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Alessandro Pigati

Publications and source records attributed to Alessandro Pigati.

At least 19 recordsLinked to original sources
arXiv2026

Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds

Given an anisotropic integrand $F:\text{Gr}_k(\mathbb{R}^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem, and we also remove a convexity assumption on $F$. As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds.

math.AP↗
arXiv2025

Topology of non-collapsed three-dimensional RCD spaces

We show that non-collapsed $\text{RCD}(K,3)$ spaces without boundary are orbifolds whose topological singularities are locally finite and locally homeomorphic to cones over $\mathbb{RP}^2$, and that the topology of such spaces is stable under non-collapsed Gromov-Hausdorff convergence. We study the notion of non-orientability on these spaces as a key part of our analysis and show that the property of non-orientability (on uniformly sized balls) is stable under non-collapsed Gromov-Hausdorff convergence. Finally, we show that any non-orientable non-collapsed $\text{RCD}(K,3)$ space without boundary admits a ramified double cover which is itself an orientable non-collapsed $\text{RCD}(K,3)$ space without boundary, and that such ramified double cover is stable under non-collapsed Gromov-Hausdorff convergence.

math.DG↗
arXiv2025

Anisotropic min-max via phase transitions

We develop a PDE-based approach to the min-max construction of nontrivial integer rectifiable varifolds that are stationary with respect to anisotropic surface energies on closed Riemannian manifolds, in codimension one. Specifically, we study the anisotropic analogue of the Allen-Cahn energy and establish a Modica-type gradient bound for its critical points. Using this in conjunction with certain estimates for stable solutions, we then prove that the energy densities of stable or bounded-Morse-index critical points of its rescalings concentrate along an integer rectifiable varifold that is stationary for the underlying anisotropic integrand. As a consequence, we construct a (possibly singular) anisotropic min-max hypersurface via Allen-Cahn, obtaining an analogue of the result of Hutchinson-Tonegawa in the anisotropic setting.

math.DG↗
arXiv2025

Nonabelian Yang-Mills-Higgs and Plateau's problem in codimension three

We investigate the asymptotic behavior of the $\mathrm{SU}(2)$-Yang-Mills-Higgs energy $E(\Phi,A)=\int_M|d_A\Phi|^2+|F_A|^2$ in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's $\Gamma$-convergence. More precisely, for a compact manifold with boundary $M$ and any family of pairs $\Phi_m\in\Omega^0(M;\mathfrak{su}(2))$ and $A_m\in \Omega^1(M;\mathfrak{su}(2))$ indexed by a mass parameter $m\to\infty$, satisfying $$E(\Phi_m,A_m)\leq Cm\quad\text{and}\quad\lim_{m\to\infty}\frac{1}{m}\int_M(m-|\Phi_m|)^2=0,$$ we prove that the $(n-3)$-currents dual to $\frac{1}{2\pi m}\mathrm{tr}(d_{A_m}\Phi_m\wedge F_{A_m})$ converge subsequentially to a relative integral $(n-3)$-cycle $T$ of mass \begin{equation} \mathbb{M}(T)\leq \liminf_{m\to\infty}\frac{1}{4\pi m}E(\Phi_m,A_m), \end{equation} and show conversely that any integral $(n-3)$-current $T$ with $[T]=0\in H_{n-3}(M,\partial M;\mathbb{Z})$ admits such an approximation, with equality in the above inequality. In the special case of pairs $(\Phi_m,A_m)$ satisfying the generalized monopole equation $*d_{A_m}\Phi_m=F_{A_m}\wedge \Theta$ for a calibration form $\Theta\in \Omega^{n-3}(M)$, we deduce that the limit $\nu=\lim_{m\to\infty}\frac{1}{2\pi m}|d_{A_m}\Phi_m|^2$ of the Dirichlet energy measures satisfies $\nu\leq |T|$, with equality if and only if $T$ is calibrated by $\Theta$, giving evidence for predictions of Donaldson-Segal in the settings of $G_2$-manifolds and Calabi-Yau $3$-folds.

