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Pedro Gaspar

Publications and source records attributed to Pedro Gaspar.

12 recordsLinked to original sources
arXiv2026

Equivariant Allen--Cahn Solutions and the Existence of Cohomogeneity $2$ Minimal Surfaces

We develop a regularity theory for equivariant Allen--Cahn solutions on closed Riemannian manifolds with a Lie group acting isometrically. When the cohomogeneity of the action is between $3$ and $7$, we show that a sequence of equivariant Allen--Cahn solutions with uniformly bounded energy and equivariant index converge to embedded minimal hypersurfaces with optimal regularity, meaning that the singular set is at least codimension $7$ and lies in the union of all non-principal orbits. When the cohomogeneity is $2$ and the action has no exceptional orbits, we show the same result but the minimal hypersurfaces may be immersed. As a result, any closed Riemmanian manifold with cohomogeneity $2$ Lie group action and no exceptional orbits admits a minimal hypersurface with optimal regularity. A key tool is the regularity theory of Chodosh--Mantoulidis, building on the work of Wang--Wei. However, we adapt their arguments to a modified Allen--Cahn equation with a drift Laplacian. We also show that appropriate index bounds hold for the limiting minimal hypersurface when it is smooth. We also extend the variational constructions of solutions of the Allen--Cahn equation of Guaraco and Gaspar--Guaraco by defining an equivariant mountain pass invariant, as well as the equivariant Allen--Cahn $p$-widths. This builds on the work of Gromov and is the Allen--Cahn parallel to Wang's equivariant volume spectrum in the Almgren-Pitts setting. We show that in the limit as $\epsilon$ tends to $0$, the equivariant Allen--Cahn $p$-widths converge to the equivariant $p$-widths, as defined by Wang.

math.DG↗
arXiv2025

When can minimal hypersurfaces be connected by mean curvature flow?

From the perspective of Morse theory, it is natural to investigate gradient flow trajectories between critical points. In this short note, we explore the minimal hypersurface analogue of this phenomenon and present examples that suggest additional topological and variational obstructions to the existence of connecting mean curvature flows.

math.DG↗
arXiv2025

Volume spectrum of fiber bundles and the widths of Berger spheres

We establish that for a fiber bundle $\pi: E \to B$, which is a Riemannian submersion, the volume spectrum of $E$ is bounded above by the product of the volume spectrum of $B$ and the volume of the largest fiber. Specifically, we prove the following inequality: $$\omega_p(E,g_E) \leq \left( \sup_{b \in B} \operatorname{vol}_{g_E}(\pi^{-1}(b)) \right) \omega_p(B,g_B). $$ Furthermore, we extend this result to the phase transition spectrum. In addition, we also obtain lower bounds for the isoperimetric profile of Riemannian fibrations with totally geodesic, spherical fibers in terms of the isoperimetric profile of the product of the base and a sphere. By exploiting connections between volume spectrum, least area minimal surfaces, and the isoperimetric profile, we employ these bounds to compute the low widths of Berger spheres and product of spheres. Notably, our analysis reveals that for sufficiently small $\tau$, the equatorial sphere $S^2$ in the Berger sphere $S^3_{\tau}$ (a $S^1-$bundle over $S^2(\frac{1}{2})$ with fiber length $2\pi \tau$) attains the Simon-Smith $1,2,3,4$ widths but fails to attain any lower widths, in both the Almgren-Pitts setting and the Allen-Cahn setting.

math.DG↗
arXiv2025

Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

We construct infinitely many distinct hypersurfaces with prescribed mean curvature (PMC) for a large class of prescribing functions when $(M^{n+1}, g)$ is a closed smooth manifold containing a minimal surface that is strictly stable (or more generally, admits a contracting neighborhood). In particular, we construct infinitely many distinct PMCs when $H_n(M, \mathbb{Z}_2) \neq 0$, or if $(M, g)$ does not satisfy the Frankel property. Our construction synthesizes ideas from Song's construction of infinitely many minimal surfaces in the non-generic setting, Dey's construction of multiple constant mean curvature surfaces, and Sun--Wang--Zhou's min-max construction of free boundary PMCs.

math.DG↗
arXiv2023

Morse theory for the Allen-Cahn functional

In this article, we use Morse-theoretic techniques to construct connections between low energy critical submanifolds of the Allen-Cahn energy functional in the 3-sphere via the negative gradient flow.

math.DG↗
arXiv2021

Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in spheres, by K. Choi and C. Mantoulidis, as well as the rigidity of stationary solutions with low Morse index proved by F. Hiesmayr.

math.DG↗
arXiv2020

Ground states of semilinear elliptic equations

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.

math.AP↗
arXiv2019

Solutions of the Allen-Cahn equation on closed manifolds in the presence of symmetry

We prove that given a minimal hypersurface $Γ$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $Γ$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2Δu +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets converge to $Γ$. This extends the results of Pacard-Ritoré (in the case of closed manifolds and zero mean curvature).

math.DG↗
arXiv2018

The Weyl Law for the phase transition spectrum and density of limit interfaces

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.

math.DG↗
arXiv2017

The second inner variation of energy and the Morse index of limit interfaces

In this article we study the second variation of the energy functional associated to the Allen-Cahn equation on closed manifolds. Extending well known analogies between the gradient theory of phase transitions and the theory of minimal hypersurfaces, we prove the upper semicontinuity of the eigenvalues of the stability operator and consequently obtain upper bounds for the Morse index of limit interfaces which arise from solutions with bounded energy and index without assuming any multiplicity or orientability condition on these hypersurfaces. This extends some recent results of N. Le and F. Hiesmayr.

math.DG↗
arXiv2016

The Allen-Cahn equation on closed manifolds

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.

math.DG↗