https://neckpinch.com/authors/brian-white/New arXiv papers by Brian White, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2605.09657arxiv:2605.09657Sun, 10 May 2026 00:00:00 GMTDavid Hoffman, Francisco Martin, Brian White — We construct new expanders for mean curvature flow that are smoothly asymptotic to cones arising from certain shrinkers. For each such cone, we prove the existence of expanders of arbitrarily large genus. Thus, for a fixed incoming shrinker, the genus of the outgoing expander can be chosen much larger than the genus before the singularity, contrary to Ilmanen's genus-reduction conjecture.Mean curvature flowRicci solitonsSingularities https://arxiv.org/abs/2601.13787arxiv:2601.13787Tue, 20 Jan 2026 00:00:00 GMTBrian White — This paper proves that, in mean curvature flow of a compact surface in a complete 3-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact.Mean curvature flowSingularities https://arxiv.org/abs/2502.20340arxiv:2502.20340Thu, 27 Feb 2025 00:00:00 GMTDavid Hoffman, Francisco Martin, Brian White — We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.Mean curvature flowRicci solitonsSingularities https://arxiv.org/abs/2411.16889arxiv:2411.16889Mon, 25 Nov 2024 00:00:00 GMTFrancisco Martín, Mariel Sáez, Raphael Tsiamis, Brian White — We complete the classification of semigraphical translators for mean curvature flow in ℝ³ that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2406.18703arxiv:2406.18703Wed, 26 Jun 2024 00:00:00 GMTTom Ilmanen, Brian White — For each g≥ 3, we prove existence of a compact, connected, smoothly embedded, genus-g surface M_g with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the tangent flow at the singularity is given by a shrinker with genus (g-1) and with two ends. Furthermore, we show that if g is sufficiently large, then M_g fattens at the first singular time. As g→∞, the shrinker converges to a multiplicity 2 plane.Mean curvature flowRicci solitonsSingularities https://arxiv.org/abs/2401.13966arxiv:2401.13966Thu, 25 Jan 2024 00:00:00 GMTBrian White — Consider a pair of smooth, possibly noncompact, properly immersed hypersurfaces moving by mean curvature flow, or, more generally, a pair of weak set flows. We prove that if the ambient space is Euclidean space and if the distance between the two surfaces is initially nonzero, then the surfaces remain disjoint at all subsequent times. We prove the same result when the ambient space is a complete Riemannian manifold of nonzero injectivity radius, provided the curvature tensor (of the ambient space) and all its derivatives are bounded.Mean curvature flowBoundaries & non-compact https://arxiv.org/abs/2308.02210arxiv:2308.02210Fri, 04 Aug 2023 00:00:00 GMTDavid Hoffman, Francisco Martín, Brian White — We construct a family of complete, properly embedded, annular translators M such that M lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as z→ -∞ to four vertical planes y= ± b\ and y= ± B\, where 0<b ≤ B < ∞. We call b and B the inner width and the (outer) width of the translator. We show that for each b ≥ π/2 and each s>0, there is a translator in the family with inner width b and with necksize s. (We also show that there are no translators with inner width <π/2 having the properties of the examples we construct.)Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2304.11447arxiv:2304.11447Sat, 22 Apr 2023 00:00:00 GMTD. Hoffman, F. Martin, B. White — In this paper, we describe new annular examples of complete translating solitons for the mean curvature flow and how they are related to a family of translating graphs, the Δ-wings. In addition, we will prove several related results that answer questions that arise naturally in this investigation. These results apply to translators in general, not just to graphs or annuli.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2204.01983arxiv:2204.01983Tue, 05 Apr 2022 00:00:00 GMTBrian White — For a manifold-with-boundary moving by mean curvature flow, the entropy at a later time is bounded by the entropy at an earlier time plus a boundary term. This paper controls the boundary term in a geometrically natural way. In particular, it shows (under mild hypotheses)that the entropy of a compact translator is less than or equal to the entropy of the boundary plus the maximal cone density of the boundary.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact