arXiv subjects

Mattia Freguglia

Publications and source records attributed to Mattia Freguglia.

8 recordsLinked to original sources

arXiv2026

Another look at a notion of fractional mass in codimension two

We study a notion of fractional $s$-mass for codimension-two currents on closed Riemannian manifolds, defined via energy minimization with a prescribed Jacobian constraint. We prove equi-coercivity and $Γ$-convergence, with respect to the flat topology, of the $s$-mass on general codimension-two currents. We also prove several additional results for fixed $s$. We establish improved regularity for $s$-harmonic maps that are minimizing among competitors with vanishing Jacobian and show that their singular set has Minkowski dimension at most $n-3$. Moreover, we show that the $s$-mass defined via weak linking, as recently introduced by the authors, agrees with the prescribed Jacobian formulation used here, clarifying the extent to which the $s$-mass depends, or ultimately does not depend, on the way singularities are prescribed.

math.DG↗

arXiv2025

$Γ$-convergence for higher order nonlocal phase transitions

For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case.

math.AP↗

arXiv2025

Min-max theory and Yamabe metrics on conical four-manifolds

We prove existence of Yamabe metrics on four-manifolds possessing finitely-many conical points with $\mathbb{Z}_2$-group, using for the first time a min-max scheme in the singular setting. In our variational argument we need to deform continuously regular bubbles into singular ones, while keeping the Yamabe energy sufficiently low. For doing this, we exploit recent positive mass theorems in the conical setting and study how the mass of the conformal blow-up diverges as the blow-up point approaches the singular set.

math.DG↗

arXiv2025

$Γ$-Limsup estimate for a nonlocal approximation of the Willmore functional

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation ot the fractional Allen-Cahn energies, and we prove the corresponding $Γ$-limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen-Cahn energies.

math.AP↗

arXiv2025

Coercivity and Gamma-convergence of the $p$-energy of sphere-valued Sobolev maps

We consider sequences of maps from an $(n+m)$-dimensional domain into the $(n-1)$-sphere, which satisfy a natural $p$-energy growth, as $p$ approaches $n$ from below. We prove that, up to subsequences, the Jacobians of such maps converge in the flat topology to an integral $m$-current, and that the $p$-energy Gamma-converges to the mass of the limit current. As a corollary, we deduce that the Jacobians of $p$-energy minimizing maps converge to an integral $m$-current that is area-minimizing in a suitable cobordism class, depending on the boundary datum. Moreover, we obtain new estimates for the minimal $p$-energy of maps with prescribed singularities.

math.AP↗

arXiv2025

A nonlocal approximation of the area in codimension two

For $s\in (0,1)$ we introduce a notion of fractional $s$-mass on $(n-2)$-dimensional closed, orientable surfaces in $\R^n$. Moreover, we prove its $Γ$-convergence, with respect to the flat topology, and pointwise convergence to the $(n-2)$-dimensional area.

math.DG↗

arXiv2024

Yamabe metrics on conical manifolds

We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin's classical result. Interestingly, the singular nature of the metric determines a different condition on the dimension, compared to the regular case. We derive asymptotic expansions on the Yamabe quotient by adding a proper and implicit lower-order correction to standard bubbles, whose contribution to the expansion of the quotient can be determined combining the decomposition of symmetric two-tensor fields and Fourier analysis on the conical links.

math.DG↗

arXiv2022

On a conjecture of De Giorgi about the phase-field approximation of the Willmore functional

In 1991 De Giorgi conjectured that, given $λ>0$, if $μ_\varepsilon$ stands for the density of the Allen-Cahn energy and $v_\varepsilon$ represents its first variation, then $\int [v_\varepsilon^2 + λ] dμ_\varepsilon$ should $Γ$-converge to $cλ\mathrm{Per}(E) + k \mathcal{W}(Σ)$ for some real constant $k$, where $\mathrm{Per}(E)$ is the perimeter of the set $E$, $Σ=\partial E$, $\mathcal{W}(Σ)$ is the Willmore functional, and $c$ is an explicit positive constant. A modified version of this conjecture was proved in space dimensions $2$ and $3$ by Röger and Schätzle, when the term $\int v_\varepsilon^2 \, dμ_\varepsilon$ is replaced by $ \int v_\varepsilon^2 {\varepsilon}^{-1} dx$, with a suitable $k>0$. In the present paper we show that, surprisingly, the original De Giorgi conjecture holds with $k=0$. Further properties on the limit measures obtained under a uniform control of the approximating energies are also provided.

math.DG↗