Andrea Marchese

The Steiner tree problem revisited through rectifiable G-currents

Abstract The Steiner tree problem can be stated in terms of finding a connected set of minimal length containing a given set of finitely many points. We show how to formulate it as a mass-minimization problem for 1-dimensional currents with coefficients in a suitable normed group. The representation used for these currents allows to state a calibration principle for this problem. We also exhibit calibrations in some examples.

Kirszbraun’s extension theorem fails for Almgren’s multiple valued functions

We prove that in general it is not possible to extend a Lipschitz multiple valued function without increasing the Lipschitz constant, i.e. we show that there is no analog of Kirszbraun’s extension theorem for Almgren’s multiple valued functions.

Quantitative minimality of strictly stable extremal submanifolds in a flat neighbourhood

Abstract In this paper we extend the results of A strong minimax property of nondegenerate minimal submanifolds, by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of A selection principle for the sharp quantitative isoperimetric inequality, by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.

On the structure of flat chains modulo p

Abstract In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of ( m - 1 ) {(m-1)} -dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.

On the building dimension of closed cones and Almgren’s stratification principle

In this paper we disprove a conjecture stated in (4) on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family C of closed cones and a set S such that every blow-up of S at every point x 2 S is contained in some element of C, is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of C?