https://neckpinch.com/authors/andreas-savas-halilaj/New arXiv papers by Andreas Savas-Halilaj, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2502.09199arxiv: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://arxiv.org/abs/2403.10739arxiv:2403.10739Fri, 15 Mar 2024 00:00:00 GMTAndreas Savas-Halilaj, Knut Smoczyk — We consider the graphical mean curvature flow of maps bf f:ℝ^m→ℝⁿ, m≥ 2, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [10]. In the case of uniformly area decreasing maps bf f:ℝ^m→ℝ², m≥ 2, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compact https://arxiv.org/abs/2307.05088arxiv: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 https://arxiv.org/abs/2201.05523arxiv:2201.05523Fri, 14 Jan 2022 00:00:00 GMTRenan Assimos, Andreas Savas-Halilaj, Knut Smoczyk — We consider the graphical mean curvature flow of strictly area decreasing maps f:M→ N, where M is a compact Riemannian manifold of dimension m>1 and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature BRic_M of M is bounded from below by the sectional curvature σ_N of N. In addition, we obtain smooth convergence to a minimal map if Ric_M≥sup\0,sup_Nσ_N\. These results significantly improve known results on the graphical mean curvature flow in codimension 2.Mean curvature flowLong-time existence & convergence