https://neckpinch.com/authors/mariel-saez/New arXiv papers by Mariel Sáez, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2505.17892arxiv:2505.17892Fri, 23 May 2025 00:00:00 GMTRomina M. Arroyo, Gabriela P. Ovando, Mariel Sáez — In this work we study the existence of solutions to the Mean Curvature Flow for which the initial condition has the structure of a two-dimensional Lie subgroup within a Lie group of dimension three. We consider Lie groups with a fixed left-invariant metric and first observe that if the Lie group is unimodular, then every Lie subgroup is a minimal surface (hence a trivial solution). For this reason we focus on non-unimodular Lie groups, finding the evolution of every Lie subgroup of dimension 2 (within a 3 dimensional Lie group). These evolutions are self-similar for abelian subgroups (i.e. evolve by isometries), but not self-similar in the other cases.Mean curvature flow3-manifoldsSurfacesHomogeneous spacesMinimal & CMC surfaces 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/2410.06942arxiv:2410.06942Wed, 09 Oct 2024 00:00:00 GMTMaria Fernanda Espinal, Mariel Sáez — In this paper we classify rotationally symmetric conformally flat admissible solitons to the k-Yamabe flow, a fully non-linear version of the Yamabe flow. For n≥ 2k we prove existence of complete expanding, steady and shrinking solitons and describe their asymptotic behavior at infinity. For n<2k we prove that steady and expanding solitons are not admissible. The proof is based on the careful analysis of an associated dynamical system.Yamabe flowRicci solitonsRotational symmetry & warped products 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/2305.02378arxiv:2305.02378Wed, 03 May 2023 00:00:00 GMTRomina M. Arroyo, Gabriela P. Ovando, Raquel Perales, Mariel Sáez — This work is a survey of the most relevant background material to motivate and understand the construction and classification of translating solutions to mean curvature flow on a family of solvmanifolds. We introduce the mean curvature flow and some known results in the field. In particular we explore the notion of translating solution in the Euclidean space and extensions into other Riemannian manifolds. We also include a discussion on solvmanifolds and some elements of its geometry that are relevant to our work. We finish by posing the equations that describe translating solutions to mean curvature flow on our family of 3-dimensional solvmanifolds with some additional assumptions. This project emerged at the "Latin American and Caribbean Workshop on Mathematics and Gender" held at Casa Matemática Oaxaca (CMO) from May 15-20, 2022.Mean curvature flowSelf-shrinkers & translators3-manifoldsHomogeneous spaces https://arxiv.org/abs/2110.12026arxiv:2110.12026Fri, 22 Oct 2021 00:00:00 GMTPanagiota Daskalopoulos, Mariel Saez — In this paper we study the uniqueness of graphical mean curvature flow. We consider as initial conditions graphs of locally Lipschitz functions and prove that in the one dimensional case solutions are unique without any further assumptions. This result is then generalized for rotationally symmetric solutions. In the general n- dimensional case, we prove uniqueness under additional conditions: we require a { \em uniform lower bound } on the second fundamental form and the height function of the initial condition. The latter result extends to initial conditions that are proper graphs over subdomains of ℝⁿ.Mean curvature flowStability & uniquenessRotational symmetry & warped products