Preliminaries for the construction of the center manifold.
Notation 17.1
(Distance and nonoriented distance between m-planes). Throughout this part, \(\pi _0\) continues to denote the plane \({\mathbb {R}}^m \times \{0\}\), with the standard orientation given by \(\mathbf {\pi }_0 = e_1 \wedge \ldots \wedge e_{m}\). Given a k-dimensional plane \(\pi \) in \({\mathbb {R}}^{m+n}\), we will in fact always identify \(\pi \) with a simple unit k-vector \(\mathbf {\pi }= v_1 \wedge \ldots \wedge v_k\) orienting it (thereby making a distinction when the same plane is given opposite orientations). By a slight abuse of notation, given two k-planes \(\pi _1\) and \(\pi _2\), we will sometimes write \(|\pi _1 - \pi _2|\) in place of \(|\mathbf {\pi }_1 - \mathbf {\pi }_2|\), where the norm is induced by the standard inner product in \(\Lambda _k({\mathbb {R}}^{m+n})\). Furthermore, for a given integer rectifiable current T, we recall the definition of \(|\mathbf {T}(y) - \pi _0|_{no}\) from (13.1). More in general, if \(\pi _1\) and \(\pi _2\) are two k-planes, we can define \(|\pi _1 - \pi _2|_{no}\) by
$$\begin{aligned} |\pi _1 - \pi _2|_{no} := \min \left\{ |\mathbf {\pi }_1 - \mathbf {\pi }_2|, \, |\mathbf {\pi }_1 + \mathbf {\pi }_2| \right\} \,. \end{aligned}$$
It is understood that \(|\pi _1 - \pi _2|_{no}\) does not depend on the choice of the orientations \(\mathbf {\pi }_1\) and \(\mathbf {\pi }_2\).
Definition 17.2
(Excess and height). Given an integer rectifiable m-dimensional current T which is a representative \({\mathrm{mod}}(p)\) in \({\mathbb {R}}^{m+n}\) with finite mass and compact support and an m-plane \(\pi \), we define the nonoriented excess of T in the ball \({\mathbf {B}}_r (x)\) with respect to the plane \(\pi \) as
$$\begin{aligned} {\mathbf {E}}^{no} (T,{\mathbf {B}}_r (x),\pi )&:= \left( 2\omega _m\,r^m\right) ^{-1}\int _{{\mathbf {B}}_r (x)} |\mathbf {T} - \pi |_{no}^2 \, d\Vert T\Vert \,. \end{aligned}$$
(17.1)
The height function in a set \(A \subset {\mathbb {R}}^{m+n}\) with respect to \(\pi \) is
$$\begin{aligned} {\mathbf {h}}(T,A,\pi ) := \sup _{x,y\,\in \,\mathrm {spt}(T)\,\cap \, A} |{\mathbf {p}}_{\pi ^\perp }(x)-{\mathbf {p}}_{\pi ^\perp }(y)|\, . \end{aligned}$$
Definition 17.3
(Optimal planes). We say that an m-dimensional plane \(\pi \) optimizes the nonoriented excess of T in a ball \({\mathbf {B}}_r (x)\) if
$$\begin{aligned} {\mathbf {E}}^{no} (T,{\mathbf {B}}_r (x)):=\min _\tau {\mathbf {E}}^{no} (T, {\mathbf {B}}_r (x), \tau ) = {\mathbf {E}}^{no} (T,{\mathbf {B}}_r (x),\pi ) \end{aligned}$$
(17.2)
and if, in addition:
$$\begin{aligned} \text{ among } \text{ all } \text{ other } \pi '\text { s.t. (}17.2\text {) holds, } |\pi -\pi _0| \text { is minimal.} \end{aligned}$$
(17.3)
Observe that in general the plane optimizing the nonoriented excess is not necessarily unique and \({\mathbf {h}}(T, {\mathbf {B}}_r (x), \pi )\) might depend on the optimizer \(\pi \). Since for notational purposes it is convenient to define a unique “height” function \({\mathbf {h}}(T, {\mathbf {B}}_r (x))\), we call a plane \(\pi \) as in (17.2) and (17.3) optimal if in addition
$$\begin{aligned} {\mathbf {h}}(T,{\mathbf {B}}_r(x)) := \min \big \{{\mathbf {h}}(T,{\mathbf {B}}_r (x),\tau ): \tau \text{ satisfies } \text{( }17.2\text{) } \text{ and } \text{( }17.3\text{) }\big \} = {\mathbf {h}}(T,{\mathbf {B}}_r(x),\pi )\,, \end{aligned}$$
(17.4)
i.e. \(\pi \) optimizes the height among all planes that optimize the nonoriented excess. However (17.4) does not play any further role apart from simplifying the presentation.
Remark 17.4
Observe that there are two differences with [DLS16a, Definition 1.2]: first of all here we consider the nonoriented excess; secondly we have the additional requirement (17.3). In fact the point of (17.3) is to ensure that the planes \(\pi \) “optimizing the nonoriented excess” always satisfy \(|\pi - \pi _0| = |\pi - \pi _0|_{no}\).
We are now ready to formulate the main assumptions of the statements in this section.
