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Otis Chodosh

Publications and source records attributed to Otis Chodosh.

At least 19 recordsLinked to original sources
arXiv2026

Minimal hypersurfaces of Morse index one

We prove that a complete, connected, embedded, minimal hypersurface in $\mathbb{R}^{n+1}$ with finite total curvature and Morse index one is the higher-dimensional catenoid.

math.DG↗
arXiv2026

Immersions with small normal curvature

We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem and an existence result for minimizers in this context.

math.DG↗
arXiv2025

Revisiting generic mean curvature flow in $\mathbb{R}^3$

Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R^3 and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler--Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture -- without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors.

math.DG↗
arXiv2025

Generic regularity for minimizing hypersurfaces in dimension 11

We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension $11$ in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, $n+1 \geq 12$, we prove in the same two contexts that area-minimizing hypersurfaces have at most an $n-10-ε_n$ dimensional singular set after an arbitrarily $C^\infty$-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively.

math.DG↗
arXiv2025

Weiss monotonicity and capillary hypersurfaces

Previous work of the authors established the rigorous limiting behavior of minimizing capillary surfaces to minimizers of the Alt--Caffarelli functional as the capillary angle tends to zero. We prove here that in this limit, the capillary area-density converges to the Weiss energy density. We apply this to obtain angle-independent curvature estimates and regularity results for capillary minimizers.

math.AP↗
arXiv2025

The p-widths of a polygon

The $p$-widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each $p$-width of a polygon in $\mathbb{R}^2$ is achieved by a union of billiard trajectories. We also compute the $p$-widths of the equilateral triangle for $p=1,\dots,4$ and square for $p=1,\dots,3$.

math.DG↗
arXiv2025

Improved regularity for minimizing capillary hypersurfaces

We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least $4$, and this estimate improves if the capillary angle is close to $0$, $\fracπ{2}$, or $π$. For capillary angles that are close to $0$ or $π$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.

math.DG↗
arXiv2025

3-Manifolds with positive scalar curvature and bounded geometry

We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be $\mathbb R^3$. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.

math.DG↗
arXiv2024

Mean curvature flow with generic low-entropy initial data II

We prove that the mean curvature flow of a generic closed embedded hypersurface in $\mathbb{R}^4$ or $\mathbb{R}^5$ with entropy $\leq 2$, or with entropy $\leq λ(\mathbb{S}^1)$ if in $\mathbb{R}^6$, encounters only generic singularities.

math.DG↗
arXiv2024

Mean Curvature Flow from Conical Singularities

We prove Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface $M^n\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens.

math.DG↗
arXiv2024

Complete Riemannian 4-manifolds with uniformly positive scalar curvature

We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on certain (non-compact) $4$-manifolds. In particular, such a metric on the interior of a compact contractible $4$-manifold uniquely distinguishes the standard $4$-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic $\mathbb{R}^4$'s that do not admit such a metric and that any (non-compact) tame $4$-manifold has a smooth structure that does not admit such a metric.

math.DG↗
arXiv2024

Improved generic regularity of codimension-1 minimizing integral currents

Let $Γ$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. We show that there exist arbitrarily small perturbations $Γ'$ of $Γ$ with the property that minimizing integral $n$-currents with boundary $Γ'$ are smooth away from a set of Hausdorff dimension $\leq n-9-\varepsilon_n$, where $\varepsilon_n \in (0, 1]$ is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales.

math.DG↗
arXiv2024

Mean curvature flow with generic initial data

We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in $\mathbb{R}^{4}$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons.

math.DG↗