https://neckpinch.com/authors/luciano-mari/New arXiv papers by Luciano Mari, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2502.09199/arxiv:2502.09199Thu, 13 Feb 2025 00:00:00 GMTMarco Magliaro, Luciano Mari, Fernanda Roing, Andreas Savas-Halilaj — In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere S²ⁿ⁺¹ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a non-minimal, complete example with topology S²ⁿ⁻¹ × R. The example wraps around a Clifford torus S²ⁿ⁻¹ × S¹ along each end, it has reflection and rotational symmetry and its mean curvature changes sign on each end. Indeed, we prove that a complete 2-dimensional soliton with non-negative mean curvature outside a compact set must be a covering of a Clifford torus. Concluding, we obtain a pinching theorem under suitable conditions on the second fundamental form.Mean curvature flowCurvature pinchingSurfacesRotational symmetry & warped products https://neckpinch.com/papers/2307.05088/arxiv:2307.05088Tue, 11 Jul 2023 00:00:00 GMTLuciano Mari, Jose Danuso Rocha de Oliveira, Andreas Savas-Halilaj, Renivaldo Sodre de Sena — In this paper we study conformal solitons for the mean curvature flow in hyperbolic space ℍⁿ⁺¹. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field −∂₀. We classify cylindrical and rotationally symmetric examples, finding appropriate analogues of grim-reaper cylinders, bowl and winglike solitons. Moreover, we address the Plateau and the Dirichlet problems at infinity. For the latter, we provide the sharp boundary convexity condition to guarantee its solvability, and address the case of noncompact boundaries contained between two parallel hyperplanes of ∂_(∞)ℍⁿ⁺¹. We conclude by proving rigidity results for bowl and grim-reaper cylinders.Mean curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniquenessBoundaries & non-compactRotational symmetry & warped productsMinimal & CMC surfaces