arXiv research

Proves unique continuation for area minimizing currents.

Study improves boundary smoothness for area-minimizing currents with complex boundaries.

Constructs area-minimizing submanifolds with fractal singularities.

Proves interior singular set dimension for area-minimizing currents in smooth submanifolds.

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…

Study on flat singularities of area-minimizing currents in codimension one.

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

New results show area-minimizing surfaces have fewer singularities than expected.

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 a suitable modification of White's original theorem for area minimizing currents…

Rectifies flat singular points for area-minimizing currents.

Upper bound on singular set dimension for area-minimizing currents.

This a survey on a series of recent papers in collaboration with Emanuele Spadaro on the regularity of area-minimizing currents in codimension higher than $1$.

Study area minimizing currents in conformal cones, solving Dirichlet problems.

A 3D area-minimizing current in R^5 has a 2-fold essential singularity.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

The study bounds Hausdorff measure of flat singular points in area-minimizing currents.

This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing cu…

Study on flat singular points of area-minimizing currents, defining a singularity degree.

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…

We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Rie…

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensiona…

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 discreteness of the singular set for the following three classes of $2$-dimensional curren…

Solves a problem posed by Brezis and Mironescu about least mass of area-minimizing currents.

Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.

Study shows unique tangent cones for area-minimizing currents at boundary points.

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of $C^{1,α}$ submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…

Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.

Study on high-codimensional minimal surfaces in hyperbolic space.

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

Unique solutions found for Plateau problems in smooth and continuous calibrations.

This lecture notes are an expanded version of the course given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain the main steps of a new proof of the partial regularity of area minimizing integer rectifiable currents in higher …

Analyzes singularities of area minimizing hypersurfaces modulo p, completing the structure analysis.

It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

The paper proves existence and partial regularity for Legendrian area-minimizing currents.

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…

The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.

In analogy with Almgren's Theorem for area minimizing currents of general dimension and codimension, we prove that an $m$-dimensional semicalibrated current in a $(n+m)$-dimensional $C^{3,\varepsilon_0}$ manifold, semicalibrated by a $C^{2,\varepsilon_0}$ $m$-form, has singular set of Hausdorff dimension at most $m-2$.

Analyzes branch points of area-minimizing currents with non-2 planar frequency.

New limits of minimal surface systems have surprising large interior parts.

We prove that tangent cones at singular boundary points of a two-dimensional current almost area minimizing are unique. Following the ideas exposed by White in [8], the result is achieved by combining a suitable epiperimetric inequality and an almost-monotonicity formula for the mass at boundary points.

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has…

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over$\mathbb{R}$; and every…

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and Whit…

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations a…