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Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold

Abstract

We construct a branched center manifold in a neighborhood of a singular point of a 2-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of 2-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of 3-dimensional area minimizing cones.

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Acknowledgements

The research of Camillo De Lellis and Luca Spolaor has been supported by the ERC grant RAM (Regularity for Area Minimizing currents), ERC 306247.

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Appendices

Appendix A: Density and height bound

In this appendix we record two estimates which are standard for area minimizing currents and can be extended with routine arguments to the three cases of Definition 0.1. Both statements are valid for general m without additional efforts and we therefore do not restrict to \(m=2\) here. Consistently with [8, 13] we introduce the parameter \({\mathbf {\Omega }}\), which equals

  • \(\mathbf {A}= \Vert A_\Sigma \Vert _{C^0}\) in case (a) of Definition 0.1;

  • \(\max \{\Vert d\omega \Vert _{C^0}, \Vert A_\Sigma \Vert _{C^0}\}\) in case (b);

  • \(C_0 R^{-1}\) in case (c).

Lemma A.1

There is a positive geometric constant c(mn) with the following property. If T is a current as in Definition 0.1, where \({\mathbf {\Omega }}\le c (m,n)\), then

$$\begin{aligned} \Vert T\Vert ({\mathbf {B}}_\rho (p)) \ge \omega _m (\Theta (T, p) - \textstyle {\frac{1}{4}}) \rho ^m \ge \omega _m \textstyle {\frac{3}{4}} \rho ^m \qquad \forall p\in \mathrm{spt}(T), \forall r\in \mathrm{{dist}}(p, \partial U)\, . \end{aligned}$$
(A.1)

Proof

By [13, Proposition 1.2] \(\Vert T\Vert \) is an integral varifold with bounded mean curvature in the sense of Allard, where \(C_0 {\mathbf {\Omega }}\) bounds the mean curvature for some geometric constant \(C_0\). It follows from Allard’s monotonicity formula that \(e^{C_0 {\mathbf {\Omega }}r} \Vert T\Vert ({\mathbf {B}}_r (x))\) is monotone nondecreasing in r, from which the first inequality in (A.1) follows. The second inequality is implied by \(\Theta (T, p)\ge 1\) for every \(p\in \mathrm{spt}(T)\): this holds because the density is an upper semicontinuous function which takes integer values \(\Vert T\Vert \)-almost everywhere. \(\square \)

For the proof of the next statement we refer to [9, Theorem A.1]: in that theorem T satisfies the stronger assumption of being area minimizing (thus covering only case (a) of Definition 0.1), but a close inspection of the proof given in [9] shows that the only property of area minimizing currents relevant to the arguments is the validity of the density lower bound (A.1).

Theorem A.2

Let Q, m and n be positive integers. Then there are \({\varepsilon }>0, c>0\) and C geometric constants with the following property. Assume that \(\pi _0 = {\mathbb {R}}^m\times \{0\}\subset {\mathbb {R}}^{m+n}\) and that:

  1. (h1)

    T is an integer rectifiable m-dimensional current as in Definition 0.1 with \(U = {\mathbf {C}}_r (x_0)\) and \({\mathbf {\Omega }}\le c\);

  2. (h2)

    , and \(E:= {\mathbf {E}}(T, {\mathbf {C}}_r (x_0)) < {\varepsilon }\).

Then there are \(k\in {\mathbb {N}}\), points \(\{y_1, \ldots , y_k\}\subset {\mathbb {R}}^{m+n}\) and positive integers \(Q_1, \ldots , Q_k\) such that:

  1. (i)

    having set \(\sigma := C E^{{1}/{2m}}\), the open sets \(\mathbf {S}_i := {\mathbb {R}}^m \times (y_i +\, ]-r \sigma , r \sigma [^n)\) are pairwise disjoint and \(\mathrm{spt}(T)\cap {\mathbf {C}}_{r (1- \sigma |\log E|)} (x_0) \subset \cup _i \mathbf {S}_i\);

  2. (ii)

    \(\forall i\in \{1, \ldots , k\}\).

  3. (iii)

    for every \(p\in \mathrm{spt}(T)\cap {\mathbf {C}}_{r(1-\sigma |\log E|)} (x_0)\) we have \(\Theta (T, p) < \max \{Q_i\} + \frac{1}{2}\).

Appendix B: Proof of Proposition 2.4

In order to prove the Proposition we recall the following classical fact about the existence of conformal coordinates. As in the rest of the paper, e denotes the standard euclidean metric.

