https://neckpinch.com/authors/rafael-lopez/New arXiv papers by Rafael López, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.00943arxiv:2607.00943Wed, 01 Jul 2026 00:00:00 GMTRafael Belli, Rafael López — In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group Nil₃ that are invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector fields and the three families of invariant surfaces (vertical translations, horizontal translations, and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.Other curvature flowsGauss curvature flow3-manifolds https://arxiv.org/abs/2605.17097arxiv:2605.17097Sat, 16 May 2026 00:00:00 GMTRafael Belli, Rafael López — We classify invariant surfaces in the 3-dimensional solvable Lie group sol that act as solitons for the Gauss curvature flow. We consider solitons associated with the canonical basis of Killing vector fields F₁, F₂, F₃\, where F₁ and F₂ generate horizontal translations and F₃ generates the scaling isometry. We establish rigidity results for F₃-invariant surfaces, proving that specific totally geodesic vertical planes are the only F₁- and F₂-solitons. For F₁-invariant surfaces, we establish the main geometric properties of F₂- and F₃-solitons in both the extrinsic and intrinsic Gauss curvature.Other curvature flowsGauss curvature flow3-manifoldsHomogeneous spacesStability & uniqueness https://arxiv.org/abs/2605.14385arxiv:2605.14385Thu, 14 May 2026 00:00:00 GMTIvan Krznarić, Rafael López — We study the inverse curve shortening flow in the hyperbolic plane h². We classify all solitons with respect to parabolic and conformal vector fields of h². In the upper half-plane model of h², we prove that parabolic solitons are all graphs on the y-axis, whereas conformal solitons are graphs on the x-axis. We study the concavity of these solitons and when they approach the coordinate axes.Curve shortening https://arxiv.org/abs/2605.13023arxiv:2605.13023Wed, 13 May 2026 00:00:00 GMTIvan Krznarić, Rafael López — We define a new notion of translations in the hyperbolic plane and explicitly solve the equation of the curve shortening flow. Next, we consider the class of ancient convex solutions and solve the equation of the curve shortening flow when the curvature function is given by separation of variables. Lastly, we prove some area estimates for closed ancient solutions of the curve shortening flow.Curve shorteningAncient solutions https://arxiv.org/abs/2508.17321arxiv:2508.17321Sun, 24 Aug 2025 00:00:00 GMTMuhittin Evren Aydin, Rafael López — A λ-translator is a surface in Euclidean space ℝ³ whose Gauss curvature K satisfies K=⟨ N, v ⟩ +λ, where N is the Gauss map, v is a fixed direction, and λ∈ ℝ. In this paper, we classify all λ-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.Other curvature flowsGauss curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2507.12863arxiv:2507.12863Thu, 17 Jul 2025 00:00:00 GMTRafael López — In this paper, we consider λ-translating solitons and λ-shrinkers of the Gauss curvature flow in Euclidean space. We prove that planes and circular cylinders are the only λ-translating solitons with constant mean curvature. We also prove that planes, spheres and circular cylinders are the only λ-shrinkers with constant mean curvature. We give a classification of the λ-translating solitons and λ-shrinkers with one constant principal curvature.Other curvature flowsGauss curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2405.15957arxiv:2405.15957Fri, 24 May 2024 00:00:00 GMTRafael López, Marian Ioan Munteanu — Translators in the special linear group SL(2,ℝ) are surfaces whose mean curvature H and unit normal vector N satisfy H=⟨ N,X⟩, where X is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is 4 and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2402.14727arxiv:2402.14727Thu, 22 Feb 2024 00:00:00 GMTRafael López, Marian Ioan Munteanu — A soliton of the mean curvature flow in the product space mathbbs²×ℝ as a surface whose mean curvature H satisfies the equation H=⟨ N,X⟩, where N is the unit normal of the surface and X is a Killing vector field. In this paper we consider the vector field tangent to the fibers and the vector field associated to a rotations about an axis of mathbbs², respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or under a group of rotations of mathbbs².Mean curvature flow https://arxiv.org/abs/2402.05772arxiv:2402.05772Thu, 08 Feb 2024 00:00:00 GMTAntonio Bueno, Rafael López — We study translators of the mean curvature flow in the product space h²×r. In h²×r there are three types of translations: vertical translations due to the factor r and parabolic and hyperbolic translations from h². A grim reaper in h²×r is a translator invariant by a one-parameter group of translations. The variety of translators and translations in h²×r makes that the family of grim reapers particularly rich. In this paper we give a full classification of the grim reapers of h²×r with a description of their geometric properties. In some cases, we obtain explicit parametrizations of the surfaces.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2201.05347arxiv:2201.05347Fri, 14 Jan 2022 00:00:00 GMTMuhittin Evren Aydin, Rafael López — A K^α-translator is a surface in Euclidean space r³ that moves by translations in a spatial direction and under the K^α-flow, where K is the Gauss curvature and α is a constant. We classify all K^α-translators that are rotationally symmetric. In particular, we prove that for each α there is a K^α-translator intersecting orthogonally the rotation axis. We also describe all K^α-translators invariant by a uniparametric group of helicoidal motions and the translators obtained by separation of variables.Other curvature flowsGauss curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products