The curve shortening flow with density of a spherical curve in codimension two
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Journal of Evolution Equations
The curve shortening flow with density of a spherical curve in codimension two Francisco Viñado- Lereu
Abstract. In the present paper we carry out a systematic study about the flow of a spherical curve by the 3 , g , ξ) mean curvature flow with density in a 3-dimensional rotationally symmetric space with density (Mw w where the density ξ decomposes as sum of a radial part ϕ and an angular part ψ. We analyse how either 3 , g ) conditions the behaviour of the flow when the solution the parabolicity or the hyperbolicity of (Mw w goes to infinity.
1. Introduction A n + 1-dimensional manifold with density (M, g, ξ ) is a Riemannian manifold (M, g) and a function ξ : M → R. In this type of manifold we may calculate the weighted volume or volume with density of the k-dimensional immersed submanifolds ι : P k → M n+1 as: Vξ (P) := eξ ◦ι dvg P , (1) P
where g P ≡ ι g is the induced metric over the manifold P by the immersion ι. We shall denote by dvξ, P , dvξ or eξ dvg P the volume element associated with a density. In this context we have a natural generalization of the mean curvature vector of a submanifold as the negative L 2 -gradient of the k-dimensional functional of volume with density. We shall call to this vector field mean curvature vector with density, and it shall be denoted by Hξ . It has the form: ⊥ Hξ := H − ∇ M ξ ,
(2)
⊥ where H is the mean curvature vector of the submanifold and ∇ M ξ is the orthogonal projection of the gradient ∇ M ξ of ξ in (M, g M ) onto the normal bundle. In the Mathematics Subject Classification: 53C44, 35R01 Keywords: Mean curvature flow, Manifolds with density. Research partially supported by Universitat Jaume I research project UJI-B2018-35 and by MINECO research project MTM2017-84851-C2-2-P. The author has been supported by a postdoctoral grant from Plan de promoción de la investigación de la Universitat Jaume I del año 2018 Acción 3.2. POSDOCA/2018/32 - grupo 041.
J. Evol. Equ.
F. Viñado- Lereu
particular case where k = n = 1, which is the case of curves on a surface, we shall change H by k the geodesic curvature vector and we shall denote by kξ the new vector field, we shall call this vector field the geodesic curvature vector with density. This fact motivates us to study the following flow: ⎧ ⎨ ∂F ( p, t) = Hξ , (3) F( p, t) ∂t ⎩ F( p, 0) = F ( p), 0
where F0 : P k → M n+1 is a k-dimensional immersed submanifold. This is the analogue of the mean curvature flow in the context of the geometry with density. This flow is called the mean curvature flow with density (ξ MCF for short). In the particular case where k = 1, the case of curves, this problem is also called the curve shortening flow with density. Some works performed in this context are [3,15,16,19]; let us remark that these authors did not necessarily use this name for the flow. Other authors had indirectly explored this problem to study the mean curvature flow of submanifolds with some symmetries [13,20]. All these works were done for hypersurfaces (k = n). Given
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