arXiv subjects

Salvatore Stuvard

Publications and source records attributed to Salvatore Stuvard.

24 records · Page 2Linked to original sources

arXiv2020

Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited

We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's $Q$-valued functions. We study some relevant properties of such $Q$-multisections and apply the theory to provide an elementary and purely geometric proof of a delicate reparametrization theorem for multi-valued graphs which plays an important role in the regularity theory for higher codimension area minimizing currents à la Almgren-De Lellis-Spadaro.

math.AP↗

arXiv2018

Soap films with gravity and almost-minimal surfaces

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.

math.AP↗

arXiv2017

Multiple valued Jacobi fields

We develop a multivalued theory for the stability operator of (a constant multiple of) a minimally immersed submanifold $Σ$ of a Riemannian manifold $\mathcal{M}$. We define the multiple valued counterpart of the classical Jacobi fields as the minimizers of the second variation functional defined on a Sobolev space of multiple valued sections of the normal bundle of $Σ$ in $\mathcal{M}$, and we study existence and regularity of such minimizers. Finally, we prove that any $Q$-valued Jacobi field can be written as the superposition of $Q$ classical Jacobi fields everywhere except for a relatively closed singular set having codimension at least two in the domain.

math.AP↗

arXiv2017

Rectifiability of the singular set of multiple valued energy minimizing harmonic maps

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map.

math.AP↗

arXiv2017

On the structure of flat chains modulo $p$

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 $(m-1)$-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 $1$.

math.AP↗

arXiv2017

On the lower semicontinuous envelope of functionals defined on polyhedral chains

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by $H \colon \mathbb{R} \to \left[ 0,\infty \right)$ an even, subadditive, and lower semicontinuous function with $H(0)=0$, and by $Φ_H$ the functional induced by $H$ on polyhedral $m$-chains, namely \[ Φ_{H}(P) := \sum_{i=1}^{N} H(θ_{i}) \mathcal{H}^{m}(σ_{i}), \quad\mbox{for every }P=\sum_{i=1}^{N} θ_{i} [[ σ_{i} ]] \in\mathbf{P}_m(\mathbb{R}^n), \] we prove that the lower semicontinuous envelope of $Φ_H$ coincides on rectifiable $m$-currents with the $H$-mass \[ \mathbb{M}_{H}(R) := \int_E H(θ(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R= [[ E,τ,θ]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). \]

math.AP↗