Assumption 17.5
\(\varepsilon _0\in ]0,1]\) is a fixed constant and \(\Sigma \subset {\mathbf {B}}_{7\sqrt{m}} \subset {\mathbb {R}}^{m+n}\) is a \(C^{3,\varepsilon _0}\) \((m+\bar{n})\)-dimensional submanifold with no boundary in \({\mathbf {B}}_{7\sqrt{m}}\). We moreover assume that, for each \(q\in \Sigma \), \(\Sigma \) is the graph of a \(C^{3, \varepsilon _0}\) map \(\Psi _q: T_q\Sigma \cap {\mathbf {B}}_{7\sqrt{m}} \rightarrow T_q\Sigma ^\perp \). We denote by \({\mathbf {c}} (\Sigma )\) the number \(\sup _{q\in \Sigma } \Vert D\Psi _q\Vert _{C^{2, \varepsilon _0}}\). \(T^0\) is an m-dimensional integer rectifiable current of \({\mathbb {R}}^{m+n}\) which is a representative \({\mathrm{mod}}(p)\) and with support in \(\Sigma \cap {{\bar{{\mathbf {B}}}}}_{6\sqrt{m}}\). \(T^0\) is area-minimizing \({\mathrm{mod}}(p)\) in \(\Sigma \) and moreover
$$\begin{aligned} \Vert T^0\Vert ({\mathbf {B}}_{6\sqrt{m} \rho })&\le \big (\omega _m Q (6\sqrt{m})^m + \varepsilon _2^2\big )\,\rho ^m \quad \forall \rho \le 1, \end{aligned}$$
(17.6)
$$\begin{aligned} {\mathbf {E}}^{no}\left( T^0,{\mathbf {B}}_{6\sqrt{m}}\right)&={\mathbf {E}}^{no}\left( T^0,{\mathbf {B}}_{6\sqrt{m}},\pi _0\right) , \end{aligned}$$
(17.7)
$$\begin{aligned} {\varvec{m}}_0&:= \max \left\{ {\mathbf {c}} (\Sigma )^2, {\mathbf {E}}^{no}\left( T^0,{\mathbf {B}}_{6\sqrt{m}}\right) \right\} \le \varepsilon _2^2 \le 1\, . \end{aligned}$$
(17.8)
Here, Q is a positive integer with \(2\le Q \le \lfloor \frac{p}{2}\rfloor \), and \(\varepsilon _2\) is a positive number whose choice will be specified in each subsequent statement.
Constants depending only upon \(m,n,\bar{n}\) and Q will be called geometric and usually denoted by \(C_0\).
Remark 17.6
Note that (17.8) implies
, where \(A_\Sigma \) denotes, as usual, the second fundamental form of \(\Sigma \) and \(C_0\) is a geometric constant. Observe further that for \(q\in \Sigma \) the oscillation of \(\Psi _q\) is controlled in \(T_q \Sigma \cap {\mathbf {B}}_{6\sqrt{m}}\) by
.
In what follows we set \(l:= n - \bar{n}\). To avoid discussing domains of definitions it is convenient to extend \(\Sigma \) so that it is an entire graph over all \(T_q \Sigma \). Moreover we will often need to parametrize \(\Sigma \) as the graph of a map \(\Psi : {\mathbb {R}}^{m+{\bar{n}}}\rightarrow {\mathbb {R}}^l\). However we do not assume that \({\mathbb {R}}^{m+{\bar{n}}}\times \{0\}\) is tangent to \(\Sigma \) at any q and thus we need the following lemma.
Lemma 17.7
There are positive constants \(C_0 (m,\bar{n}, n)\) and \(c_0 (m, \bar{n}, n)\) such that, provided \(\varepsilon _2 < c_0\), the following holds. If \(\Sigma \) is as in Assumption 17.5, then we can (modify it outside \({\mathbf {B}}_{6\sqrt{m}}\) and) extend it to a complete submanifold of \({\mathbb {R}}^{m+n}\) which, for every \(q\in \Sigma \), is the graph of a global \(C^{3,\varepsilon _0}\) map \(\Psi _q : T_q \Sigma \rightarrow T_q \Sigma ^\perp \) with
. \(T^0\) is still area-minimizing \({\mathrm{mod}}(p)\) in the extended manifold and in addition we can apply a global affine isometry which leaves \(\mathbb {\mathbb {R}}^m \times \{0\}\) fixed and maps \(\Sigma \) onto \(\Sigma '\) so that
and \(\Sigma '\) is the graph of a \(C^{3, \varepsilon _0}\) map \(\Psi : {\mathbb {R}}^{m+\bar{n}} \rightarrow {\mathbb {R}}^l\) with \(\Psi (0)=0\) and
.
From now on we assume w.l.o.g. that \(\Sigma ' = \Sigma \). The next lemma is a standard consequence of the theory of area-minimizing currents (we include the proofs of Lemma 17.7 and Lemma 17.8 in Section 18 for the reader’s convenience).
Lemma 17.8
There are positive constants \(C_0 (m,n, \bar{n},Q)\) and \(c_0 (m,n,\bar{n}, Q)\) with the following property. If \(T^0\) is as in Assumption 17.5, \(\varepsilon _2 < c_0\) and
, then:
In particular, for each \(x\in B_{11\sqrt{m}/2} (0, \pi _0)\) there is a point \(q\in \mathrm {spt}(T)\) with \({\mathbf {p}}_{\pi _0} (q)=x\).
Construction of the center manifold.
