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Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ ${\Omega }^{+},{\Omega }^{-}\subset R^{n+1}$, and$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$,$$r>0$$ $r>0$, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ ${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{inf}{H}^n\left(\right(\left(\partial B\right(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup (\left(\partial B\right(x,r)\cap H^{-})\setminus {\Omega }^{-}\left)\right),$where the infimum is taken over all open affine half-spaces$$H^{+}$$ $H^{+}$such that$$x \in \partial H^{+}$$ $x\in \partial H^{+}$and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ $H^{-}=R^{n+1}\setminus \overline{H^{+}}$. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ $x\in R^{n+1}$where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ ${\int }_0^1{\epsilon }_n{(x,r)}^2\frac{dr}{r}<\infty$is$$n$$ $n$-rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ${\Omega }^{+}$,$$\Omega ^{-}$$ ${\Omega }^{-}$are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ ${\Omega }^{+}\cup {\Omega }^{-}$satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ $x\in \partial {\Omega }^{+}\cup \partial {\Omega }^{-}$and$$r>0$$ $r>0$, we denote by$$\alpha ^{\pm }(x,r)$$ ${\alpha }^{\pm }(x,r)$the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ ${\Omega }^{\pm }\cap \partial B(x,r)$. We show that, up to a set of$$\mathcal{H}^{n}$$ $H^n$measure zero,$$x$$ $x$is a tangent point for both$$\partial \Omega ^{+}$$ $\partial {\Omega }^{+}$and$$\partial \Omega ^{-}$$ $\partial {\Omega }^{-}$if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ ${\int }_0^{1}min(1,{\alpha }^{+}(x,r)+{\alpha }^{-}(x,r)-2)\frac{dr}{r}<\infty .$The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ ${\epsilon }^2$conjecture of Carleson. 

Abstract We show that for an area minimizingm‐dimensional integral currentTof codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most . This provides a strengthening of the existing ‐dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by‐product of the proof, we establish an improvement on the persistence of singularities along the sequence of center manifolds taken to approximateTalong blow‐up scales. 

Abstract We consider integral area-minimizing 2-dimensional currents$$T$$ $T$in$$U\subset \mathbf {R}^{2+n}$$ $U\subset R^{2+n}$with$$\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$, where$$Q\in \mathbf {N} \setminus \{0\}$$ $Q\in N\setminus \left\{0\right\}$and$$\Gamma $$ $\Gamma$is sufficiently smooth. We prove that, if$$q\in \Gamma $$ $q\in \Gamma$is a point where the density of$$T$$ $T$is strictly below$$\frac{Q+1}{2}$$ $\frac{Q+1}{2}$, then the current is regular at$$q$$ $q$. The regularity is understood in the following sense: there is a neighborhood of$$q$$ $q$in which$$T$$ $T$consists of a finite number of regular minimal submanifolds meeting transversally at$$\Gamma $$ $\Gamma$(and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for$$Q=1$$ $Q=1$. As a corollary, if$$\Omega \subset \mathbf {R}^{2+n}$$ $\Omega \subset R^{2+n}$is a bounded uniformly convex set and$$\Gamma \subset \partial \Omega $$ $\Gamma \subset \partial \Omega$a smooth 1-dimensional closed submanifold, then any area-minimizing current$$T$$ $T$with$$\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$$ $\partial T=Q〚\Gamma 〛$is regular in a neighborhood of $$\Gamma $$ $\Gamma$. 

Abstract We establish a theory ofQ‐valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currentsmod(p)whenp = 2Q, and to establish a first general partial regularity theorem for everypin any dimension and codimension . © 2020 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC. 

Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure.