Translating solitons in Riemannian products

TL;DR

This paper investigates translating solitons in Riemannian product manifolds, characterizing their properties, especially in rotationally invariant cases with non-positive curvature, and uses them to derive non-existence results.

Contribution

It introduces a framework for understanding translating solitons in Riemannian products, including their characterization and applications as barriers.

Findings

Translating solitons are minimal for a weighted volume functional.

They naturally appear in a monotonicity formula for mean curvature flow.

Complete classification of rotationally invariant solitons under certain curvature conditions.

Abstract

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.

Equations

$$c{X}^{\perp }=\mathbf{H},$$

$${\mathcal{A}}_{c\eta }[M]={\int }_M{e}^{c\eta }\mathrm{d}M$$

$$\Psi :({\omega }_{\ast },{\omega }^{\ast })\times M\to \bar{M}$$

$$\frac{d\Psi }{d\tau }={\Psi }_{\ast }\frac{\partial }{\partial \tau }=\mathbf{H},$$

$${\bf H}=\bigg{(}\sum_{i=1}^{m}\bar{\nabla}_{\Psi_{\tau*}{\sf e}_{i}}\Psi_{\tau*}{\sf e}_{i}\bigg{)}^{\perp}$$

$$\Phi (s,(t,x))=(s+t,x),(t,x)\in \bar{M},s\in \mathbb{R}$$

$${\Psi }_{\tau }(M)={\Phi }_{\sigma (\tau )}(\psi (M)),$$

$${\tilde{\Psi }}_{\tau }(M)=\psi (M)$$

$$\Psi (\tau ,x)=\Phi (\sigma (\tau ),\tilde{\Psi }(\tau ,x)),$$

$$\Psi (\tau ,x)=(c,x),\tau \in \mathbb{R},$$

$$\Psi (\tau ,x)=(u(\tau ,x),x).$$

$$\frac{\partial u}{\partial\tau}=W\,{\rm div}\bigg{(}\frac{\nabla u}{W}\bigg{)},$$

$$W=\sqrt{1+|\nabla u{|}^2}$$

$$c_{\tau }X=c_{\tau }{\Psi }_{\tau \ast }T+\mathbf{H},$$

$${\bar{\nabla }}^{\perp }\mathbf{H}+c_{\tau }II(T,\cdot )=0,$$

$$I{I}_{-\mathbf{H}}+\frac{c_{\tau }}{2}{\pounds }_{T}g=0,$$

$$c{X}^{\perp }=\mathbf{H}$$

$$H=c⟨X,N⟩,$$

$$\mathbf{H}=HN.$$

$$I{I}_{-\mathbf{H}}+\frac{c}{2}{\pounds }_{T}g=0,$$

$${\bar{\nabla }}^{\perp }\mathbf{H}+cII(T,\cdot )=0,$$

$$0=c⟨{\bar{\nabla }}_{{\psi }_{\ast }U}X,{\psi }_{\ast }V⟩+c⟨{\bar{\nabla }}_{{\psi }_{\ast }V}X,{\psi }_{\ast }U⟩$$

$$=c⟨{\bar{\nabla }}_{{\psi }_{\ast }U}{\psi }_{\ast }T,{\psi }_{\ast }V⟩+c⟨{\bar{\nabla }}_{{\psi }_{\ast }V}{\psi }_{\ast }T,{\psi }_{\ast }U⟩+⟨{\bar{\nabla }}_{{\psi }_{\ast }U}\mathbf{H},{\psi }_{\ast }V⟩+⟨{\bar{\nabla }}_{{\psi }_{\ast }V}\mathbf{H},{\psi }_{\ast }U⟩$$

$$=c⟨{\nabla }_{U}T,V⟩+c⟨{\nabla }_{V}T,U⟩+2I{I}_{-\mathbf{H}}(U,V).$$

$$I{I}_{-\mathbf{H}}+\frac{c}{2}{\pounds }_{T}g=0.$$

$$0=c({\bar{\nabla }}_{{\psi }_{\ast }U}X{)}^{\perp }=c({\bar{\nabla }}_{{\psi }_{\ast }U}{\psi }_{\ast }T{)}^{\perp }+c({\bar{\nabla }}_{{\psi }_{\ast }U}{X}^{\perp }{)}^{\perp }$$

