is clearly contained in the previous corollary (with the additional feature that the Hölder exponent β is equal to 1− δ, i.e. is arbitrarily close to 1). In order to make the paper self-contained, we will include also a proof of Proposition 1.2 and Corollary 1.4 in the appendix. 1.4. Two technical lemmas. We conclude this section with the following two lemmas which will be needed in the sequel. Consider two functions f : D ⊂ π0 → π⊥ 0 and f ′ : D′ ⊂ π → π⊥, with associated systems of coordinates x and x′, respectively, and x′(p) = A · x(p) for every p ∈ R. If for every q′ ∈ D′ there exists a unique q ∈ D such that (q′, f ′(q′)) = A · (q, f(q)) and vice versa, then it follows that graphπ0(f) = graphπ(f ′), where graphπ0(f) := { (q, f(q)) ∈ D × π⊥ 0 } and graphπ(f ′) := { (q′, f ′(q′)) ∈ D′ × π⊥ } . The following lemma compares norms of functions (and of differences of functions) having the same graphs in two nearby system of coordinates. Lemma 1.6. There are constants c0, C > 0 with the following properties. Assume that (i) ‖A− Id‖ ≤ c0, r ≤ 1; (ii) (q, u) ∈ π0 × π⊥ 0 is given and f, g : B 2r(q) → R are Lipschitz functions such that Lip(f),Lip(g) ≤ c0 and |f(q) − u| + |g(q) − u| ≤ c0 r. Then, in the system of coordinates x′ = A · x, for (q′, u′) = A · (q, u), the following holds: (a) graphπ0(f) and graphπ0(g) are the graphs of two Lipschitz functions f ′ and g′, whose domains of definition contain both Br(q ′); (b) ‖f ′ − g‖L1(Br(q′)) ≤ C ‖f − g‖L1(B2r(q)); (c) if f ∈ C(B2r(q)), then f ′ ∈ C(Br(q)), with the estimates ‖f ′ − u‖C3 ≤ Φ (‖A− Id ‖, ‖f − u‖C3) , (1.5) ‖Df ‖C0 ≤ Ψ (‖A− Id ‖, ‖f − u‖C3) ( 1 + ‖Df‖C0 ) , (1.6) where Φ and Ψ are smooth functions. Proof. Let P : Rm×n → R and Q : Rm×n → R be the usual orthogonal projections. Set π = A(π0) and consider the maps F,G : B2r(q) → π⊥ and I, J : B2r(q) → π given by F (x) = Q(A((x, f(x))) and G(x) = Q(A((x, g(x))), I(x) = P (A((x, f(x))) and J(x) = P (A((x, g(x))). Obviously, if c0 is sufficiently small, I and J are injective Lipschitz maps. Hence, graphπ0(f) and graphπ0(g) coincide, in the new coordinates, with the graphs of the functions f ′ and g′ defined respectively in D := I(B2r(q)) and D̃ := J(B2r(q)) by f ′ = F ◦I−1 and g′ = G◦I−1. If c0 is chosen sufficiently small, then we can find a constant C such that Lip(I), Lip(J), Lip(I−1), Lip(J−1) ≤ 1 + C c0, (1.7) CENTER MANIFOLD: A CASE STUDY 5 and |I(q) − q′|, |J(q) − q′| ≤ C c0 r. (1.8) Clearly, (1.7) and (1.8) easily imply (a). Conclusion (c) is a simple consequence of the inverse function theorem. Finally we claim that, for small c0, |f ′(x′) − g′(x′)| ≤ 2 |f(I−1(x′)) − g(I−1(x′))| ∀ x′ ∈ Br(q), (1.9) from which, using the change of variables formula for biLipschitz homeomorphisms and (1.7), (b) follows. In order to prove (1.9), consider any x′ ∈ Br(q), set x := I−1(x′) and p1 := (x, f(x)) ∈ π0 × π⊥ 0 , p2 := (x, g(x)) ∈ π0 × π⊥ 0 and p3 := (x′, g′(x′)) ∈ π × π⊥. Obviously |f ′(x′)− g′(x′)| = |p1 − p3| and |f(x)− g(x)| = |p1 − p2|. Note that, g(x) = f(x) if and only if g′(x′) = f ′(x′), and in this case (1.9) follows trivially. If this is not the case, the triangle with vertices p1, p2 and p3 is non-degenerate. Let θi be the angle at pi. Note that, Lip(g) ≤ c0 implies |90◦ − θ2| ≤ Cc0 and ‖A− Id‖ ≤ c0 implies |θ1| ≤ Cc0, for some dimensional constant C. Since θ3 = 180 ◦ − θ1 − θ2, we conclude as well |90◦ − θ3| ≤ Cc0. Therefore, if c0 is small enough, we have 1 ≤ 2 sin θ3, so that, by the Sinus Theorem, |f ′(x′) − g′(x′)| = |p1 − p3| = sin θ2 sin θ3 |p1 − p2| ≤ 2 |p1 − p2| = 2 |f(x) − g(x)|, thus concluding the claim. ¤ The following is an elementary lemma on polynomials. Lemma 1.7. For every n,m ∈ N, there exists a constant C(m,n) such that, for every polynomial R of degree at most n in R and every positive r > 0, |DR(q)| ≤ C rm+k ˆ Br(q) |R| for all k ≤ n and all q ∈ R. (1.10) Proof. We rescale and translate the variables by setting S(x) = R(rx + q). The lemma is then reduced to show that n ∑