Translating solitons in Riemannian products

By J. D. de Lira, F. Martín. Journal of Differential Equations, 2018

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In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R} \times P, {\rm d}t^2+g_0)$ and the Killing field is $X=\partial_t$. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in $\mathbb{R} \times P$ can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in $\mathbb{R} \times P$. When $g_0$ is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/20538445df62270216037ce453e8742cbc2b5f72

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields (2017, background): “raphs conﬁned in appropriate regions of a product space R×P. See Theorem 10.2. Some examples and a further elaborated discussion on translating solitons in general Riemannian products may be found at [35]. In the next section we present, in the codimension one case for simplicity, how the equation c”