Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

Alı́as, Luis J.; Lira, Jorge H. de; Rigoli, Marco

doi:10.1007/s12220-019-00186-3

Cited by 28 publications

(51 citation statements)

References 46 publications

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“…[14][15][16][17][18][19][20]), and conformal vector fields (cf. [21][22][23][24]). It is known that these vector fields of special type influence not only the geometry, but also the topology of the Riemannian spaces.…”

Section: Introductionmentioning

confidence: 99%

A Note on Geodesic Vector Fields

et al. 2020

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The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it is shown that the geodesicity property of vector fields is also useful in characterizing not only spheres, but also Euclidean spaces.

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Section: Introductionmentioning

confidence: 99%

A Note on Geodesic Vector Fields

et al. 2020

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“…Note that, when the external force W is a closed conformal vector field and φ ≡ 1, the corresponding flow (1.8) has been systematically studied in [36] and, very recently, the general MCF solitons in the presence of conformal vector fields are studied in [1]. However, we also know from [36] that the existence of a closed conformal vector field W is rather restrictive for the ambient Riemannian manifold (N, g).…”

Section: Introductionmentioning

confidence: 99%

The blow-up of the conformal mean curvature flow

Zhang

2019

Sci. China Math.

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In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space R n . This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in R n , the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in R n with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.2000 Mathematics Subject Classification. Primary 53A30; Secondary 53B25. Key words and phrases. conformal mean curvature flow, conformal external force, blow-up of the curvature, round point.

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“…Although relation between self-similar solutions of general curvature flows and their singularities is unclear now, there have been some study on rigidity of closed self-similar solutions of curvature flows, for instance, [12], [9], etc. Recently self-similar solutions of the mean curvature flows were introduced on manifolds endowed with a conformal vector field [1], such as Riemannian cone manifolds [8] and warped product manifolds [14,1]. In this paper we study closed strictly convex self-similar solutions of a class of curvature flows in Riemannian warped products and obtain the uniqueness of closed strictly convex self-similar solutions in hemispheres and hyperbolic spaces.…”

Section: Introductionmentioning

confidence: 99%

“…In R n+1 , λ(r)∂ r is just the position vector. So in the spirit of [14,1], we call solutions of (1.1) self-similar solutions to the following curvature flow…”

Section: Introductionmentioning

confidence: 99%

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Self-similar solutions of curvature flows in warped products

Gao

2019

Differential Geometry and its Applications

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In this paper we study self-similar solutions in warped products satisfying F − F =ḡ(λ(r)∂r , ν), where F is a nonnegative constant and F is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such F . We also obtain a similar uniqueness result in hyperbolic space H 3 for Gauss curvature F and F ≥ 1.2010 Mathematics Subject Classification. 53C44, 53C40.

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Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

Cited by 28 publications

References 46 publications

A Note on Geodesic Vector Fields

A Note on Geodesic Vector Fields

The blow-up of the conformal mean curvature flow

Self-similar solutions of curvature flows in warped products

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