Related papers: Regularity of nonlocal minimal cones in dimension …
We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by…
In this paper, by constructing area-nonincreasing retractions, we prove area-minimizing properties of some cones over minimal embeddings of R-spaces.
We find the first examples of triply periodic minimal surfaces of which the intrinsic symmetries are all of horizontal type.
In this paper, we show that circular $(s,t)$-Furstenberg sets in $\mathbb R^2$ have Hausdorff dimension at least $$\max\{\frac{t}3+s,(2t+1)s-t\} \text{ for all $0<s,t\le 1$}.$$ This result extends the previous dimension estimates on…
We prove that the determinantal complexity of a hypersurface of degree $d > 2$ is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least $5$. As a result, we obtain that the…
We consider an area-minimizing integral current of dimension $m$ and codimension at least $2$ and fix an arbitrary interior singular point $q$ where at least one tangent cone is flat. For any vanishing sequence of scales around $q$ along…
We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result…
In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a…
In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension…
We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of…
This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle…
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small…
In this paper, we study the differential inclusion associated to the minimal surface system for two-dimensional graphs in $\mathbb{R}^{2 + n}$. We prove regularity of $W^{1,2}$ solutions and a compactness result for approximate solutions of…
We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group…
We prove that the number s(n) of disjoint minimal graphs supported on domains in R^n is bounded by e(n+1)^2. In the two-dimensional case we show that s(2) is at most three (the conjectured number is two).
We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least $2\pi$ have minimal area among all nonpositively curved…
In this paper we find, for any arbitrary finite topological type, a compact Riemann surface $\mathcal{M},$ an open domain $M\subset\mathcal{M}$ with the fixed topological type, and a conformal complete minimal immersion $X:M\to\R^3$ which…
An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact…
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…
We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a…