From now we will always work with the current T of Lemma 17.8. We specify next some notation which will be recurrent in the paper when dealing with cubes of \(\pi _0\). For each \(j\in {\mathbb {N}}\), \({{\mathscr {C}}}^j\) denotes the family of closed cubes L of \(\pi _0\) of the form
$$\begin{aligned}{}[a_1, a_1+2\ell ] \times \ldots \times [a_m, a_m+ 2\ell ] \times \{0\}\subset \pi _0\, , \end{aligned}$$
(17.13)
where \(2\,\ell = 2^{1-j} =: 2\,\ell (L)\) is the side-length of the cube, \(a_i\in 2^{1-j}{\mathbb {Z}}\) \(\forall i\) and we require in addition \(-4 \le a_i \le a_i+2\ell \le 4\). To avoid cumbersome notation, we will usually drop the factor \(\{0\}\) in (17.13) and treat each cube, its subsets and its points as subsets and elements of \({\mathbb {R}}^m\). Thus, for the center \(x_L\) of L we will use the notation \(x_L=(a_1+\ell , \ldots , a_m+\ell )\), although the precise one is \((a_1+\ell , \ldots , a_m+\ell , 0, \ldots , 0)\). Next we set \({{\mathscr {C}}}:= \bigcup _{j\in {\mathbb {N}}} {{\mathscr {C}}}^j\). If H and L are two cubes in \({{\mathscr {C}}}\) with \(H\subset L\), then we call L an ancestor of H and H a descendant of L. When in addition \(\ell (L) = 2\ell (H)\), H is a son of L and L the father of H.
Definition 17.9
A Whitney decomposition of \([-4,4]^m\subset \pi _0\) consists of a closed set \({\mathbf {\Gamma }}\subset [-4,4]^m\) and a family \({\mathscr {W}}\subset {{\mathscr {C}}}\) satisfying the following properties:
-
(w1)
\({\mathbf {\Gamma }}\cup \bigcup _{L\in {\mathscr {W}}} L = [-4,4]^m\) and \({\mathbf {\Gamma }}\) does not intersect any element of \({\mathscr {W}}\);
-
(w2)
the interiors of any pair of distinct cubes \(L_1, L_2\in {\mathscr {W}}\) are disjoint;
-
(w3)
if \(L_1, L_2\in {\mathscr {W}}\) have nonempty intersection, then \(\frac{1}{2}\ell (L_1) \le \ell (L_2) \le 2\, \ell (L_1)\).
Observe that (w1)–(w3) imply
$$\begin{aligned} \mathrm{sep}\, ({\mathbf {\Gamma }}, L) := \inf \{ |x-y|: x\in L, y\in {\mathbf {\Gamma }}\} \ge 2\ell (L) \quad \text{ for } \text{ every } L\in {\mathscr {W}}. \end{aligned}$$
(17.14)
However, we do not require any inequality of the form \(\mathrm{sep}\, ({\mathbf {\Gamma }}, L) \le C \ell (L)\), although this would be customary for what is commonly called a Whitney decomposition in the literature.
The algorithm for the construction of the center manifold involves several parameters which depend in a complicated way upon several quantities and estimates. We introduce these parameters and specify some relations among them in the following
Assumption 17.10
\(C_e,C_h,\beta _2,\delta _2, M_0\) are positive real numbers and \(N_0\) is a natural number for which we assume always
$$\begin{aligned}&\beta _2 = 4\,\delta _2 = \min \left\{ \frac{1}{2m}, \frac{\gamma _1}{100}\right\} , \nonumber \\&\quad \text{ where } \gamma _1 \text{ is } \text{ the } \text{ exponent } \text{ in } \text{ the } \text{ estimates } \text{ of } \text{ Theorem }~15.1, \end{aligned}$$
(17.15)
$$\begin{aligned}&\quad M_0 \ge C_0 (m,n,\bar{n},Q) \ge 4\, \quad \text{ and }\quad \sqrt{m} M_0 2^{7-N_0} \le 1\, . \end{aligned}$$
(17.16)
As we can see, \(\beta _2\) and \(\delta _2\) are fixed. The other parameters are not fixed but are subject to further restrictions in the various statements, respecting the following “hierarchy”. As already mentioned, “geometric constants” are assumed to depend only upon \(m, n, \bar{n}\) and Q. The dependence of other constants upon the various parameters \(p_i\) will be highlighted using the notation \(C = C (p_1, p_2, \ldots )\).
Assumption 17.11
(Hierarchy of the parameters) In all the coming statements:
-
(a)
\(M_0\) is larger than a geometric constant (cf. (17.16)) or larger than a costant \(C (\delta _2)\), see Proposition 17.29;
-
(b)
\(N_0\) is larger than \(C (\beta _2, \delta _2, M_0)\) (see for instance (17.16) and Proposition 17.32);
-
(c)
\(C_e\) is larger than \(C(\beta _2, \delta _2, M_0, N_0)\) (see the statements of Proposition 17.13, Theorem 17.19 and Proposition 17.29);
-
(d)
\(C_h\) is larger than \(C(\beta _2, \delta _2, M_0, N_0, C_e)\) (see Propositions 17.13 and 17.26 );
-
(e)
\(\varepsilon _2\) is smaller than \(c(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\) (which will always be positive).
The functions C and c will vary in the various statements: the hierarchy above guarantees however that there is a choice of the parameters for which all the restrictions required in the statements of the next propositions are simultaneously satisfied. To simplify our exposition, for smallness conditions on \(\varepsilon _2\) as in (e) we will use the sentence “\(\varepsilon _2\) is sufficiently small”.
Thanks to Lemma 17.8, for every \(L\in {{\mathscr {C}}}\), we may choose \(y_L\in \pi _0^\perp \) so that \(p_L := (x_L, y_L)\in \mathrm {spt}(T)\) (recall that \(x_L\) is the center of L). \(y_L\) is in general not unique and we fix an arbitrary choice. A more correct notation for \(p_L\) would be \(x_L + y_L\). This would however become rather cumbersome later, when we deal with various decompositions of the ambient space in triples of orthogonal planes. We thus abuse the notation slightly in using (x, y) instead of \(x+y\) and, consistently, \(\pi _0\times \pi _0^\perp \) instead of \(\pi _0 \oplus \pi _0^\perp \).
Definition 17.12
(Refining procedure). For \(L\in {{\mathscr {C}}}\) we set \(r_L:= M_0 \sqrt{m} \,\ell (L)\) and \({\mathbf {B}}_L := {\mathbf {B}}_{64 r_L} (p_L)\). We next define the families of cubes \({{\mathscr {S}}}\subset {{\mathscr {C}}}\) and \({{\mathscr {W}}}= {{\mathscr {W}}}_e \cup {{\mathscr {W}}}_h \cup {{\mathscr {W}}}_n \subset {{\mathscr {C}}}\) with the convention that \({{\mathscr {S}}}^j = {{\mathscr {S}}}\cap {{\mathscr {C}}}^j, {{\mathscr {W}}}^j = {{\mathscr {W}}}\cap {{\mathscr {C}}}^j\) and \({{\mathscr {W}}}^j_{\square } = {{\mathscr {W}}}_\square \cap {{\mathscr {C}}}^j\) for \(\square = h,n, e\). We define \({{\mathscr {W}}}^i = {{\mathscr {S}}}^i = \emptyset \) for \(i < N_0\). We proceed with \(j\ge N_0\) inductively: if no ancestor of \(L\in {{\mathscr {C}}}^j\) is in \({{\mathscr {W}}}\), then
-
(EX)
\(L\in {{\mathscr {W}}}^j_e\) if \({\mathbf {E}}^{no} (T, {\mathbf {B}}_L) > C_e {\varvec{m}}_0\, \ell (L)^{2-2\delta _2}\);
-
(HT)
\(L\in {{\mathscr {W}}}_h^j\) if \(L\not \in {\mathscr {W}}_e^j\) and
;
-
(NN)
\(L\in {{\mathscr {W}}}_n^j\) if \(L\not \in {{\mathscr {W}}}_e^j\cup {{\mathscr {W}}}_h^j\) but it intersects an element of \({{\mathscr {W}}}^{j-1}\);
if none of the above occurs, then \(L\in {{\mathscr {S}}}^j\). We finally set
$$\begin{aligned} {\mathbf {\Gamma }}:= [-4,4]^m \setminus \bigcup _{L\in {{\mathscr {W}}}} L = \bigcap _{j\ge N_0} \bigcup _{L\in {{\mathscr {S}}}^j} L. \end{aligned}$$
(17.17)
Observe that, if \(j>N_0\) and \(L\in {{\mathscr {S}}}^j\cup {{\mathscr {W}}}^j\), then necessarily its father belongs to \({{\mathscr {S}}}^{j-1}\).
Proposition 17.13
(Whitney decomposition). Let Assumptions 17.5 and 17.10 hold and let \(\varepsilon _2\) be sufficiently small. Then \(({\mathbf {\Gamma }}, {\mathscr {W}})\) is a Whitney decomposition of \([-4,4]^m \subset \pi _0\). Moreover, for any choice of \(M_0\) and \(N_0\), there is \(C^\star := C^\star (M_0, N_0)\) such that, if \(C_e \ge C^\star \) and \(C_h \ge C^\star C_e\), then
$$\begin{aligned} {{\mathscr {W}}}^{j} = \emptyset \qquad \text{ for } \text{ all } j\le N_0+6. \end{aligned}$$
(17.18)
Finally, the following estimates hold with \(C = C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\):
We will prove Proposition 17.13 in Section 19. Next, we fix two important functions \(\vartheta ,\varrho : {\mathbb {R}}^m \rightarrow {\mathbb {R}}\).
Assumption 17.14
\(\varrho \in C^\infty _c (B_1)\) is radial, \(\int \varrho =1\) and \(\int |x|^2 \varrho (x)\, dx = 0\). For \(\lambda >0\) \(\varrho _\lambda \) denotes, as usual, \(x\mapsto \lambda ^{-m} \varrho (\frac{x}{\lambda })\). \(\vartheta \in C^\infty _c \big ([-\frac{17}{16}, \frac{17}{16}]^m, [0,1]\big )\) is identically 1 on \([-1,1]^m\).
\(\varrho \) will be used as convolution kernel for smoothing maps z defined on m-dimensional planes \(\pi \) of \({\mathbb {R}}^{m+n}\). In particular, having fixed an isometry A of \({\mathbb {R}}^m\) onto \(\pi \), the smoothing will be given by \([(z \circ A) * \varrho _\lambda ] \circ A^{-1}\). Observe that since \(\varrho \) is radial, our map does not depend on the choice of the isometry and we will therefore use the shorthand notation \(z*\varrho _\lambda \).
Definition 17.15
(\(\pi \)-approximations). Let \(L\in {{\mathscr {S}}}\cup {{\mathscr {W}}}\) and \(\pi \) be an m-dimensional plane. If
fulfills the assumptions of Theorem 16.1 in the cylinder \({\mathbf {C}}_{32 r_L} (p_L, \pi )\), then the resulting map u given by the theorem, which is defined on \(B_{8r_L} (p_L, \pi )\) and takes values either in \({{\mathcal {A}}}_Q(\pi ^\perp )\) (if \(Q < \frac{p}{2}\)) or in \({\mathscr {A}}_Q (\pi ^\perp )\) (if \(Q= \frac{p}{2}\)) is called a \(\pi \)-approximation of T in \({\mathbf {C}}_{8 r_L} (p_L, \pi )\). The map \({\hat{h}}:B_{7r_L} (p_L, \pi ) \rightarrow \pi ^\perp \) given by \({\hat{h}}:= (\varvec{\eta }\circ u)* \varrho _{\ell (L)}\) will be called the smoothed average of the \(\pi \)-approximation.
