##
**Regularity of area minimizing currents. III: Blow-up.**
*(English)*
Zbl 1345.49053

This is third and last of a series of papers in which the authors give a new, shorter proof of a slightly improved version of Almgren’s partial regularity of area minimizing currents in Riemannian manifolds.

Assumption 0.1. Let \(\varepsilon_0 \in [0,1], m,\overline{n}\in \mathbb{N}\backslash \{0\}\) and \(l\in \mathbb{N}\). Denote by \((M)\) \(\Sigma \subset \mathbb{R}^{m+n} =\mathbb{R}^{m+\overline{n}+l}\) an embedded \((m+\overline{n})\)-dimensional submanifold of class \(C^{3,\varepsilon_0}\) and by \(T\) an integral current of dimension \(m\) with compact support \(\mathrm{spt}(T)\subset \Sigma\), which is area minimizing in \(\Sigma\).

For a given \(T\) and \(\Sigma\) as in Assumption 0.1, define \(\mathrm{Reg}(T)=\{x\in \mathrm{spt}(T): \mathrm{spt}(T)\cap \mathbb{B}_r(x)\) is a \(C^{3,\varepsilon_0}\) submanifold for some \(r>0\)}.

Next, define \(\mathrm{Sing}(T) = \mathrm{spt}(T)\setminus (\mathrm{spt}(\partial T)\cup \mathrm{Reg} (T))\).

The authors obtain an estimate on the Hausdorff dimension \(\mathrm{dim}_H(\mathrm{Sing}(T))\) of \(\mathrm{Sing}(T)\) by proving:

Theorem 0.3. \(\mathrm{dim}_H(\mathrm{Sing}(T))\leq m-2\) for any \(m,\overline{n}, l, T\) and \(\Sigma \) as in Assumption 0.1.

The authors start from the following Assumption 0.4. (Contradiction). There exist \(m\geq 2, \overline{n}, l, \Sigma\) and \(T\) as in Assumption 0.1 such that \(\mathcal{H}^{m-2+\alpha}(\mathrm{Sing}(T))>0\) for some \(\alpha>0\). Then they make a careful blow-up analysis split up into the following steps.

0.1. Flat tangent planes. They reduce the flat blow-ups around a given point (assume that this point is the origin). The blow-ups will be chosen so that the size of the singular set satisfies an appropriate uniform estimate.

0.2. Intervals of flattening. For appropriate rescalings of the current around the origin, one can use the center manifold defined and constructed in the second part of this paper, to obtain a good approximation of the average of the sheets of the current at some given scale. The authors introduce a stopping condition for the center manifolds and define appropriate intervals of flattening \(I_j=[s_j,t_j]\). For each \(j\) the authors construct a different center manifold \(\mathcal{M}_j\) and approximate the (rescaled) current with a suitable multi-valued map on the normal bundle of \(\mathcal{M}_j\).

0.3. Finite order of contact. The authors prove that the minimizing current has a finite order of contact with the center manifold. They introduce a variant of the frequency function and prove its monotonicity and boundedness. The corresponding analysis relies on the variational formulas for the images of multiple valued maps (see [the authors, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 4, 1239–1269 (2015; Zbl 1343.49073)]) and on some careful estimates.

0.4. Convergence to Dir-minimizer and contradiction. One can blow-up the Lipschitz approximations from the center manifold \(\mathcal{M}_j\) in order to get a limiting Dir-minimizing function on a flat \(m\)-dimensional domain. Then one shows that the singularities of the rescaled currents converge to singularities of that limiting Dir-minimizer, leading to the proof of Theorem 0.3.

The titles of the sections are: Flat tangent cones; Intervals of flattening; Frequency function and first variations; Error estimates; Boundedness of the frequency; Final blow-up sequence and capacitary argument; Harmonicity of the limit; Appendix A: Some technical lemmas.

Editorial remark: for parts I and II, see [the authors, Geom. Funct. Anal. 24, No. 6, 1831–1884 (2014; Zbl 1307.49043); Ann. Math. (2) 183, No. 2, 499–575 (2016; Zbl 1345.49052)].

Assumption 0.1. Let \(\varepsilon_0 \in [0,1], m,\overline{n}\in \mathbb{N}\backslash \{0\}\) and \(l\in \mathbb{N}\). Denote by \((M)\) \(\Sigma \subset \mathbb{R}^{m+n} =\mathbb{R}^{m+\overline{n}+l}\) an embedded \((m+\overline{n})\)-dimensional submanifold of class \(C^{3,\varepsilon_0}\) and by \(T\) an integral current of dimension \(m\) with compact support \(\mathrm{spt}(T)\subset \Sigma\), which is area minimizing in \(\Sigma\).

