Invariant translators of the solvable group
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Invariant translators of the solvable group Giuseppe Pipoli1 Received: 10 July 2019 / Accepted: 25 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We classify the translators to the mean curvature flow in the three-dimensional solvable group Sol3 that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular, we show that Sol3 admits graphical translators defined on a half plane, in contrast with a rigidity result of Shahriyari (Geom Dedicata 175:57– 64, 2015) for translators in the Euclidean space. Moreover, we exhibit some nonexistence results. Keywords Translators · Mean curvature flow · Sol group Mathematics Subject Classification 53A10 · 53C42 · 53C44
1 Introduction The aim of this paper is to classify all the translators of the three-dimensional solvable group Sol3 that are invariant under the action of a one-parameter group of symmetries. A hypersurface M in a general ambient space is said to be a soliton to the mean curvature flow without changing its shape. More precisely, it means that there exists { if it evolves } G = 𝜑t || t ∈ ℝ a one-parameter group of isometries of the ambient space such that the evolution by mean curvature flow of M is given by the family of hypersurfaces (1)
Mt = 𝜑t (M).
When the ambient space is a Lie group equipped with a left invariant metric, we can choose G consisting of left translations. In this case, we say that M is a translator. Translators provide interesting examples of explicit eternal solutions of the mean curvature flow, and as shown by Huisken and Sinestrari [8], they have a fundamental role in the analysis of type II singularities.
* Giuseppe Pipoli [email protected] 1
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, via Vetoio 1, 67100 L’Aquila, Italy
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It is well known that the property of being a translator can be viewed as a prescribed mean curvature problem: let V be the Killing vector field associated with G, then M evolves moving according to (1) if and only if (2)
H = ḡ (𝜈, V),
where 𝜈 and H are, respectively, the unit normal vector field and the mean curvature of M , while ḡ is the metric in the ambient space. In this case, we will say that M is a translator in the direction of V. A proof of (2) can be found in [9]. The ambient space considered in this paper is the three-dimensional solvable group Sol3 . It is one of the eight geometries of Thurston, the one with the least number of isometries. It can be seen as a Riemannian Lie group defined in the following way: on ℝ3 equipped with the usual coordinates, we consider the group operation, ) ( (x1 , y1 , z1 ) ⋆ (x2 , y2 , z2 ) = x1 + e−z1 x2 , y1 + ez1 y2 , z1 + z2 , and the left invariant Riemannian metric
ḡ = e2z dx2 + e−2z dy2 + dz2 . It follows that an orthonormal basis of left invariant vector fields is
E1 = e−z
𝜕 , 𝜕x
E2 = ez
𝜕 , 𝜕y
E3 =
𝜕 . 𝜕z
T
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