arXiv subjects

Jared Marx-Kuo

Publications and source records attributed to Jared Marx-Kuo.

17 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↗

arXiv2026

The p-widths of the Hemisphere

We compute the p-widths, $\{\omega_p\}$, for the hemisphere with the standard round metric. This provides the first example of a manifold with boundary for which the $p$-widths are known for all $p$.

math.DG↗

arXiv2025

Mountain Pass Critical Points of the Volume Constrained Area Functional

We construct mountain pass critical points of the perimeter functional on sets of fixed volume. For a generic metric, this gives rise to a smooth almost embedded hypersurface with non-zero constant mean curvature. Our work utilizes recent techniques of Mazurwoski--Zhou \cite{mazurowski2024infinitely}, and a new result on the connectedness of Cacciopoli sets: any two smooth Cacciopoli sets can be connected by an $\mathbf{F}$-continuous map.

math.AP↗

arXiv2025

Almgren's Three-Legged Starfish

In this note we use classical tools from min-max and hyperbolic geometry to substantiate a folklore example in Almgren--Pitts min-max theory, the three-legged starfish metric on a 2-sphere, whose systolic length, Almgren--Pitts width, and Gromov--Guth width are attained by ``figure-eight'' geodesics. We also recover a hyperbolic geometry fact about ``figure-eight'' geodesics using min-max.

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↗

arXiv2025

The p-widths of $RP^2$

We compute the p-widths, $\{\omega_p\}$, for the real projective plane with the standard metric.

math.DG↗

arXiv2024

Diameter, Area, and Mean curvature

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean curvature and the area of the manifold. We apply our diameter bounds to minimal, constant mean curvature, and prescribed mean curvature surfaces arising in min-max constructions.

math.DG↗

arXiv2024

Index, Intersections, and Multiplicity of Min-Max Geodesics

We prove upper bounds for the Morse index and number of intersections of min-max geodesics achieving the $p$-widths of a closed surface. A key tool in our analysis is a proof that for a generic set of metrics, the tangent cone at any vertex of any finite union of closed immersed geodesics consists of exactly two lines. We also construct examples to demonstrate that multiplicity one does not hold generically in this setting. Specifically, we construct an open set of metrics on $S^2$ for which the $p$-width is only achieved by $p$ copies of a single geodesic.

math.DG↗

arXiv2024

The Isospectral Problem for p-widths: An Application of Zoll Metrics

We pose the isospectral problem for the $p$-widths: Is a riemannian manifold $(M^n, g)$ uniquely determined by its $p$-widths, $\{\omega_p(M,g)\}_{p=1}^{\infty}$? We construct many counterexamples on $S^2$ using Zoll metrics and the fact that geodesic $p$-widths are given by unions of immersed geodesics.

math.DG↗

arXiv2024

An Inverse Problem for Renormalized Area: Determining the Bulk Metric with Minimal Surfaces

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to recover the expansion of asymptotically hyperbolic, conformally compact metrics which are partially even to high order. We use a rigidity argument to determine the conformal infinity of the metric via the renormalized area. We then consider renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.

math.DG↗

arXiv2023

Second Inner Variations of Energy and Index of Codimension $2$ Minimal Submanifolds

We compute the second inner variation of the Abelian Yang--Mills--Higgs and Ginzburg--Landau energies. Given a sequence of critical points with energy measures converging to a codimension $2$ minimal submanifold, we use the second inner variation formula to bound the morse index of the submanifold by the index of the critical points. The key tools are the convergence of the energy measures and the stress-energy tensors of solutions to Abelian Yang--Mills--Higgs and Ginzburg--Landau equations.