Lemma B.1

For every \(k\in {\mathbb {N}}\) and \(\alpha , \beta \in ]0,1[\) there are positive constants \(C_0\) and \(c_0\) with the following properties. Let g be a \(C^{k,\beta }\) Riemannian metric on the unit disk \(B_2 \subset {\mathbb {R}}^2\) with \(\Vert g-e\Vert _{C^{0,\alpha }} \le c_0\). Then there exists an orientation preserving diffeomorphism \(\Lambda :\Omega \rightarrow B_2\) and a positive function \(\lambda : \Omega \rightarrow {\mathbb {R}}\) such that

  1. (i)

    \(\Lambda ^\sharp g = \lambda e\);

  2. (ii)

    \(\Vert \Lambda - \mathrm{Id}\Vert _{C^{1,\alpha }} + \Vert \lambda -1\Vert _{C^{0,\alpha }}\le C_0 \Vert g-e\Vert _{C^{0, \alpha }}\);

  3. (ii)

    \(\Vert \Lambda - \mathrm{Id}\Vert _{C^{k+1,\beta }} + \Vert \lambda -1\Vert _{C^{k,\beta }} \le C_0 \Vert g-e\Vert _{C^{k,\beta }}\).

Although the statement above is a well-known fact (and it follows, for instance, from the treatment of the problem given in [18, Addendum 1 to Chapter 9]), we have not been able to find a classical reference for it. However a complete proof can be found in the Appendix of [5].

Proof of Proposition 2.4

After rescaling we can assume that \(\rho \ge 2^Q\). We fix Q and drop subscripts in \({\mathfrak {B}}_{Q, 2}\). Observe also that, if we rescale by a large factor R, the constants \(C_i\) in Definition 1.4 can then replaced by the constants \(C_i R^{-\alpha }\). Hence, without loss of generality we can assume that \(C_i\) is sufficiently small.

Let \({\varvec{\Phi }}: {\mathfrak {B}}\rightarrow {\mathbb {R}}^{n+2}\) be the graphical parametrization of the branching and recall that \(g= {\varvec{\Phi }}^\sharp e\). Fix a point \((z_0,w_0)\in {\mathfrak {B}}\setminus \{0\}\), let \(r:= |z_0|/2\) and observe that on \(B_r (z_0,w_0)\) we can use z as a chart and compute the metric tensor explicitely as

$$\begin{aligned} g_{ij} (z,w) = \delta _{ij} + \partial _i u (z,w) \partial _j u (z,w) =: \delta _{ij} + \sigma _{ij} \, . \end{aligned}$$

It then follows easily that

$$\begin{aligned} |D^j \sigma (z)|\le&C_0 C_i^2 |z|^{2\alpha - j}\qquad \text{ for } j\in \{0,1,2\} \end{aligned}$$
(B.1)
$$\begin{aligned} _{\alpha , B_r (z_0, w_0)} \le&C_0 C_i^2 r^{\alpha -2}\, . \end{aligned}$$
(B.2)

Step 1. Next consider the map \({\mathbf {W}}:{\mathbf {C}}= {\mathbb {R}}^2\supset B_2 \rightarrow {\mathfrak {B}}\) defined by \({\mathbf {W}}(z):=(z^{Q},z)\). We set

$$\begin{aligned} \bar{g} = {\mathbf {W}}^\sharp g = ({\varvec{\Phi }}\circ {\mathbf {W}})^\sharp e\, . \end{aligned}$$

We then infer that (following Einstein’s convention on repeated indices)

$$\begin{aligned} \bar{g}_{ij} (z) = Q^2 |z|^{2Q-2} \delta _{ij} + \sigma _{kl} ({\mathbf {W}}(z)) \partial _i {\mathbf {W}}_l \partial _j {\mathbf {W}}_k\, , \end{aligned}$$

and we set

$$\begin{aligned} \tau (z) := (Q^2 |z|^{2Q-2})^{-1} \bar{g} (z)\, . \end{aligned}$$

We then easily see that

$$\begin{aligned} |\tau (z) - e| \le C_0|z|^{-(2Q-2)} |D {\mathbf {W}}(z)|^2 |{\mathbf {W}}(z)| \le C_0 C_i^2 |z|^{2Q\alpha }\, . \end{aligned}$$