Definition 17.16
(Reference plane \(\pi _L\)). For each \(L\in {{\mathscr {S}}}\cup {{\mathscr {W}}}\) we let \({{\hat{\pi }}}_L\) be an optimal plane in \({\mathbf {B}}_L\) (cf. Definition 17.3) and choose an m-plane \(\pi _L\subset T_{p_L} \Sigma \) which minimizes \(|{{\hat{\pi }}}_L-\pi _L|\).
The following lemma, which will be proved in Section 19, deals with graphs of multivalued functions f in several systems of coordinates.
Lemma 17.17
Let the assumptions of Proposition 17.13 hold and assume \(C_e \ge C^\star \) and \(C_h \ge C^\star C_e\) (where \(C^\star \) is the constant of Proposition 17.13). For any choice of the other parameters, if \(\varepsilon _2\) is sufficiently small, then
satisfies the assumptions of Theorem 16.1 for any \(L\in {{\mathscr {W}}}\cup {{\mathscr {S}}}\). Moreover, if \(f_L\) is a \(\pi _L\)-approximation, denote by \({\hat{h}}_L\) its smoothed average and by \(\bar{h}_L\) the map \({\mathbf {p}}_{T_{p_L}\Sigma } ({\hat{h}}_L)\), which takes values in the plane \(\varkappa _L := T_{p_L} \Sigma \cap \pi _L^\perp \), i.e. the orthogonal complement of \(\pi _L\) in \(T_{p_L} \Sigma \). If we let \(h_L\) be the map \(x \in B_{7 r_L}(p_L, \pi _L)\mapsto h_L (x):= (\bar{h}_L (x), \Psi _{p_L} (x, \bar{h}_L (x)))\in \varkappa _L \times T_{p_L} \Sigma ^\perp \), then there is a smooth map \(g_L: B_{4r_L} (p_L, \pi _0)\rightarrow \pi _0^\perp \) such that
.
For the sake of simplicity, in the future we will sometimes regard \(g_L\) as a map \(g_L :B_{4r_L}(x_L,\pi _0) \rightarrow \pi _0^\perp \) rather than as a map \(g_L :B_{4r_L}(p_L,\pi _0) \rightarrow \pi _0^\perp \). In particular, we will sometimes consider \(g_L(x)\) with \(x \in B_{4r_L}(x_L,\pi _0)\) even though the correct writing is the more cumbersome \(g_L((x,y_L))\).
Definition 17.18
(Interpolating functions). The maps \(h_L\) and \(g_L\) in Lemma 17.17 will be called, respectively, the tilted L-interpolating function and the L-interpolating function. For each j let \({{\mathscr {P}}}^j := {{\mathscr {S}}}^j \cup \bigcup _{i=N_0}^j {{\mathscr {W}}}^i\) and for \(L\in {{\mathscr {P}}}^j\) define \(\vartheta _L (y):= \vartheta (\frac{y-x_L}{\ell (L)})\). Set
$$\begin{aligned} {{\hat{\varphi }}}_j := \frac{\sum _{L\in {{\mathscr {P}}}^j} \vartheta _L\, g_L}{\sum _{L\in {{\mathscr {P}}}^j} \vartheta _L} \qquad \text{ on } ]-4,4[^m, \end{aligned}$$
(17.21)
let \({\bar{\varphi }}_j (y)\) be the first \(\bar{n}\) components of \({\hat{\varphi }}_j (y)\) and define \(\varphi _j (y) := \big ({\bar{\varphi }}_j (y), \Psi (y, {\bar{\varphi }}_j (y))\big )\), where \(\Psi \) is the map of Lemma 17.7. \(\varphi _j\) will be called the glued interpolation at the step j.
Theorem 17.19
(Existence of the center manifold). Assume that the hypotheses of the Lemma 17.17 hold and let \(\kappa := \min \{\varepsilon _0/2, \beta _2/4\}\). For any choice of the other parameters, if \(\varepsilon _2\) is sufficiently small, then
-
(i)
and
, with \(C = C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\);
-
(ii)
if \(L\in {{\mathscr {W}}}^i\) and H is a cube concentric to L with \(\ell (H)=\frac{9}{8} \ell (L)\), then \(\varphi _j = \varphi _k\) on H for any \(j,k\ge i+2\);
-
(iii)
\(\varphi _j\) converges in \(C^3\) to a map \({\mathbf {\varphi }}\) and \({{\mathcal {M}}}:= \mathrm {Gr}({\mathbf {\varphi }}|_{]-4,4[^m} )\) is a \(C^{3,\kappa }\) submanifold of \(\Sigma \).
