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Peter McGrath

Publications and source records attributed to Peter McGrath.

At least 19 recordsLinked to original sources
arXiv2026

Nonorientable Minimal Surfaces Embedded in the Round $4$-sphere

We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in $\mathbb{R}^n$ for $n\geq 4$. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in $\mathbb{R}^4$, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in $\mathbb{R}^4$ without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.

math.DG↗
arXiv2025

New Homogeneous Solutions for the One-Phase Free Boundary Problem

For each sufficiently large integer $k$, we construct a domain in the round $2$-sphere with $k$ boundary components which is the link of a cone in $\mathbb{R}^3$ admitting a homogeneous solution to the one-phase free boundary problem. This answers a question of Jerison-Kamburov, and also disproves a conjecture of Souam left open in earlier work. The method exploits a new connection with minimal surfaces, which we also use to construct an infinite family of homogeneous solutions in dimension four.

math.AP↗
arXiv2025

Large topology asymptotics for spectrally extremal minimal surfaces in $\mathbb{B}^3$ and $\mathbb{S}^3$

In recent work with Kusner, we developed a method, based on the equivariant optimization of Laplace and Steklov eigenvalues, for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres. We used the method to construct many new minimal embeddings in $\mathbb{S}^3$ with area below $8π$, and many new free boundary minimal embeddings in $\mathbb{B}^3$ with area below $2π$. In this paper, we study the geometry of these surfaces in more detail, with an emphasis on studying sharp area estimates and varifold limits in the large Euler characteristic regime. This allows us to confirm some well-known conjectures regarding the space of low-area minimal surfaces in $\mathbb{S}^3$ in this class of examples and the special role played by Lawson's $ξ_{γ,1}$ surfaces. We also confirm analogous statements in $\mathbb{B}^3$ and identify a family of free boundary minimal surfaces in $\mathbb{B}^3$ most closely resembling $ξ_{γ,1}$.

math.DG↗
arXiv2024

New minimal surface doublings of the Clifford torus and contributions to questions of Yau

The purpose of this article is three-fold. First, we apply a general theorem from our earlier work to produce many new minimal doublings of the Clifford Torus in the round three-sphere. This construction generalizes and unifies prior doubling constructions for the Clifford Torus, producing doublings with catenoidal bridges arranged along parallel copies of torus knots. Ketover has also constructed similar minimal surfaces by min-max methods as suggested by Pitts-Rubinstein, but his methods apply only to surfaces which are lifts of genus two surfaces in lens spaces, while ours are not constrained this way. Second, we use this family to prove a new, quadratic lower bound for the number of embedded minimal surfaces in $\mathbb{S}^3$ with prescribed genus. This improves upon bounds recently given by Ketover and Karpukhin-Kusner-McGrath-Stern, and contributes to a question of Yau about the structure of the space of minimal surfaces in $\mathbb{S}^3$ with fixed genus. Third, we verify Yau's conjecture for the first eigenvalue of minimal surfaces in $\mathbb{S}^3$ in the following cases. First, for all minimal surface doublings of the equatorial two-sphere constructible by our earlier general theorem. Second, for all the Clifford Torus doublings constructed in this article.

math.DG↗
arXiv2024

Embedded minimal surfaces in $\mathbb{S}^3$ and $\mathbb{B}^3$ via equivariant eigenvalue optimization

In 1970, Lawson solved the topological realization problem for minimal surfaces in the sphere, showing that any closed orientable surface can be minimally embedded in $\mathbb{S}^3$. The analogous problem for surfaces with boundary was posed by Fraser and Li in 2014, and it has attracted much attention in recent years, stimulating the development of many new constructions for free boundary minimal surfaces. In this paper, we resolve this problem by showing that any compact orientable surface with boundary can be embedded in $\mathbb{B}^3$ as a free boundary minimal surface with area below $2π$. Furthermore, we show that the number of minimal surfaces in $\mathbb{S}^3$ of prescribed topology and area below $8π$, and the number of free boundary minimal surfaces in $\mathbb{B}^3$ with prescribed topology and area below $2π$, grow at least linearly with the genus. This is achieved via a new method for producing minimal surfaces of prescribed topology in low-dimensional balls and spheres, based on the optimization of Laplace and Steklov eigenvalues in the presence of a discrete symmetry group. As a key ingredient, we develop new techniques for proving the existence of maximizing metrics, which can be used to resolve the existence problem in many symmetric situations and provide at least partial existence results for classical eigenvalue optimization problems.

math.DG↗
arXiv2022

Free Boundary Minimal Annuli Immersed in the Unit 3-Ball

Using the linearized doubling methodology we carry out a PDE gluing construction of a discrete family of non-rotational properly immersed free boundary minimal annuli in the Euclidean unit 3-ball. The surfaces we construct resemble equatorial disks joined by half-catenoidal bridges at the boundary.

math.DG↗
arXiv2022

Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers

In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Σ$, a minimal surface $\breve{M}$ resembling two copies of $Σ$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $φ$ on $Σ$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|\breve{M}|$ of $\breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|\breve{M}| < 2 | Σ|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)\times \mathbb{Z}_2)$-symmetric backgrounds $(Σ, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Σ=\mathbb{S}^2 \subset N=\mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.

math.DG↗
arXiv2022

On Steklov Eigenspaces for Free Boundary Minimal Surfaces in the Unit Ball

We develop new methods to compare the span $\mathcal{C}(Σ)$ of the coordinate functions on a free boundary minimal submanifold $Σ$ embedded in the unit $n$-ball $\mathbb{B}^n$ with its first Steklov eigenspace $\mathcal{E}_{σ_1}(Σ)$. Using these methods, we show that $\mathcal{C}(A)=\mathcal{E}_{σ_1}(A)$ for any embedded free boundary minimal annulus $A$ in $\mathbb{B}^3$ invariant under the antipodal map, and thus prove that $A$ is congruent to the critical catenoid. We also confirm that $\mathcal{C}=\mathcal{E}_{σ_1}$ for any free boundary minimal surface embedded in $\mathbb{B}^3$ with the symmetries of many known or expected examples, including: examples of any positive genus from stacking at least three disks; two infinite families of genus $0$ examples with dihedral symmetry, as well as a finite family with the various Platonic symmetries; and examples of any genus by desingularizing several disks that meet at equal angles along a diameter of the ball.

math.DG↗
arXiv2021

Symmetry and Isoperimetry for Riemannian Surfaces

For a domain $Ω$ in a geodesically convex surface, we introduce a scattering energy $\mathcal{E}(Ω)$, which measures the asymmetry of $Ω$ by quantifying its incompatibility with an isometric circle action. We prove several sharp quantitative isoperimetric inequalities involving $\mathcal{E}(Ω)$ and characterize the domains with vanishing scattering energy by their convexity and rotational symmetry. We also give a new proof of the sharp Sobolev inequality for Riemannian surfaces which is independent of the isoperimetric inequality.

math.DG↗
arXiv2021

On the Canham Problem: Bending Energy Minimizers for any Genus and Isoperimetric Ratio

Building on work of Mondino-Scharrer, we show that among closed, smoothly embedded surfaces in $\mathbb{R}^3$ of genus $g$ and given isoperimetric ratio $v$, there exists one with minimum bending energy $\mathcal{W}$. We do this by gluing $g+1$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $Δ(Δ+2)φ=0$ on the $(g+1)$-punctured sphere $\mathbb{S}^2$ to construct a comparison surface of genus $g$ with arbitrarily small isoperimetric ratio $v\in (0, 1)$ and $\mathcal{W} < 8π$.

math.DG↗
arXiv2018

Sharp Area Bounds for Free Boundary Minimal Surfaces in Conformally Euclidean Balls

We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove analogous results for a class of conformally euclidean ambient spaces. This follows work of Brendle and Fraser-Schoen in the euclidean setting.

math.DG↗
arXiv2018

A Characterization of the Critical Catenoid

We show that an embedded minimal annulus $Σ^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique free boundary minimal annulus in $B^n$ with lowest Steklov eigenvalue equal to 1. We also give more general criteria which imply that a free boundary minimal surface in $B^3$ invariant under a group of reflections has lowest Steklov eigenvalue 1.

math.DG↗
arXiv2017

Minimal Surfaces in the Round Three-Sphere by Doubling the Equatorial Two-Sphere, II

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the Linearized Doubling (LD) methodology developed before and the LD solutions constructed here can be modified readily for use to doubling constructions of rotationally symmetric minimal surfaces with asymmetric sides (work in progress). In particular they allow us to develop in this forthcoming work doubling constructions for the catenoid in Euclidean three-space, the critical catenoid in the unit ball, and the spherical shrinker of the mean curvature flow. Our constructions here allow for sequences of minimal surfaces where the catenoidal bridges tend to be "densely distributed", that is do not miss any open set of $\mathbb{S}^2_{eq}$ in the limit. This in particular leads to interesting observations which seem to suggest that it may be impossible to construct embedded minimal surfaces with isolated singularities by concentrating infinitely many catenoidal necks at a point.

math.DG↗
arXiv2016

Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant cones are rotationally symmetric.

math.DG↗
arXiv2016

Free Boundary Minimal Surfaces in the Unit Ball With Low Cohomogeneity

We study free boundary minimal surfaces in the unit ball of low cohomogeneity. For each pair of positive integers $(m,n)$ such that $m, n >1$ and $m+n\geq 8$, we construct a free boundary minimal surface $Σ_{m, n} \subset B^{m+n}$(1) invariant under $O(m)\times O(n)$. When $m+n<8$, an instability of the resulting equation allows us to find an infinite family $\{Σ_{m,n, k}\}_{k\in \mathbb{N}}$ of such surfaces. In particular, $\{Σ_{2, 2, k}\}_{k\in \mathbb{N}}$ is a family of solid tori which converges to the cone over the Clifford Torus as $k$ goes to infinity. These examples indicate that a smooth compactness theorem for Free Boundary Minimal Surfaces due to Fraser and Li does not generally extend to higher dimensions. For each $n\geq 3$, we prove there is a unique nonplanar $SO(n)$-invariant free boundary minimal surface (a "catenoid") $Σ_n \subset B^n(1)$. These surfaces generalize the "critical catenoid" in $B^3(1)$ studied by Fraser and Schoen.

math.DG↗
arXiv2015

Closed Mean Curvature Self-Shrinking Surfaces of Generalized Rotational Type

For each $n\geq 2$ we construct a new closed embedded mean curvature self-shrinking hypersurface in $\mathbb{R}^{2n}$. These self-shrinkers are diffeomorphic to $S^{n-1}\times S^{n-1}\times S^1$ and are $SO(n)\times SO(n)$ invariant. The method is inspired by constructions of Hsiang and these surfaces generalize self-shrinking "tori" diffeomorphic to $S^{n-1}\times S^1$ constructed by Angenent.

math.DG↗