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 area minimizing currents in conformal cones, solving Dirichlet problems.

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…

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.

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

New examples show flat singular sets can be arbitrarily complex.

Rectifies flat singular points 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 $1$.

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

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

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…

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

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…

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 high-codimensional minimal surfaces in hyperbolic space.

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…

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.

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

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

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

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

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

Minimal surfaces in a ball have limited area.

Study shows negatively curved manifolds' spherical volume equals minimal surface area.

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…

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

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…

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…

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…

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 …

In what follows we give a quick tour through the field of minimal submanifolds, starting at the definition and the classical results and ending up with current areas of research.

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

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 prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, …

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.