Definition 17.20
(Whitney regions). The manifold \({{\mathcal {M}}}\) in Theorem 17.19 is called a center manifold of T relative to \(\pi _0\), and \(({\mathbf {\Gamma }}, {{\mathscr {W}}})\) the Whitney decomposition associated to \({{\mathcal {M}}}\). Setting \({\mathbf {\Phi }}(y) := (y,{\mathbf {\varphi }}(y))\), we call \({\mathbf {\Phi }}({\mathbf {\Gamma }})\) the contact set. Moreover, to each \(L\in {{\mathscr {W}}}\) we associate a Whitney region \({{\mathcal {L}}}\) on \({{\mathcal {M}}}\) as follows:
-
(WR)
\({{\mathcal {L}}}:= {\mathbf {\Phi }}(H\cap [-\frac{7}{2},\frac{7}{2}]^m)\), where H is the cube concentric to L with \(\ell (H) = \frac{17}{16} \ell (L)\).
We will present a proof of Theorem 17.19 in Section 20
The \({\mathcal {M}}\)-normal approximation and related estimates.
In what follows we assume that the conclusions of Theorem 17.19 apply and denote by \({{\mathcal {M}}}\) the corresponding center manifold. For any Borel set \({{\mathcal {V}}}\subset {{\mathcal {M}}}\) we will denote by \(|{{\mathcal {V}}}|\) its \({{\mathcal {H}}}^m\)-measure and will write \(\int _{{\mathcal {V}}}f\) for the integral of f with respect to
. \({\mathcal {B}}_r (q)\) denotes the geodesic open balls in \({{\mathcal {M}}}\).
Assumption 17.21
We fix the following notation and assumptions.
-
(U)
\({{\mathbf {U}}}:=\big \{x\in {\mathbb {R}}^{m+n} : \exists !\, y = {\mathbf {p}}(x) \in {{\mathcal {M}}} \text{ with } |x- y| <1 \text{ and } (x-y)\perp {{\mathcal {M}}}\big \}\).
-
(P)
\({\mathbf {p}}: {{\mathbf {U}}}\rightarrow {{\mathcal {M}}}\) is the map defined by (U).
-
(R)
For any choice of the other parameters, we assume \(\varepsilon _2\) to be so small that \({\mathbf {p}}\) extends to \(C^{2, \kappa }({{\bar{{{\mathbf {U}}}}}})\) and \({\mathbf {p}}^{-1} (y) = y + \overline{B_1 (0, (T_y {{\mathcal {M}}})^\perp )}\) for every \(y\in {{\mathcal {M}}}\).
-
(L)
We denote by \(\partial _l {{\mathbf {U}}}:= {\mathbf {p}}^{-1} (\partial {{\mathcal {M}}})\) the lateral boundary of \({{\mathbf {U}}}\).
The following is then a corollary of Theorem 17.19 and the construction algorithm; see Section 21 for the proof.
Corollary 17.22
Under the hypotheses of Theorem 17.19 and of Assumption 17.21 we have:
-
(i)
,
, and
;
-
(ii)
for every \(q\in L\in {{\mathscr {W}}}\), where \(C= C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\);
-
(iii)
\(\langle T, {\mathbf {p}}, q\rangle = Q \llbracket {q}\rrbracket \) for every \(q\in {\mathbf {\Phi }}({\mathbf {\Gamma }})\).
The next main goal is to couple the center manifold of Theorem 17.19 with a good approximating map defined on it.
Definition 17.23
(\({{\mathcal {M}}}\)-normal approximation). An \({{\mathcal {M}}}\)-normal approximation of T is given by a pair \(({{\mathcal {K}}}, F)\) with the following properties. \({{\mathcal {K}}}\subset {{\mathcal {M}}}\) is closed and contains \({\mathbf {\Phi }}\big ({\mathbf {\Gamma }}\cap [-\frac{7}{2}, \frac{7}{2}]^m\big )\). Moreover:
-
(a)
If \(Q = \frac{p}{2}\), F is a Lipschitz map which takes values in \({\mathscr {A}}_Q({\mathbb {R}}^{m+n})\) and satisfies the requirements of [DLHMS, Assumption 11.1].
-
(b)
If \(Q < \frac{p}{2}\), F is a Lipschitz map which takes values in \({{\mathcal {A}}}_Q({\mathbb {R}}^{m+n})\) and has the special form \(F (x) = \sum _i \llbracket {x+N_i (x)}\rrbracket \).
In both cases we require that
-
(A1)
\(\mathrm {spt}({{\mathbf {T}}}_F) \subset \Sigma \);
-
(A2)
,
where \({{\mathbf {T}}}_F\) is the integer rectifiable current induced by F; see [DLHMS, Definition 11.2]. The map N (for the case \(Q = \frac{p}{2}\) see [DLHMS, Assumption 11.1]) is the normal part of F.
In the definition above it is not required that the map F approximates efficiently the current outside the set \({\mathbf {\Phi }}\big ({\mathbf {\Gamma }}\cap [-\frac{7}{2}, \frac{7}{2}]^m\big )\). However, all the maps constructed will approximate T with a high degree of accuracy in each Whitney region: such estimates are detailed in the next theorem, the proof of which will be tackled in Section 21.
Theorem 17.24
(Local estimates for the \({{\mathcal {M}}}\)-normal approximation). Let \(\gamma _2 := \frac{\gamma }{4}\), with \(\gamma \) the constant of Theorem 15.1. Under the hypotheses of Theorem 17.19 and Assumption 17.21, if \(\varepsilon _2\) is suitably small (depending upon all other parameters), then there is an \({{\mathcal {M}}}\)-normal approximation \(({{\mathcal {K}}}, F)\) such that the following estimates hold on every Whitney region \({{\mathcal {L}}}\) associated to a cube \(L\in {{\mathscr {W}}}\), with constants \(C = C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\):
$$\begin{aligned} |{{\mathcal {L}}}\setminus {{\mathcal {K}}}| + \Vert {{\mathbf {T}}}_F - T\Vert _p ({\mathbf {p}}^{-1} ({{\mathcal {L}}}))\le & {} C {\varvec{m}}_0^{1+\gamma _2} \ell (L)^{m+2+\gamma _2}, \end{aligned}$$
(17.23)
$$\begin{aligned} \int _{{{\mathcal {L}}}} |DN|^2\le & {} C {\varvec{m}}_0\,\ell (L)^{m+2-2\delta _2}\, . \end{aligned}$$
(17.24)
Moreover, for any \(a>0\) and any Borel \({{\mathcal {V}}}\subset {{\mathcal {L}}}\), we have (for \(C=C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\))
where \(\square =s\) in case \(p=2Q\), and it is empty otherwise.
