Related papers: Uniqueness of 2-dimensional minimal cones in $\mat…
In this article we prove the measure stability for all 2-dimensional Almgren minimal cones in $\mathbb{R}^n$, and the Almgren (resp. topological) sliding stability for the 2-dimensional Almgren (resp. topological) minimal cones in…
This article is dedicated to discuss the sliding stability and the uniqueness property for the 2-dimensional minimal cone YXY in R4. This problem is motivated by the classification of singularities for Almgren minimal sets, a model for…
In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…
In this article we prove that the union of two almost orthogonal planes in R4 is Almgren-minimal. This gives an example of a one parameter family of minimal cones, which is a phenomenon that does not exist in R3. This work is motivated by…
In this paper we discuss various minimality properties for the orthogonal product of two 1-dimensional $\Y$ sets, and some related problems. This is motivated by an attempt to give the classification of singularities for 2-dimensional…
In this article we prove that for a large class of 2-dimensional minimal cones (including almost all 2-dimensional minimal cones that we know), the almost orthogonal union of any two of them is still a minimal cone. Comparing to existing…
We discuss the global regularity for 2 dimensional minimal sets that are near a $\T$ set, that is, whether every global minimal set in $\R^n$ that looks like a $\T$ set at infinity is a $\T$ set or not. The main point is to use the…
In this article we introduce a definition of topological minimal sets, which is a generalization of that of Mumford-Shah-minimal sets. We prove some general properties as well as two existence theorems for topological minimal sets. As an…
We discuss the global regularity of 2 dimensional minimal sets that are near a union of two planes, and prove that every global minimal set in R^4 that looks like a union of two almost orthogonal planes at infinity is a cone. The main point…
In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible…
We give a new proof and a partial generalization of Jean Taylor's result [Ta] that says that Almgren almost-minimal sets of dimension 2 in $\R^3$ are locally $C^{1+\alpha}$-equivalent to minimal cones. The proof is rather elementary, but…
We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren's notion of minimality, which is well suited in the context of…
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…
In [15], Jean Taylor has proved a regularity theorem away from boundary for Almgren almost minimal sets of dimension two in $\mathbb{R}^{3}$. It is quite important for understanding the soap films and the solutions of Plateau's problem away…
We consider stable minimal surfaces of genus 1 in Euclidean space and in Riemannian manifolds. Under the condition of covering stability (all finite covers are stable) we show that a genus 1 finite total curvature minimal surface in…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its…
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…
We give a different and probably more elementary proof of a good part of Jean Taylor's regularity theorem for Almgren almost-minimal sets of dimension 2 in $\R^3$. We use this opportunity to settle some details about almost-minimal sets,…