Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

Xu, Runzhang; Liu, Tao

doi:10.1016/j.jmaa.2019.04.057

Cited by 7 publications

(1 citation statement)

References 39 publications

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“…In [22], Chen and Qiu proved that any complete m -dimensional spacelike self-shrinkers in pseudo-Euclidean spaces of index n must be affine planes, and that there exists no complete m -dimensional spacelike translating soliton in . Subsequently, Xu and Liu [44] classified m -dimensional complete spacelike translating solitons in by affine techniques and classical gradient estimates, and they obtained a Bernstein-type theorem when the translating vector is spacelike. In addition, Lambert and Lotay [32] proved long-time existence and convergence results for spacelike solitons to mean curvature flow in that are entire or defined on bounded domains and satisfy Neumann or Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

On the Geometry of Spacelike Mean Curvature Flow Solitons Immersed in a GRW Spacetime

DE LIMA,

GOMES,

SANTOS

et al. 2023

J. Aust. Math. Soc.

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We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$ , with base $I\subset \mathbb R$ , Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$ . For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$ , which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

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