$$=cII(T,U)+({\nabla }_{{\psi }_{\ast }U}\mathbf{H}{)}^{\perp }=cII(T,U)+{\nabla }_{{\psi }_{\ast }U}^{\perp }\mathbf{H}$$

$$\eta (x)=(\pi \circ \psi )(x),x\in M,$$

$${\mathcal{A}}_{c\eta }[\psi ,\Omega ]={\mathrm{vol}}_{c\eta }(\psi (\Omega ))={\int }_{\Omega }{e}^{c\eta }\mathrm{d}M,$$

$$H=c⟨X,N⟩$$

$${\delta }^2{\mathcal{A}}_{c\eta }[\psi ,\Omega ]\cdot (f,f)=-{\int }_M{e}^{c\eta }f{L}_{X}f\mathrm{d}M,f\in C_0^{\infty }(\Omega ),$$

$$L_{X}u={\Delta }_{-c\eta }u+(|II{|}^2+{\mathrm{Ric}}_{\bar{M}}(N,N))u$$

Full text

J. H. de Lira is supported by PRONEX/FUNCAP/CNPq PR2-0101-00089.01.00-15 and CNPq/Edital Universal 409689/2016-5. F. Martín is partially supported by the MINECO/FEDER grant MTM2014-52368-P and by the Leverhulme Trust grant IN-2016-019.

1. Introduction

The study of solitons of the mean curvature flow is intimately associated to the nature of singularities of this flow. In this paper, we focus on families of solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. Those one parameter families can be seen, up to inner diffeomorphisms, as ancient solutions of the mean curvature flow which evolve by ambient isometries.

A special case occurs when the Riemannian manifold $(\bar{M}^{n+1},g)$ is the Riemannian product $(\mathbb{R}\times P,{\rm d}t^{2}+g_{0})$ where $(P^{n},g_{0})$ is a Riemannian $n$-manifold and ${\rm d}t^{2}$ denotes the Euclidean metric of the real line. In this class of manifolds, the flow generated by the coordinate vector field $X=\partial_{t}$ is parallel and defines a one-parameter flow of isometries we will refer to as vertical translations. Hence, we say that a submanifold $M$ in $\mathbb{R}\times P$ is a translating soliton of the mean curvature flow if

[TABLE]

where $c\in\mathbb{R}$ is a constant that indicates the velocity of the flow. When $M$ is a hypersurface, the previous equation is equivalent to $H=c\langle X,N\rangle$, where $N$ means the Gauss map of $M$ and we have denoted $g(\cdot,\cdot)=\langle\cdot,\cdot\rangle$.

Translating solitons have been extensively studied in the particular case $P=\mathbb{R}^{n}$ (see for instance [3, 4, 12, 13]). In the Euclidean setting, if the initial immersion has nonnegative mean curvature, then it is known [7] that any limiting flow of a type-II singularity has convex time slices hypersurfaces $M_{\tau}$. Furthermore, either $M_{\tau}$ is a strictly convex translating soliton or (up to rigid motion) $M_{\tau}=\mathbb{R}^{n-k}\times\Sigma_{\tau}^{k}$, where $\Sigma_{\tau}^{k}$ is a lower $k$-dimensional strictly convex translating soliton in $\mathbb{R}^{k+1}$.

Similarly to what happens in the Euclidean space, a translating soliton $M$ in $\mathbb{R}\times P$ can be seen as a minimal surface for the weighted volume functional

[TABLE]

where $\eta$ represents the height Euclidean function, that is, the restriction of the natural coordinate $t\in\mathbb{R}$ to $M$. Furthermore, in Subsection 2.2 we shall see that the translating soliton equation appears naturally in the context of a monotonicity formula associated to the mean curvature flow in $\mathbb{R}\times P$.

The last decade has witnessed the construction of many families of new translating solitons in $\mathbb{R}^{3}$ using different techniques, see [6, 8, 14, 15, 16, 18]. Clutterbuck, Schnürer and Schulze in [5] (see also [3]) proved that there exists an entire, rotationally symmetric, strictly convex graphical translator in $\mathbb{R}^{n+1}$, for $n\geq 2$