Differentiating the identity which defines \(\tau \) we also get

$$\begin{aligned} |D\tau (z)| \le&C_0 |z|^{-(2Q-1)} |D {\mathbf {W}}(z)|^2 |\sigma ({\mathbf {W}}(z))| + C_0 |z|^{-(2Q-2)} |D^2 {\mathbf {W}}(z)\\&| |D {\mathbf {W}}(z)| |\sigma ({\mathbf {W}}(z))|+ C_0 |z|^{-(2Q-2)} |D{\mathbf {W}}(z)|^2 |D\sigma ({\mathbf {W}}(z))| |z|^{Q-1}\\ \le&C_0 C_i^2 |z|^{2Q\alpha -1}\, . \end{aligned}$$

Analogous computations lead then to the estimates

$$\begin{aligned} |D^j(\tau -e)|(z)\le&C_0 C_i^2 |z|^{2 Q\alpha -j}\qquad \text{ for } j\in \{0,1,2\} \end{aligned}$$
(B.3)
$$\begin{aligned} _{\alpha , B_s (z)} \le&C_0 C_i^2 |z|^{2 Q\alpha -2-\alpha }\qquad \text{ for } s = |z|/2. \end{aligned}$$
(B.4)

Interpolating between the \(C^1\) and the \(C^0\) bound, we easily conclude that

$$\begin{aligned} {[}\tau ]_{2Q\alpha , B_{2r}\setminus B_r}\le C_0 C_i^2\, . \end{aligned}$$

Note in particular that \(\tau \) (unlike g) can be extended to a nondegenerate \(C^{0,Q\alpha }\) metric to the origin.

Since \(C_i\) can be assumed sufficiently small, we can apply Lemma B.1 to find an orientation preserving diffeomorphism \(\Lambda :\Omega \rightarrow B_2\) and a function \(\lambda : \Omega \rightarrow {\mathbb {R}}^+\) such that

$$\begin{aligned} \Lambda ^\sharp \tau =&{\bar{\lambda }} e \end{aligned}$$
(B.5)
$$\begin{aligned} \Vert \Lambda - \mathrm{Id}\Vert _{C^{1, 2Q\alpha }} + \Vert {\bar{\lambda }} - 1\Vert _{C^{0, 2Q\alpha }} \le&C_0 C_i\, . \end{aligned}$$
(B.6)

Observe that, without loss of generality, we can assume that \(0\in \Omega \) and \(\Lambda (0)=0\). In particular (B.6) implies that, for \(C_i\) suitably small, \(B_1 \subset \Omega \) and hence we will regard \(\Lambda \) and \(\lambda \) as defined on \(B_1\). Next divide \(\Lambda \) by \({\bar{\lambda }} (0)^{{1}/{2}}\) and keep, by abuse of notation, the same symbols for the resulting map and the resulting conformal factor in (B.5). After this normalization we achieve that \({\bar{\lambda }} (0)=1\) and that the estimates (B.6) still hold with a larger \(C_0\). Moreover, \({\bar{\lambda }} (0)=1\) implies that \(D\Lambda (0) \in SO (2)\): composing \(\Lambda \) with an appropriate rotation we can then assume that \(D \Lambda (0)\) is the identity. This implies that

$$\begin{aligned} |{\bar{\lambda }} (z)-1| \le&C_0 C_i |z|^{2Q\alpha } \end{aligned}$$
(B.7)
$$\begin{aligned} |D^j (\Lambda (z) - z)| \le&C_0 C_i |z|^{1+2Q\alpha -j} \qquad \text{ for } j\in \{0,1\}\, . \end{aligned}$$
(B.8)

Step 2. We next wish to estimates the higher derivatives of both \(\Lambda \) and \({\bar{\lambda }}\). We adopt the following procedure. We fix a point \(p\ne 0\) and let \(r:= |p|/2\). We then apply a simple scaling argument to rescale \(B_r (p)\) to a ball of radius 2 so that we can apply Lemma B.1. If we rescale back to \(B_r (p)\) it is then easy to see that we find maps \(\Lambda _p: \Omega _p \rightarrow B_r (p)\), \(\lambda _p: \Omega _p \rightarrow {\mathbb {R}}^+\) with the properties properties:

$$\begin{aligned}&\Lambda _p^\sharp \tau = \lambda _p g \end{aligned}$$
(B.9)
$$\begin{aligned}&\Vert \Lambda _p - \mathrm{Id}\Vert _{C^{1,2Q\alpha }} + \Vert \lambda _p-1\Vert _{C^{0,2Q\alpha }}\le C_0 C_i \end{aligned}$$
(B.10)
$$\begin{aligned}&[\Lambda _p - \mathrm{Id}]_{3,\alpha } + [\lambda _p -1]_{2,\alpha } \le C_0 C_i r^{2Q\alpha -2-\alpha }\, . \end{aligned}$$
(B.11)