math.DG↗
arXiv2024

Lower Ricci Curvature Bounds and the Orientability of Spaces

We study orientability in spaces with Ricci curvature bounded below. Building on the theory developed by Honda, we establish equivalent characterizations of orientability for Ricci limit and RCD spaces in terms of the orientability of their manifold part. We prove a new stability theorem and, as a corollary, we deduce that four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable. As a global counterpart of the latter, we show that four-manifolds with nonnegative Ricci curvature and Euclidean volume growth are orientable.

math.DG↗
arXiv2024

Decay of excess for the abelian Higgs model

In this article we prove that entire critical points $(u,\nabla)$ of the self-dual $U(1)$-Yang-Mills-Higgs functional $E_1$, with energy $$E_1(u,\nabla;B_R):=\int_{B_R}\left[|\nabla u|^2+\frac{(1-|u|^2)^2}{4}+|F_\nabla|^2\right]\leq(2\pi+\tau(n)) \omega_{n-2}R^{n-2}$$ for all $R>0$, have unique blow-down. Moreover, we show that they are two-dimensional in ambient dimension $2\leq n\leq4$, or in any dimension $n\ge2$ assuming that $(u,\nabla)$ is a local minimizer, thus establishing a co-dimension-two analogue of Savin's theorem. The main ingredient is an Allard-type improvement of flatness.

math.AP↗
arXiv2024

A Variational Theory for The Area of Legendrian Surfaces

We study a new notion of critical point for the area of surfaces under the Legendrian constraint, called parametrized Hamiltonian stationary Legendrian varifolds (PHSLVs). We establish several fundamental properties of these objects, including their sequential compactness and an optimal regularity result, showing that they are smooth immersions away from a locally finite set of branch points and Schoen Wolfson conical singularities. This generalizes in particular the regularity theory of Schoen Wolfson for minimizers to general critical points. This theory can be used to show two new variational results: every minmax operation with the area of (closed, immersed) Legendrian surfaces in a closed Sasakian 5-dimensional manifold is achieved by a Hamiltonian stationary map with this regularity; also, the minimal area in any given exact isotopy class of Legendrian immersions of $S^2$ is realized by such a map. Along the way, we prove an effective monotonicity formula for general two-dimensional stationary varifolds in the Legendrian setting, as well as the closure of integral stationary varifolds among rectifiable ones, in spite of the lack of compactness of the latter.

math.DG↗
arXiv2024

Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M_i^n,g_i,p_i)\to (X^n,d)$. We confirm a conjecture proposed by Colding and Naber in dimension $n=4$, showing that the cross-sections of tangent cones at a given point $x\in X^4$ are all homeomorphic to a fixed spherical space form $S^3/\Gamma_x$, and $\Gamma_x$ is trivial away from a $0$-dimensional set. In dimensions $n>4$, we show an analogous statement at points where all tangent cones are $(n-4)$-symmetric. Furthermore, we prove that $(n-3)$-symmetric noncollapsed Ricci limits are topological manifolds, thus confirming a particular case of a conjecture due to Cheeger, Colding, and Tian. Our analysis relies on two key results, whose importance goes beyond their applications in the study of cross-sections of noncollapsed Ricci limit spaces: (i) A new manifold recognition theorem for noncollapsed ${\rm RCD}(-2,3)$ spaces. (ii) A cone rigidity result ruling out noncollapsed Ricci limit spaces of the form $\mathbb{R}^{n-3}\times C(\mathbb{RP}^2)$.