From (17.22) to (17.24) it is not difficult to infer analogous “global versions” of the estimates.
Corollary 17.25
(Global estimates). Let \({{\mathcal {M}}}'\) be the domain
and N the map of Theorem 17.24. Then, (again with \(C = C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\))
$$\begin{aligned} |{{\mathcal {M}}}'\setminus {{\mathcal {K}}}| + \Vert {{\mathbf {T}}}_F - T\Vert _p ({\mathbf {p}}^{-1} ({{\mathcal {M}}}'))\le & {} C {\varvec{m}}_0^{1+\gamma _2}, \end{aligned}$$
(17.27)
$$\begin{aligned} \int _{{{\mathcal {M}}}'} |DN|^2\le & {} C {\varvec{m}}_0\, . \end{aligned}$$
(17.28)
Separation and domains of influence of large excess cubes.
We now analyze more in detail the consequences of the various stopping conditions for the cubes in \({{\mathscr {W}}}\). We first deal with \(L\in {{\mathscr {W}}}_h\).
Proposition 17.26
(Separation). There is a constant \(C^\sharp (M_0) > 0\) with the following property. Assume the hypotheses of Theorem 17.24 and in addition \(C_h^{2m} \ge C^\sharp C_e\). If \(\varepsilon _2\) is sufficiently small, then the following conclusions hold for every \(L\in {{\mathscr {W}}}_h\):
-
(S1)
\(\Theta (T, q) \le Q - \frac{1}{2}\) for every \(q\in {\mathbf {B}}_{16 r_L} (p_L)\);
-
(S2)
\(L\cap H= \emptyset \) for every \(H\in {{\mathscr {W}}}_n\) with \(\ell (H) \le \frac{1}{2} \ell (L)\);
-
(S3)
for every \(x\in {\mathbf {\Phi }}(B_{2 \sqrt{m} \ell (L)} (x_L, \pi _0))\), where \(\square =s\) if \(p=2Q\) or \(\square = \;\) otherwise.
A simple corollary of the previous proposition is the following.
Corollary 17.27
Given any \(H\in {{\mathscr {W}}}_n\) there is a chain \(L =L_0, L_1, \ldots , L_j = H\) such that:
-
(a)
\(L_0\in {{\mathscr {W}}}_e\) and \(L_i\in {{\mathscr {W}}}_n\) for all \(i>0\);
-
(b)
\(L_i\cap L_{i-1}\ne \emptyset \) and \(\ell (L_i) = \frac{1}{2} \ell (L_{i-1})\) for all \(i>0\).
In particular, \(H\subset B_{3\sqrt{m}\ell (L)} (x_L, \pi _0)\).
We use this last corollary to partition \({{\mathscr {W}}}_n\).
Definition 17.28
(Domains of influence). We first fix an ordering of the cubes in \({{\mathscr {W}}}_e\) as \(\{J_i\}_{i\in {\mathbb {N}}}\) so that their sidelengths do not increase. Then \(H\in {{\mathscr {W}}}_n\) belongs to \({{\mathscr {W}}}_n (J_0)\) (the domain of influence of \(J_0\)) if there is a chain as in Corollary 17.27 with \(L_0 = J_0\). Inductively, \({{\mathscr {W}}}_n (J_r)\) is the set of cubes \(H\in {{\mathscr {W}}}_n \setminus \cup _{i<r} {{\mathscr {W}}}_n (J_i)\) for which there is a chain as in Corollary 17.27 with \(L_0 = J_r\).
Splitting before tilting.
The following proposition contains a “typical” splitting-before-tilting phenomenon: the key assumption of the theorem (i.e. \(L\in {{\mathscr {W}}}_e\)) is that the excess does not decay at some given scale (“tilting”) and the main conclusion (17.30) implies a certain amount of separation between the sheets of the current (“splitting”); see Section 22 for the proof.
Proposition 17.29
(Splitting I). There are functions \(C_1 (\delta _2), C_2 (M_0, \delta _2)\) such that, if \(M_0 \ge C_1 ( \delta _2)\), \(C_e \ge C_2 (M_0, \delta _2)\), if the hypotheses of Theorem 17.24 hold and if \(\varepsilon _2\) is chosen sufficiently small, then the following holds. If \(L\in {{\mathscr {W}}}_e\), \(q\in \pi _0\) with \(\mathrm {dist}(L, q) \le 4\sqrt{m} \,\ell (L)\) and \(\Omega = {\mathbf {\Phi }}(B_{\ell (L)/4} (q, \pi _0))\), then (with \(C, C_3 = C(\beta _2, \delta _2, M_0, N_0, C_e, C_h)\)):
$$\begin{aligned}&C_e {\varvec{m}}_0\ell (L)^{m+2-2\delta _2} \le \ell (L)^m {\mathbf {E}}^{no} (T, {\mathbf {B}}_L) \le C \int _\Omega |DN|^2\, , \end{aligned}$$
(17.29)
$$\begin{aligned}&\int _{{{\mathcal {L}}}} |DN|^2 \le C \ell (L)^m {\mathbf {E}}^{no} (T, {\mathbf {B}}_L) \le C_3 \ell (L)^{-2} \int _\Omega |N|^2\, . \end{aligned}$$
(17.30)
Persistence of multiplicity Q points.
