arXiv subjects
Publications and source records attributed to Marco A. M. Guaraco.
We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.
We prove that the cone $C(Σ)$ in $\mathbb R^4$ over a suitable minimal Möbius band $Σ\subset \mathbb S^3$ is locally mass-minimising with coefficients in $\mathbb Z_2$. This proves a conjecture by Brian White in the context of models of boundary singularities of Plateau's problem in ambient dimension $4$ or higher. The key insight is that the cylindrical slices $C(Σ)\cap \{(z,w)\in \mathbb R^2 \times \mathbb R^2 : |w|=r\}$ are minimal surfaces in a flat rank-one translation quotient of $\mathbb R^3$. In fact, they correspond to non-orientable minimising helicoids previously studied by Antonio Ros. We also show that, up to rigid motions, the minimising cones $C(Σ)\times \mathbb R^{n-4}\subset \mathbb R^n$ are the only non-planar minimal hypercones with boundary $\mathbb R^{n-2}$ which are smooth away from the boundary that satisfy the minimal cylindrical slice property.
We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fröhlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen--Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.
Let $Γ$ be a compact codimension-two submanifold of $\mathbb{R}^n$, and let $L$ be a nontrivial real line bundle over $X = \mathbb{R}^n \setminus Γ$. We study the Allen--Cahn functional, \[E_\varepsilon(u) = \int_X \varepsilon \frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon}\,dx,\] on the space of sections $u$ of $L$. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to $Γ$. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as $\varepsilon \to 0$, the associated family of energy measures converges to an integer rectifiable $(n-1)$-varifold $V$. Moreover, $V$ is stationary with respect to any variation which leaves $Γ$ fixed. Away from $Γ$, this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary $Γ$. Under additional hypotheses, we can say more about $V$. When $V$ arises as a limit of critical sections with uniformly bounded Morse index, $Σ:= \operatorname{supp} \|V\|$ is a minimal hypersurface, smooth away from $Γ$ and a singular set of Hausdorff dimension at most $n-8$. If the sections are globally energy minimizing and $n = 3$, then $Σ$ is a smooth surface with boundary, $\partial Σ= Γ$ (at least if $L$ is chosen correctly), and $Σ$ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in $\mathbb{R}^3$. This also works if $4 \leq n\leq 7$ and $Γ$ is assumed to lie in a strictly convex hypersurface.
We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solutions.
We study global aspects of the mean curvature flow of non-separating hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface $Γ$. We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic $3$-manifolds fibered over $\mathrm{S}^1$ admit smooth entire foliations whose leaves are either minimal or have non-vanishing mean curvature. We also conclude the existence of outermost minimal surfaces for quasi-Fuchsian ends and study their continuity with respect to variations of the quasi-Fuchsian metric.
We study solutions of $Δu - F'(u)=0$, where the potential $F$ can have an arbitrary number of wells at arbitrary heights, including bottomless wells with subcritical decay. In our setting, ground state solutions correspond to unstable solutions of least energy. We show that in convex domains of $\mathbb{R}^N$ and manifolds with $\operatorname{Ric}\geq 0$, ground states are always of mountain-pass type and have Morse index 1. In addition, we prove symmetry of the ground states if the domain is either an Euclidean ball or the entire sphere $S^{N}$. For the Allen-Cahn equation $\varepsilon^2Δu - W'(u)=0$ on $S^{N}$, we prove the ground state is unique up to rotations and corresponds to the equator as a minimal hypersurface. We also study bifurcation at the energy level of the ground state as $\varepsilon\to 0$, showing that the first $N+1$ min-max Allen-Cahn widths of $S^{N}$ are ground states, and we prove a gap theorem for the corresponding $(N+2)$-th min-max solution.
We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh-Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. $H_n(M,\mathbb{Z}_2) = 0$, for $4 \leq n+1 \leq 7$. These provide alternative proofs of Yau's conjecture on the existence of infinitely many minimal hypersurfaces for generic metrics on each setting, using the Allen-Cahn approach.
We study global variational properties of the space of solutions to $-\varepsilon^2Δu + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if $\varepsilon$ is not small enough, in terms of the Cheeger constant of $M$, then there are no interesting solutions. However, we show that the number of min-max solutions to the equation above goes to infinity as $\varepsilon\to 0$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $\varepsilon$, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as $\varepsilon \to 0$, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with ${\rm Ric}_M>0$, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves.
Strong parallels can be drawn between the theory of minimal hypersurfaces and the theory of phase transitions. Borrowing ideas from the former we extend recent results on the regularity of stable phase transition interfaces to the finite Morse index case. As an application we present a PDE-based proof of the celebrated theorem of Almgren-Pitts, on the existence of embedded minimal hypersurfaces in compact manifolds. We compare our results with other min-max theories.