https://neckpinch.com/authors/j-m-daniels-holgate/New arXiv papers by J. M. Daniels-Holgate, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.12763arxiv:2610.12763Thu, 08 Oct 2026 00:00:00 GMTJ. M. Daniels-Holgate, Or Hershkovits, Gábor Székelyhidi — We show that for a generic compact initial surface in ℝ³, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.Mean curvature flowSingularities https://arxiv.org/abs/2510.06979arxiv:2510.06979Wed, 08 Oct 2025 00:00:00 GMTJ. M. Daniels-Holgate — When mean curvature flow evolves non-uniquely, the flow is said to fatten. The work of Ilmanen shows that any weak MCF is supported inside the fattening, and work of Hershkovits–White identified canonical weak flows supported on the boundary of the fattening, known as the outermost flows. It is natural to ask, when the flow fattens, are there weak mean curvature flows supported strictly inside the fattening? Outside of some special cases (e.g. flow from cones), this question was entirely open. We show these interior flows exist, providing a general construction for non-outermost flows as limits of solutions to the parabolic ε-Allen–Cahn. This gives the first examples of closed, non-trivial, non-canonical, integral Brakke motions. As part of this construction, we study the ε-Allen–Cahn flow from low regularity initial data, and our results demonstrate the existence of integral Brakke motions from fractal sets. This includes the existence portion of Hershkovits's work on mean curvature flow from Reifenberg sets.Mean curvature flowWeak & singular flowsStability & uniquenessPhase fields & multiphase flows https://arxiv.org/abs/2507.16805arxiv:2507.16805Tue, 22 Jul 2025 00:00:00 GMTJ. M. Daniels-Holgate, Or Hershkovits — In this paper we demonstrate that if two mean curvature flows of compact hypersurfaces M¹ₜ and M²ₜ encounter only isolated, multiplicity one, asymptotically conical singularities at the first singular time T, and if M¹_T=M²_T then M¹ₜ=M²ₜ for every t∈ [0,T]. This is seemingly the first backwards uniqueness result for any geometric flow with singularities, that assumes neither self-shrinking nor global asymptotically conical behaviour. This necessitates the development of new global tools to deal with both the core of the singularity, its asymptotic structure, and the smooth part of the flows simultaneously. As an immediate application, we show that low entropy flows in ℝ⁴ are backwards uniqueMean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2503.11522arxiv:2503.11522Fri, 14 Mar 2025 00:00:00 GMTJoshua Daniels-Holgate, Or Hershkovits — Suppose (Mⁱₜ)_(t∈ [0,T)), i=1,2, are two mean curvature flows in ℝⁿ⁺¹ encountering a multiplicity one compact singularity at time T, in such a manner that for every k, the Hausdorff distance between the two flows, d_H, satisfies d_H(M¹ₜ,M²ₜ)/(T-t)^k → 0. We demonstrate that M¹ₜ=M²ₜ for every t. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where M¹ₜ is itself a self-similarly shrinking flow.Mean curvature flowSingularities https://arxiv.org/abs/2312.00759arxiv:2312.00759Fri, 01 Dec 2023 00:00:00 GMTOtis Chodosh, J. M. Daniels-Holgate, Felix Schulze — 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 Mⁿ⊂ ℝⁿ⁺¹, 2≤ n≤ 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.Mean curvature flowSingularitiesWeak & singular flowsBoundaries & non-compact