We next state two important properties triggered by the existence of \(q\in \mathrm {spt}(T)\) with \(\Theta (T,q)=Q\), both related to the splitting before tilting. Their proofs will be discussed in Section 23.
Proposition 17.30
(Splitting II). Let the hypotheses of Theorem 17.19 hold and assume \(\varepsilon _2\) is sufficiently small. For any \(\alpha , {\bar{\alpha }}, {{\hat{\alpha }}} >0\), there is \(\varepsilon _3 = \varepsilon _3 (\alpha , {\bar{\alpha }}, {{\hat{\alpha }}}, \beta _2, \delta _2, M_0, N_0, C_e, C_h) >0\) as follows.
When \(Q<\frac{p}{2}\), if for some \(s\le 1\)
$$\begin{aligned}&\sup \big \{\ell (L): L\in {{\mathscr {W}}}, L\cap B_{3s} (0, \pi _0) \ne \emptyset \big \} \le s\, , \end{aligned}$$
(17.31)
$$\begin{aligned}&{{\mathcal {H}}}^{m-2+\alpha }_\infty \big (\{\Theta (T, \cdot ) = Q\}\cap {\mathbf {B}}_{s}\big ) \ge {\bar{\alpha }} s^{m-2+\alpha }, \end{aligned}$$
(17.32)
and \(\min \big \{s, {\varvec{m}}_0\big \}\le \varepsilon _3\), then,
$$\begin{aligned} \sup \big \{ \ell (L): L\in {{\mathscr {W}}}_e \text{ and } L\cap B_{19 s/16} (0, \pi _0)\ne \emptyset \big \} \le {\hat{\alpha }} s\, . \end{aligned}$$
When \(Q=\frac{p}{2}\), the same conclusion can be reached if (17.32) is replaced by
$$\begin{aligned} {{\mathcal {H}}}^{m-1+\alpha }_\infty \big (\{\Theta (T, \cdot ) = Q\}\cap {\mathbf {B}}_{s}\big ) \ge {\bar{\alpha }} s^{m-1+\alpha }\, . \end{aligned}$$
(17.33)
Proposition 17.31
(Persistence of Q-points). Assume the hypotheses of Proposition 17.29 hold. For every \(\eta _2>0\) there are \(\bar{s}, {\bar{\ell }} > 0\), depending upon \(\eta _2, \beta _2, \delta _2, M_0, N_0, C_e\) and \(C_h\), such that, if \(\varepsilon _2\) is sufficiently small, then the following property holds. If \(L\in {{\mathscr {W}}}_e\), \(\ell (L)\le {{\bar{\ell }}}\), \(\Theta (T, q) = Q\) and \(\mathrm {dist}({\mathbf {p}}_{\pi _0} ({\mathbf {p}}(q)), L) \le 4 \sqrt{m} \,\ell (L)\), then
where \(\square =s\) if \(p=2Q\) or \(\square = \;\) otherwise.
Comparison between center manifolds.
We list here a final key consequence of the splitting before tilting phenomenon. \(\iota _{0,r}\) denotes the map \(z\mapsto \frac{z}{r}\).
Proposition 17.32
(Comparing center manifolds). There is a geometric constant \(C_0\) and a function \(\bar{c}_s (\beta _2, \delta _2, M_0, N_0, C_e, C_h) >0\) with the following property. Assume the hypotheses of Proposition 17.29, \(N_0 \ge C_0\), \(c_s := \frac{1}{64\sqrt{m}}\) and \(\varepsilon _2\) is sufficiently small. If for some \(r\in ]0,1[\):
-
(a)
\(\ell (L) \le c_s \rho \) for every \(\rho > r\) and every \(L\in {{\mathscr {W}}}\) with \(L\cap B_\rho (0, \pi _0)\ne \emptyset \);
-
(b)
\({\mathbf {E}}^{no} (T, {\mathbf {B}}_{6\sqrt{m} \rho }) < \varepsilon _2\) for every \(\rho >r\);
-
(c)
there is \(L\in {{\mathscr {W}}}\) such that \(\ell (L) \ge c_s r \) and \(L\cap {\bar{B}}_r (0, \pi _0)\ne \emptyset \);
then
-
(i)
the current
and the submanifold \(\Sigma ':= \iota _{0,r} (\Sigma )\cap {\mathbf {B}}_{7\sqrt{m}}\) satisfy the assumptions of Theorem 17.24 for some plane \(\pi \) in place of \(\pi _0\);
-
(ii)
for the center manifold \({{\mathcal {M}}}'\) of \(T'\) relative to \(\pi \) and the \({{\mathcal {M}}}'\)-normal approximation \(N'\) as in Theorem 17.24, we have
$$\begin{aligned} \int _{{{\mathcal {M}}}'\cap {\mathbf {B}}_2} |N'|^2 \ge \bar{c}_s \max \big \{{\mathbf {E}}^{no} (T', {\mathbf {B}}_{6\sqrt{m}}), {\mathbf {c}} (\Sigma ')^2\big \}\, . \end{aligned}$$
(17.35)