Regularity of area minimizing currents. III: Blow-up
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.NEWLINENEWLINEAssumption 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 \(\Sigma\).NEWLINENEWLINEFor 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 \(r>0\)\}.NEWLINENEWLINENext, define \(\mathrm{Sing}(T) = \mathrm{spt}(T)\setminus (\mathrm{spt}(\partial T)\cup \mathrm{Reg} (T))\).NEWLINENEWLINEThe authors obtain an estimate on the Hausdorff dimension \(\mathrm{dim}_H(\mathrm{Sing}(T))\) of \(\mathrm{Sing}(T)\) by proving:NEWLINENEWLINETheorem 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.NEWLINENEWLINEThe 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.NEWLINENEWLINE0.1. \textit{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.NEWLINENEWLINE0.2. \textit{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 \(\mathcal{M}_j\).NEWLINENEWLINE0.3. \textit{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.NEWLINENEWLINE0.4. \textit{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.NEWLINENEWLINEThe 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.NEWLINENEWLINENEWLINEEditorial 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)].
scientific article; zbMATH DE number 6575344