https://neckpinch.com/authors/francisco-martin/New arXiv papers by Francisco Martín, 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/2509.11473arxiv:2509.11473Sun, 14 Sep 2025 00:00:00 GMTEddygledson Souza Gama, Francisco Martín, Niels Martin Møller — We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time t→± ∞, any finite entropy, finite genus, embedded, collapsed translating soliton in ℝ³ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with L^∞-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as t→ +∞.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessBoundaries & non-compact 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/2310.06980arxiv:2310.06980Tue, 10 Oct 2023 00:00:00 GMTFrancisco Martín, Mariel Sáez, Raphael Tsiamis — We prove a conjecture by Hoffman, White, and the first author regarding the uniqueness of pitchfork and helicoid translators of the mean curvature flow in ℝ³. We employ an arc-counting argument motivated by Morse-Radó theory for translators and a rotational maximum principle. Applications to the classification of semigraphical translators in ℝ³ and their limits are discussed, strengthening compactness results of the first author with Hoffman-White and with Gama-Moller.Mean curvature flowSelf-shrinkers & translatorsLimit spaces & RCDStability & uniqueness 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/2209.01640arxiv:2209.01640Sun, 04 Sep 2022 00:00:00 GMTEddygledson Souza Gama, Francisco Martín, Niels Martin Møller — We study translating solitons for the mean curvature flow, Σ²⊆ℝ³ which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected components of slices to define asymptotic invariants ω^±(Σ)∈ℕ, which count the numbers of "wings". Analyzing these, we give a method for computing the entropies λ(Σ) via a simple formula involving the wing numbers, which in particular shows that for this class of solitons the entropy is quantized into integer steps. Finally, combining the concept of wing numbers with Morse theory for minimal surfaces, we prove the uniqueness theorem that if Σ is a complete embedded simply connected translating soliton contained in a slab with entropy λ(Σ)=3 and containing a vertical line, then Σ is one of the translating pitchforks of Hoffman-Martín-WhiteMean curvature flowSelf-shrinkers & translatorsStability & uniquenessMinimal & CMC surfaces