math.DG↗
arXiv2023

The Parabolic $U(1)$-Higgs Equations and Codimension-two Mean Curvature Flows

We develop the asymptotic analysis as $ε\to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_ε(u,\nabla)=\int_M\left(|\nabla u|^2+ε^2|F_{\nabla}|^2+\frac{(1-|u|^2)^2}{4ε^2}\right)$$ on Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows -- i.e., integral $(n-2)$-Brakke flows -- generalizing results of the last two authors from the stationary case. Given any integral $(n-2)$-cycle $Γ_0$ in $M$, these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at $Γ_0$ with additional structure, similar to those produced via Ilmanen's elliptic regularization.

math.DG↗
arXiv2023

Quantization and non-quantization of energy for higher-dimensional Ginzburg-Landau vortices

Given a family of critical points $u_ε:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_ε(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4ε^2}\right), \end{align*} on a manifold $M$, with natural energy growth $E_ε(u_ε)=O(|\logε| )$, it is known that the vorticity sets $\{|u_ε|\leq \frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{ε\to 0} \frac{e_ε(u_ε)}{π|\logε| }$ of the normalized energy measures. When $n=2$ or the solutions $u_ε$ are energy-minimizing, it is known moreover that this varifold $V$ is integral; i.e., the $(n-2)$-density $Θ_{n-2}(|V|,x)$ of $|V|$ takes values in $\mathbb{N}$ at $|V|$-a.e. $x\in M$. In the present paper, we show that for a general family of critical points with $E_ε(u_ε)=O(|\logε| )$ in dimension $n\geq 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density $Θ_{n-2}(|V|,x)$ of the limit varifold takes values in $\{1\}\cup [2,\infty)$ at $|V|$-a.e. $x\in M$, and show that this is sharp, in the sense that for any $n\geq 3$ and $θ\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_ε$ for $E_ε$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $θ$.

math.DG↗
arXiv2022

Non-degenerate minimal submanifolds as energy concentration sets: a variational approach

We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg-Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet-Jerrard-Sternberg. The same proof applies also to the $U(1)$-Yang-Mills-Higgs and to the Allen-Cahn-Hilliard energies.

math.DG↗
arXiv2021

Convergence of the self-dual $U(1)$-Yang-Mills-Higgs energies to the $(n-2)$-area functional

Given a hermitian line bundle $L\to M$ on a closed Riemannian manifold $(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family of functionals \begin{align*} &E_ε(u,\nabla):=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4ε^2}\Big) \end{align*} defined for couples $(u,\nabla)$ consisting of a section $u\inΓ(L)$ and a hermitian connection $\nabla$ with curvature $F_\nabla$. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in previous work of the second- and third-named authors that critical points in higher dimension converge as $ε\to 0$ (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the $Γ$-convergence of $E_ε$ to ($2π$ times) the codimension two area: more precisely, given a family of couples $(u_ε,\nabla_ε)$ with $\sup_εE_ε(u_ε,\nabla_ε)<\infty$, we prove that a suitable gauge invariant Jacobian $J(u_ε,\nabla_ε)$ converges to an integral $(n-2)$-cycle $Γ$, in the homology class dual to the Euler class $c_1(L)$, with mass $2 π\mathbb{M}(Γ)\le\liminf_{ε\rightarrow 0}E_ε(u_ε,\nabla_ε)$. We also obtain a recovery sequence for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the $(n-2)$-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of $E_ε$.

math.DG↗
arXiv2020

The viscosity method for min-max free boundary minimal surfaces

We adapt the viscosity method introduced by Rivière to the free boundary case. Namely, given a compact oriented surface $Σ$, possibly with boundary, a closed ambient Riemannian manifold $(\mathcal{M}^m,g)$ and a closed embedded submanifold $\mathcal{N}^n\subset\mathcal{M}$, we study the asymptotic behavior of (almost) critical maps $Φ$ for the functional \begin{align*} &E_σ(Φ):=\operatorname{area}(Φ)+σ\operatorname{length}(Φ|_{\partialΣ})+σ^4\int_Σ|{\mathrm {I\!I}}^Φ|^4\,\operatorname{vol}_Φ\end{align*} on immersions $Φ:Σ\to\mathcal{M}$ with the constraint $Φ(\partialΣ)\subseteq\mathcal{N}$, as $σ\to 0$, assuming an upper bound for the area and a suitable entropy condition. As a consequence, given any collection $\mathcal{F}$ of compact subsets of the space of smooth immersions $(Σ,\partialΣ)\to(\mathcal{M},\mathcal{N})$, assuming $\mathcal{F}$ to be stable under isotopies of this space we show that the min-max value \begin{align*} &β:=\inf_{A\in\mathcal{F}}\max_{Φ\in A}\operatorname{area}(Φ) \end{align*} is the sum of the areas of finitely many branched minimal immersions $Φ_{(i)}:Σ_{(i)}\to\mathcal{M}$ with $\partial_νΦ_{(i)}\perp T\mathcal{N}$ along $\partialΣ_{(i)}$, whose (connected) domains $Σ_{(i)}$ can be different from $Σ$ but cannot have a more complicated topology. We adopt a point of view which exploits extensively the diffeomorphism invariance of $E_σ$ and, along the way, we simplify several arguments from the original work. Some parts generalize to closed higher-dimensional domains, for which we get a rectifiable stationary varifold in the limit.

math.DG↗
arXiv2019

Minimal submanifolds from the abelian Higgs model

Given a Hermitian line bundle $L\to M$ over a closed, oriented Riemannian manifold $M$, we study the asymptotic behavior, as $ε\to 0$, of couples $(u_ε,\nabla_ε)$ critical for the rescalings \begin{align*} &E_ε(u,\nabla)=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{1}{4ε^2}(1-|u|^2)^2\Big) \end{align*} of the self-dual Yang-Mills-Higgs energy, where $u$ is a section of $L$ and $\nabla$ is a Hermitian connection on $L$ with curvature $F_{\nabla}$. Under the natural assumption $\limsup_{ε\to 0}E_ε(u_ε,\nabla_ε)<\infty$, we show that the energy measures converge subsequentially to (the weight measure $μ$ of) a stationary integral $(n-2)$-varifold. Also, we show that the $(n-2)$-currents dual to the curvature forms converge subsequentially to $2πΓ$, for an integral $(n-2)$-cycle $Γ$ with $|Γ|\leμ$. Finally, we provide a variational construction of nontrivial critical points $(u_ε,\nabla_ε)$ on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren's existence result of (nontrivial) stationary integral $(n-2)$-varifolds in an arbitrary closed Riemannian manifold.

math.DG↗
arXiv2018

The regularity of parametrized integer stationary varifolds in two dimensions

We establish an optimal regularity result for parametrized two-dimensional stationary varifolds. Namely, we show that the parametrization map is a smooth minimal branched immersion and that the multiplicity function is constant. We provide some applications of this regularity result, especially in the calculus of variations for the area functional.

math.AP↗
arXiv2017

Free boundary minimal surfaces: a nonlocal approach

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in H^{1/2}(\partial S,\mathcal N)$. These maps are critical points of the nonlocal energy \begin{equation}E(f;g):=\int_S\big|\nabla\widetilde u\big|^2\,d\text{vol}_g,\end{equation} where $\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\frac 12$-energies at each boundary component of $\partial S$ (suitably identified with the circle $\mathcal S^1$), plus a quadratic term which is continuous in the $H^s(\mathcal S^1)$ topology, for any $s\in\mathbb R$. We show the $C^{k-1,δ}$ regularity of $\frac 12$-harmonic maps. We also establish a connection between free boundary minimal surfaces and critical points of $E$ with respect to variations of the pair $(f,g)$, in terms of the Teichmüller space of $S$.

math.AP↗
arXiv2017

Existence of tangent lines to Carnot-Carathéodory geodesics

We show that length minimizing curves in Carnot-Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result for length minimizers that holds with no assumption on either the space (e.g., its rank, step, or analyticity) or the curve, and it is novel even in the setting of Carnot groups.

math.OC↗