For a given \(T\) and \(\Sigma\) as in Assumption 0.1, define \(\mathrm{Reg}(T)=\{x\in \mathrm{spt}(T): \mathrm{spt}(T)\cap \mathbb{B}_r(x)\) is a \(C^{3,\varepsilon_0}\) submanifold for some \(r>0\)}.

Next, define \(\mathrm{Sing}(T) = \mathrm{spt}(T)\setminus (\mathrm{spt}(\partial T)\cup \mathrm{Reg} (T))\).

The authors obtain an estimate on the Hausdorff dimension \(\mathrm{dim}_H(\mathrm{Sing}(T))\) of \(\mathrm{Sing}(T)\) by proving:

Theorem 0.3. \(\mathrm{dim}_H(\mathrm{Sing}(T))\leq m-2\) for any \(m,\overline{n}, l, T\) and \(\Sigma \) as in Assumption 0.1.

The authors start from the following Assumption 0.4. (Contradiction). There exist \(m\geq 2, \overline{n}, l, \Sigma\) and \(T\) as in Assumption 0.1 such that \(\mathcal{H}^{m-2+\alpha}(\mathrm{Sing}(T))>0\) for some \(\alpha>0\). Then they make a careful blow-up analysis split up into the following steps.

0.1. Flat tangent planes. They reduce the flat blow-ups around a given point (assume that this point is the origin). The blow-ups will be chosen so that the size of the singular set satisfies an appropriate uniform estimate.

0.2. Intervals of flattening. For appropriate rescalings of the current around the origin, one can use the center manifold defined and constructed in the second part of this paper, to obtain a good approximation of the average of the sheets of the current at some given scale. The authors introduce a stopping condition for the center manifolds and define appropriate intervals of flattening \(I_j=[s_j,t_j]\). For each \(j\) the authors construct a different center manifold \(\mathcal{M}_j\) and approximate the (rescaled) current with a suitable multi-valued map on the normal bundle of \(\mathcal{M}_j\).

0.3. Finite order of contact. The authors prove that the minimizing current has a finite order of contact with the center manifold. They introduce a variant of the frequency function and prove its monotonicity and boundedness. The corresponding analysis relies on the variational formulas for the images of multiple valued maps (see [the authors, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 14, No. 4, 1239–1269 (2015; Zbl 1343.49073)]) and on some careful estimates.

0.4. Convergence to Dir-minimizer and contradiction. One can blow-up the Lipschitz approximations from the center manifold \(\mathcal{M}_j\) in order to get a limiting Dir-minimizing function on a flat \(m\)-dimensional domain. Then one shows that the singularities of the rescaled currents converge to singularities of that limiting Dir-minimizer, leading to the proof of Theorem 0.3.

The titles of the sections are: Flat tangent cones; Intervals of flattening; Frequency function and first variations; Error estimates; Boundedness of the frequency; Final blow-up sequence and capacitary argument; Harmonicity of the limit; Appendix A: Some technical lemmas.

Editorial remark: for parts I and II, see [the authors, Geom. Funct. Anal. 24, No. 6, 1831–1884 (2014; Zbl 1307.49043); Ann. Math. (2) 183, No. 2, 499–575 (2016; Zbl 1345.49052)].

Reviewer: Vasile Oproiu (Iaşi)

### MSC:

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

49Q20 | Variational problems in a geometric measure-theoretic setting |

49N60 | Regularity of solutions in optimal control |

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\textit{C. De Lellis} and \textit{E. Spadaro}, Ann. Math. (2) 183, No. 2, 577--617 (2016; Zbl 1345.49053)

### References:

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[4] | C. De Lellis and E. Spadaro, ”Regularity of area minimizing currents I: gradient \(L^p\) estimates,” Geom. Funct. Anal., vol. 24, iss. 6, pp. 1831-1884, 2014. · Zbl 1307.49043 · doi:10.1007/s00039-014-0306-3 |

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[6] | C. De Lellis and E. Spadaro, ”Regularity of area-minimizing currents II: center manifold,” Ann. of Math., vol. 183, pp. 499-575, 2016. · Zbl 1345.49052 · doi:10.4007/annals.2016.183.2.2 |

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