https://neckpinch.com/authors/raphael-tsiamis/New arXiv papers by Raphael Tsiamis, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.02325arxiv:2610.02325Thu, 01 Oct 2026 00:00:00 GMTSimon Brendle, Raphael Tsiamis — We extend the classification of manifolds with positive isotropic curvature to all dimensions n ≥ 5. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact κ-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.Ricci flowRicci solitonsAncient solutions(Non)collapsingCurvature pinchingSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2604.14139arxiv:2604.14139Wed, 15 Apr 2026 00:00:00 GMTRaphael Tsiamis — For every closed set K ⊂ ℝⁿ and every m ≥ 2, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in ℝ^(m+n), with respect to a smooth Riemannian metric arbitrarily C^∞-close to the Euclidean metric, whose first-time singular set is exactly K × \0\.Mean curvature flowAncient solutions 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/2408.13982arxiv:2408.13982Mon, 26 Aug 2024 00:00:00 GMTBenjy Firester, Raphael Tsiamis — 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-Ampère 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-Ampère equation to prove rigidity for these solitons.Ricci solitons(Non)collapsing4-manifoldsKähler geometryHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2402.11803arxiv:2402.11803Mon, 19 Feb 2024 00:00:00 GMTS. Brendle, R. Tsiamis — We prove an eigenvalue estimate which holds on every properly embedded shrinker for mean curvature flow. This generalizes earlier work of Ding and Xin to the noncompact case.Mean curvature flowRicci solitonsBoundaries & non-compactEigenvalues & spectrum 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