math.DG↗

arXiv2023

Geometric Variations of an Allen-Cahn Energy on Hypersurfaces

We introduce an Allen-Cahn type functional, $\text{BE}_{\epsilon}$, that defines an energy on separating hypersurfaces, $Y$, of closed Riemannian Manifolds. We establish $\Gamma$-convergence of $\text{BE}_{\epsilon}$ to the area functional, and compute first and second variations of this functional under hypersurface pertrubations. We then compute an explicit expansion for the variational formula as $\epsilon \to 0$. A key component of this proof is the invertibility of the linearized Allen-Cahn equation about a solution, on the space of functions vanishing on $Y$. We also relate the index and nullity of $\text{BE}_{\epsilon}$ to the Allen-Cahn index and nullity of a corresponding solution vanishing on $Y$. We apply the second variation formula and index theorems to show that the family of $2p$-dihedrally symmetric solutions to Allen-Cahn on $S^1$ have index $2p - 1$ and nullity $1$.

math.DG↗

arXiv2023

A Dirichlet-to-Neumann Map for the Allen-Cahn Equation on Manifolds with Boundary

We study the asymptotic behavior of Dirichlet minimizers to the Allen--Cahn equation on manifolds with boundary, and we relate the Neumann data to the geometry of the boundary. We show that Dirichlet minimizers are asymptotically local in orders of $\epsilon$ and compute expansions of the solution to high order. A key tool is showing that the linearized allen-cahn operator is invertible at the heteroclinic solution, on functions with $0$ boundary condition. We apply our results to separating hypersurfaces in closed Riemannian manifolds. This gives a projection theorem about Allen--Cahn solutions near minimal surfaces, as constructed by Pacard--Ritore.

math.DG↗

arXiv2021

Variations of Renormalized Volume for Minimal Submanifolds of Poincare-Einstein Manifolds

We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds, $Y^m$ of arbitrary codimension in Poincare-Einstein manifolds, $M^{n+1}$. In particular, we derive formulae for the first and second variations of renormalized volume for $Y^m \subseteq M^{n+1}$ when $m < n + 1$. We apply our formulae to the codimension $1$ and the $M = \mathbb{H}^{n+1}$ case. Furthermore, we prove the existence of an asymptotic description of our minimal submanifold, $Y$, over the boundary cylinder $\partial Y \times \mathbb{R}^+$, and we further derive an $L^2$-inner-product relationship between $u_2$ and $u_{m+1}$ when $M = \mathbb{H}^{n+1}$. Our results apply to a slightly more general class of manifolds, which are conformally compact with a metric that has an even expansion up to high order near the boundary.

math.DG↗

arXiv2020

Characters of Renner Monoids and Their Hecke Algebras

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e., the Cartan type $A$ Renner monoid) was computed earlier by Dieng, Halverson, and Poladian using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type $C$ and use them to obtain an explicit description of the character table for the type $C$ Renner monoid Hecke algebra.

math.RT↗

arXiv2019

Sandpile Groups of Cayley Graphs of $\mathbb{F}_2^r$

The sandpile group of a connected graph $G$, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous works on the sandpile group of hypercubes, we study the sandpile groups of the Cayley graphs of $\mathbb{F}_2^r$, focusing on their poorly understood Sylow-$2$ component. We find the number of Sylow-$2$ cyclic factors for "generic" Cayley graphs and deduce a bound for the non-generic ones. Moreover, we provide a sharp upper bound for their largest Sylow-$2$ cyclic factors. In the case of hypercubes, we give exact formulae for the largest $n-1$ Sylow-$2$ cyclic factors. Some key ingredients of our work include the natural ring structure on these sandpile groups from representation theory, and calculation of the $2$-adic valuations of binomial sums via the combinatorics of carries.

math.CO↗

arXiv2018

Convolution Algebras for Finite Reductive Monoids

For an arbitrary finite monoid $M$ and subgroup $K$ of the unit group of $M$, we prove that there is a bijection between irreducible representations of $M$ with nontrivial $K$-fixed space and irreducible representations of $\mathcal{H}_K$, the convolution algebra of $K\times K$-invariant functions from $M$ to $F$, where $F$ is a field of characteristic not dividing $|K|$. When $M$ is reductive and $K = B$ is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of $M$ and those of $F[R]$, where $R$ is the Renner monoid of $M$. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.

math.RT↗