1 Introduction

Recall that a smooth family of hypersurfaces \(F_t : \Sigma^n \to M^{n+1}\), \(t \geqslant 0\), in a Riemannian manifold \((M,g)\) is called a solution of the mean curvature flow (M.C.F.) on \((0,T)\), \(T>0\), if \[\begin{aligned} \frac{d}{dt} F_t&=-\mathbf{H}, \quad &\text{on}&\; \Sigma \times (0,T),\\ F_0&=f, \quad &\text{on}&\; \Sigma,\\ \end{aligned}\] where \(f : \Sigma \to M\) is a given initial hypersurface and \(\mathbf{H}\) denotes the mean curvature vector of \(F_t(\Sigma)\). Suppose there exists a conformal Killing vector field \(\mathbf{X}\) on \(M\) with flow \(\varphi : M \times \mathbb{R}\to M\). A family of hypersurfaces \(F_t\) is said to be a soliton solution of the M.C.F. with respect to the conformal Killing vector field \(\mathbf{X}\) if \(\tilde{F}_t=\varphi^{-1}(F_t,t)\) is stationary in normal direction, i.e. \(\tilde{F}_t(\Sigma)\) is the fixed hypersurface \(f(\Sigma)\). In [8] it was shown that for a given initial hypersurface \(f: \Sigma \to M\) to give rise to a soliton solution of the mean curvature flow it is necessary that \[\tag{1.1} \mathbf{H}+\mathbf{X}^{\perp}=0,\] where \(\perp\) denotes the \(g\)-orthogonal projection onto the normal bundle of the hypersurface \(f : \Sigma \to M\). If \(\mathbf{X}\) is Killing, then (1.1) is also sufficient.

Soliton solutions have played an important rôle in the development of the theory of the M.C.F. Such solutions served, e.g., as tailor-made comparison solutions to investigate the development of singularities (e.g. Angenent’s self-similarly shrinking doughnut, see [3]). Actually, soliton solutions appear as blow-up of so called type II singularities of the flow of plane curves (see [2]). Moreover, soliton solutions turn out to enjoy certain stability properties and allow some insight into the behaviour of the mean curvature flow viewed as a dynamical system (see [8], [13] and [6]).

In [8] the boundary value problem for rotating soliton solutions has been discussed. The corresponding local existence result has been generalised to arbitrary Killing fields in [9]. For rotating solitons in the euclidean plane, so called yin-yang curves, a quantity was identified that remains invariant along the curve (see [9]). This invariant allowed to show that yin-yang curves share fundamental geometric properties with geodesic curves. In [9] the corresponding results have been generalised to arbitrary soliton curves on surfaces (see Figure 1.1).

In addition, it was observed in [9], that translating solitons in the euclidean plane, the so called grim reaper curves, actually are geodesics with respect to a conformally deformed Riemannian metric. Therefore the natural question arose whether soliton curves are (at least locally) always geodesic curves with respect to a modified Riemannian metric. This is not the case. On a surface \((M,g)\), the solutions of (1.1) are immersed curves on \(M\) which may be reparametrised to become geodesics of the Weyl connection \(\nabla_{g,\mathbf{X}}\) given by \[(\mathbf{Y}_1,\mathbf{Y}_2)\mapsto (D_g)_{\mathbf{Y}_1} \mathbf{Y}_2-g(\mathbf{Y}_1,\mathbf{Y}_2)\mathbf{X}+g(\mathbf{X},\mathbf{Y}_1)\mathbf{Y}_2+g(\mathbf{X},\mathbf{Y}_2)\mathbf{Y}_1,\] where we have written \(D_g\) for the Levi-Civita connection of \(g\). The equation (1.1) is parametrisation invariant and thus its solutions are naturally interpreted as the geodesics of a projective structure on \(M\). Recall that a projective structure is an equivalence class of affine torsion-free connections, where two such connections are said to be equivalent if they have the same geodesics up to parametrisation. Recently in [4], Bryant, Dunajski and Eastwood determined the necessary and sufficient local conditions for an affine torsion-free connection to be projectively equivalent to a Levi-Civita connection. Applying their results¹ it follows that the Weyl connection whose geodesics are the yin-yang curves is not projectively equivalent to a Levi-Civita connection. However Jürgen Moser conjectured² that soliton curves can at least locally be interpreted as geodesics of a Finsler metric. Recent results about Finsler metrisability of path geometries by Álvarez Paiva and Berck [1] show that this is indeed the case. Of course, one can ask analogue questions also for higher dimensional solitons. Before we do that, we generalise the notion of soliton solutions slightly.

Note that the \(\mathbf{0}\)-pseudosoliton hypersurfaces are the minimal hypersurfaces of \((M,g)\). It was observed in [13] (see also [7]) that solitons with respect to gradient vector fields correspond to minimal hypersurfaces. However it was left open if such a correspondence holds when the vector field is not the gradient of a smooth function. In this short article we provide an answer using the framework of Monge-Ampère differential systems.

In §2 we will associate to the \(\mathbf{X}\)-pseudosoliton hypersurface equation on \((M,g)\) a Monge-Ampère system on the unit tangent bundle of \(M\) whose Legendre integral manifolds, which satisfy a natural transversality condition, locally correspond to \(\mathbf{X}\)-pseudosoliton hypersurfaces on \(M\). We then show that for a gradient vector field \(\mathbf{X}=\nabla_g u\) on \(M\), the \(\mathbf{X}\)-pseudosoliton M.A.S. is equivalent to the minimal hypersurface M.A.S. on \((M,e^{-2u}g)\). This was already shown in [13], albeit expressed in different language. We complete the picture by proving the

Theorem 2.3 is wrong for \(n=1\), i.e. the case of curves on surfaces. We provide counterexamples and comment on the necessary and sufficient conditions for \(\mathbf{X}\) in the surface case. Theorem 2.3 provides an answer to the equivalence problem for specific M.A.S. in arbitrary dimension \(n+1\geqslant 3\). The equivalence problem for general M.A.S. has been studied for \(5\)-dimensional contact manifolds in [5] and in various low dimensions in [11].

Throughout the article manifolds are assumed to be connected and smoothness, i.e. infinite differentiability is assumed.

Acknowledgements

The second author is grateful to Robert Bryant for helpful discussions. Research for this article was carried out while the authors were supported by the Swiss National Science Foundation, the first author by the grant 200020-124668 and the second by the postdoctoral fellowship PBFRP2-133545.