Define \(\Xi := \Lambda \circ \Lambda _p^{-1}\) Moreover, its domain is \(B_r (p)\). Since

$$\begin{aligned} \sup _{z\in B_r (p)} |\partial _z (\Xi (z)- z)| \le C_0 r^{2 Q\alpha }\, , \end{aligned}$$

we easily conclude the higher derivative estimates

$$\begin{aligned} \Vert \partial ^k_z (\Xi (z) -z)\Vert \le C_0 C_i r^{2 Q\alpha -k} \qquad \text{ for } k\in \{1,2,3,4\}\, , \end{aligned}$$

which, by holomorphicity, are actually estimates on the full derivatives. Since \(\Lambda = \Xi \circ \Lambda _p\) we then easily conclude that

$$\begin{aligned} |D^{j+1} \Lambda (z)| + |D^j ({\bar{\lambda }} (z) -1)|\le&C_0 C_i |z|^{2Q\alpha -j} \qquad \text{ for } j\in \{0,1,2\} \end{aligned}$$
(B.12)
$$\begin{aligned} _{\alpha ,B_r (z)} + [D^2 {\bar{\lambda }}]_{\alpha , B_r (z)} \le&C_0 C_i r^{2Q\alpha -2-\alpha } \qquad \text{ for } r= |z|/2 >0\, . \end{aligned}$$
(B.13)

Finally notice that

$$\begin{aligned} (\Lambda ^\sharp \bar{g})\, (z) = Q^2 |\Lambda (z)|^{2Q-2} {\bar{\lambda }} (z) e\, . \end{aligned}$$
(B.14)

Step 3. We are finally ready to define \({\varvec{\Psi }}:= {\varvec{\Phi }}\circ {\mathbf {W}}\circ \Lambda \circ {\mathbf {W}}^{-1}\). First of all observe that

$$\begin{aligned} ({\varvec{\Psi }}^\sharp e) (z,w) = (({\mathbf {W}}^{-1})^\sharp \Lambda ^\sharp \bar{g} ) (z,w) = \frac{|\Lambda ({\mathbf {W}}^{-1} (z,w))|^{2Q-2}}{|z|^{2-2/Q}} {\bar{\lambda }} ({\mathbf {W}}^{-1} (z,w)) e_Q =: \lambda (z,w) e_Q \, . \end{aligned}$$

Since \(|{\mathbf {W}}^{-1} (z,w)| = |z|^{1/Q}\), we can also estimate

$$\begin{aligned} |\lambda (z,w) -1|\le&\,\frac{|\Lambda ({\mathbf {W}}^{-1} (z,w))|^{2Q-2}}{|z|^{2-2/Q}} |\bar{\lambda } ({\mathbf {W}}^{-1} (z,w))-1|\\&+\, C \frac{|\Lambda ({\mathbf {W}}^{-1} (z,w))|^{2Q-2} - |z|^{2-2/Q}}{|z|^{2-2/Q}}\\ \le&\,C_0 C_i^2 |{\mathbf {W}}^{-1} (z,w)|^{2Q\alpha } + C_0 |z|^{-1/Q} \left( |\Lambda ({\mathbf {W}}^{-1} (z,w))| - |{\mathbf {W}}^{-1} (z,w)|\right) \\ \le&\,C_0 C_i^2 |z|^{2\alpha } + C_0 C_i^2 |z|^{-1/Q} |{\mathbf {W}}^{-1} (z,w)|^{1+2Q\alpha } \le C_0 C_i^2 |z|^{2\alpha }\, . \end{aligned}$$

Similarly

$$\begin{aligned} |D\lambda (z,w)| \le&\,C_0 |D\bar{\lambda } ({\mathbf {W}}^{-1} (z,w))||z|^{-1} + C_0 \left| D \frac{|\Lambda ({\mathbf {W}}^{-1} (z,w))|^{2Q-2}}{|{\mathbf {W}}^{-1} (z,w)|^{2Q-2}}\right| \\ \le&\,C_0 C_i^2 |z|^{2\alpha -1} + C_0 \left| D \frac{|\Lambda ({\mathbf {W}}^{-1} (z,w)|}{|{\mathbf {W}}^{-1} (z,w)|}\right| \end{aligned}$$

