arXiv subjects

Benjy Firester

Publications and source records attributed to Benjy Firester.

11 recordsLinked to original sources

arXiv2026

The anisotropic Michael-Simon inequality

We prove a Michael-Simon inequality for varifolds of every dimension and codimension with respect to anisotropic energies. In particular, this resolves the problem for every convex even hypersurface anisotropy and, using work of Allard, yields the regularity of hypersurfaces with bounded anisotropic mean curvature in every dimension. By the results of De Philippis and Pigati, it also implies density bounds and compactness properties for rectifiable varifolds with uniformly bounded anisotropic first variation. The proof relies on a projection method for anisotropic stress measures as well as the recent resolution of the vanishing mass conjecture by Gennaioli and Rindler.

math.DG↗

arXiv2026

The anisotropic Michael-Simon inequality for surfaces in every codimension

We prove a Michael-Simon inequality for $2$-varifolds in $\mathbb{R}^N$, in arbitrary codimension, for anisotropies sufficiently close to the area functional. Building on the ideas of Almgren and De Philippis-Pigati, we introduce a projection method for anisotropic stress measures yielding geometric inequalities that apply in arbitrary dimension and codimension.

math.DG↗

arXiv2026

On Chern's conjecture for minimal submanifolds of the sphere

A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.

math.DG↗

arXiv2026

Topology of minimal surfaces in the sphere from capillarity

We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known families and produces new minimal surfaces in the sphere with rich topological structures as sphere bundles over base spaces which include space-form products, projective planes over division algebras, Stiefel manifolds, complex quadrics, and twisted products and quotients of Lie subgroups of $SO(n)$. We show these bundles are non-trivial and study their homotopy types using topological obstructions, including characteristic classes and tools from $K$-theory and stable homotopy theory. Finally, we prove uniqueness results for the rotationally invariant capillary CMC problem.

math.DG↗

arXiv2026

New minimal surfaces in the sphere from capillary minimal cones

For every $p,q\geq 1$, we construct minimal embeddings of $\mathbb{S}^p \times \mathbb{S}^q \times \mathbb{S}^1$ in $\mathbb{S}^{p + q + 2}$ by doubling the links of free-boundary minimal cones in $\mathbb{R}^{p+q+3}$ with bi-orthogonal symmetry. This solves problems posed by Hsiang-Lawson and Hsiang-Hsiang. The equivariance reduces the minimal surface equation to an ODE, and we prove the existence of capillary minimal cones for every contact angle. We obtain free-boundary solutions as limits of capillary surfaces via a singular shooting problem with infinite initial slope. As the contact angle degenerates to $0$, rescalings of the capillary cones converge to a homogeneous solution of the one-phase Bernoulli problem, further illustrating the connection between one-phase free boundaries and minimal surfaces through the capillary functional.

math.DG↗

arXiv2026

Area-minimizing capillary cones

We construct non-flat minimal capillary cones with bi-orthogonal symmetry groups for any dimension and contact angle. These cones interpolate between rescalings of a singular solution to the one-phase problem and the free-boundary cone obtained by halving a Lawson cone along a hyperplane of symmetry. The existence and uniqueness of such cones is proved by solving a nonlinear free boundary equation parametrized by the contact angle and obtaining monotonicity properties for the solutions. The constructed cones are minimizing in ambient dimension $8$ or higher, for appropriate contact angles, demonstrating that the regularity theory for minimizing capillary hypersurfaces can have singularities in codimension $7$ and completing the capillary regularity theory for contact angles near $\pi/2$. We further develop the connection between capillary hypersurfaces and solutions of the one-phase problem, consequently producing new examples of singular minimizing free boundaries for the Alt-Caffarelli functional.

math.DG↗

arXiv2026

Stability inequalities for one-phase cones

We obtain strict stability inequalities for homogeneous solutions of the one-phase Bernoulli problem. We prove that in dimension $7$ and above, cohomogeneity one solutions with bi-orthogonal symmetry are strictly stable. As a consequence, we obtain a bound on the first eigenvalue and the decay rates of Jacobi fields, with applications to the generic regularity of the one-phase problem.

math.AP↗

arXiv2025

Homogeneous optimal transport maps between oblique cones

We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi-Yau metrics on certain quasi-projective varieties.

math.AP↗

arXiv2025

On a general class of free boundary Monge-Amp\`ere equations

We solve a general class of free boundary Monge-Amp\`ere equations given by \[ \det D^2u = \lambda \dfrac{f(-u)}{g(u^\star)h(\nabla u)}\chi_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Amp\`ere eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary K\"ahler-Ricci solitons on toric Fano manifolds.

math.AP↗

arXiv2025

Uniqueness of Cylindrical Tangent Cones $C_{p,q} \times \mathbb{R}$

We show the uniqueness of the cylindrical tangent cone $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ for area-minimizing hypersurfaces in $\mathbb{R}^9$, completing the uniqueness of all tangent cones of the form $C_{p,q} \times \mathbb{R}$ proved by Simon for dimensions at least 10 and Sz\'ekelyhidi for the Simons cone.

math.DG↗

arXiv2024

Cohomogeneity two Ricci solitons with sub-Euclidean volume

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Amp\`{e}re equation for the conformal factor coupled with ODEs. We obtain explicit complete expanding solitons as well as abstract existence results for shrinking and steady solitons with boundary. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics. We also classify local solutions of this Monge-Amp\`{e}re equation to prove rigidity for these solitons.

math.DG↗