and observe that

$$\begin{aligned} \left| D \frac{|\Lambda ({\mathbf {W}}^{-1})|}{|{\mathbf {W}}^{-1} |}\right| =&\,\left| \left( \frac{D\Lambda ({\mathbf {W}}^{-1})}{|\Lambda ({\mathbf {W}}^{-1})||{\mathbf {W}}^{-1}|} - \frac{|\Lambda ({\mathbf {W}}^{-1})|}{|{\mathbf {W}}^{-1}|^3}\mathrm{Id}\right) D{\mathbf {W}}^{-1} {\mathbf {W}}^{-1}\right| \\ \le&\,C_0 |D{\mathbf {W}}^{-1}| |{\mathbf {W}}^{-1}|^{-1} \left( |D\Lambda ({\mathbf {W}}^{-1}) - \mathrm{Id}| + |{\mathbf {W}}^{-1}|\left( |\Lambda ({\mathbf {W}}^{-1}) - ({\mathbf {W}}^{-1})|\right) \right) \\ \le&\,C_0 C_i^2 |D{\mathbf {W}}^{-1}||{\mathbf {W}}^{-1}|^{2Q\alpha -2}\, . \end{aligned}$$

Recalling that \(|D{\mathbf {W}}^{-1} (z,w)|\le |z|^{1/Q-1}\), \(|{\mathbf {W}}^{-1} (z,w)|= |z|^{1/Q}\), we conclude

$$\begin{aligned} |D \lambda (z,w)|\le C_0 C_i^2 |z|^{2\alpha -1}\, . \end{aligned}$$

The estimates on the second derivative and its Hölder norm follow from similar computations.

We now come to the estimates on \({\varvec{\Psi }}\). Let \({\bar{\Lambda }} := {\mathbf {W}}\circ \Lambda \circ {\mathbf {W}}^{-1}\). Fix \((z_0, w_0)\ne 0\), let \(r:= |z_0|/2\) and use z as a local chart. It will then suffice to show that

$$\begin{aligned} |D^j ({\bar{\Lambda }} (z) -z)|\le&\,C_0 C_i |z|^{1+\alpha -l}\qquad \text{ for } j \in \{0,1,2,3\} \end{aligned}$$
(B.15)
$$\begin{aligned} _{\alpha , B_r (z_0, w_0)} \le&\,C_0 C_i |z|^{-2}\, . \end{aligned}$$
(B.16)

On the other hand since \({\bar{\Lambda }} (0,0) = (0,0)\), it actually suffices to show the first estimate for \(j=1\) to obtain it in the case \(j=0\).

We start computing the first derivatives:

$$\begin{aligned} D {\bar{\Lambda }} = D {\mathbf {W}}(\Lambda \circ {\mathbf {W}}^{-1}) D\Lambda ({\mathbf {W}}^{-1}) D{\mathbf {W}}^{-1}\, . \end{aligned}$$

Recalling that \(D{\mathbf {W}}({\mathbf {W}}^{-1})D{\mathbf {W}}^{-1} = \mathrm{Id}\), we estimate

$$\begin{aligned} |D{\bar{\Lambda }} (z) - \mathrm{Id}| \le&\,|D{\mathbf {W}}(\Lambda ({\mathbf {W}}^{-1} (z))) - D {\mathbf {W}}({\mathbf {W}}^{-1} (z))| |D\Lambda ({\mathbf {W}}^{-1} (z))| |D{\mathbf {W}}^{-1} (z)|\\&+\,|D{\mathbf {W}}({\mathbf {W}}^{-1} (z))| |D\Lambda ({\mathbf {W}}^{-1} (z))-\mathrm{Id}||D{\mathbf {W}}^{-1} (z)|\\ \le&\,C_0 |{\mathbf {W}}^{-1} (z)|^{Q-1}|\Lambda ({\mathbf {W}}^{-1} (z)) - {\mathbf {W}}^{-1} (z)| |z|^{1/Q-1}\\ \le&+\,C_0 C_i^2 |{\mathbf {W}}^{-1} (z)|^{Q-1} ||{\mathbf {W}}^{-1} (z)|^{2Q\alpha } |z|^{1/Q-1}\\ \le&\,C_0 C_i^2 |{\mathbf {W}}^{-1} (z)|^{Q+2Q\alpha } |z|^{1/Q-1} + C_0 C_i^2 |z|^{2\alpha } \le C_0 C_i^2 |z|^{2\alpha }\, . \end{aligned}$$

Similar computations give the estimates on the higher derivatives. \(\square \)

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De Lellis, C., Spadaro, E. & Spolaor, L. Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold. Ann. PDE 3, 18 (2017). https://doi.org/10.1007/s40818-017-0035-7

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Keywords

  • Area-minimizing currents
  • Regularity
  • Two-dimensional
  • Branching singularities
  • Center manifold