Invariant hypersurfaces with linear prescribed mean curvature

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Abstract

Our aim is to study invariant hypersurfaces immersed in the Euclidean space Rn+1, whose mean curvature is given as a linear function in the unit sphere Sn depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

Introduction

Let us consider an oriented hypersurface Σ immersed into Rn+1 whose mean curvature is denoted by HΣ and its Gauss map by η:ΣSnRn+1. Following [5], given a function HC1(Sn), Σ is said to be a hypersurface of prescribed mean curvature H ifHΣ(p)=H(ηp), for every point pΣ. Observe that when the prescribed function H is constant, Σ is a hypersurface of constant mean curvature (CMC).

It is a classical problem in Differential Geometry the study of hypersurfaces in Rn+1 whose principal curvatures κ1,...,κn and its Gauss map η satisfy a prescribed relation of the form Φ(κ1,...,κn)=F(η),FC1(Sn),ΦC1(Rn). This fruitful theory goes back, at least, to the famous Minkowski and Christoffel problem for ovaloids [7], [20]. Specifically, the Minkowski problem studies the existence and uniqueness of ovaloids satisfying κ1κn=F(η), i.e. with prescribed Gauss-Kronecker curvature, while the Christoffel problem focuses on the prescribed relation 1/κ1++1/κn=F(η). Note that for the function Φ(κ1,...,κn)=(κ1++κn)/n, the hypersurfaces arising are the ones governed by (1.1). For them, the existence and uniqueness of ovaloids was studied, among others, by Alexandrov and Pogorelov in the '50s, [1], [21], and more recently by Guan and Guan in [10]. Nevertheless, the global geometry of complete, non-compact hypersurfaces of prescribed mean curvature in Rn+1 has been unexplored for general choices of H until recently. In this framework, the first author jointly with Gálvez and Mira have started to develop the global theory of hypersurfaces with prescribed mean curvature in [5], taking as a starting point the well-studied global theory of CMC hypersurfaces in Rn+1. The same authors have also studied rotational hypersurfaces in Rn+1, getting a Delaunay-type classification result and several examples of rotational hypersurfaces with further symmetries and topological properties (see [6]). For prescribed mean curvature surfaces in R3, see [3] for the resolution of the Björling problem and [4] for the obtention of half-space theorems for properly immersed surfaces.

Our objective in this paper is to further investigate the geometry of complete hypersurfaces of prescribed mean curvature for a relevant choice of the prescribed function. In particular, let us consider HC1(Sn) a linear function, that is,H(x)=ax,v+λ for every xSn, where a,λR and v is a unit vector called the density vector. Note that if a=0 we are studying hypersurfaces with constant mean curvature equal to λ. Moreover, if λ=0, we are studying self-translating solitons of the mean curvature flow, case which is widely studied in the literature (see e.g. [8], [14], [15], [19], [23] and references therein). Therefore, we will assume that a and λ are not null in order to avoid these cases. Furthermore, after a homothety of factor 1/a in Rn+1, we can get a=1 without loss of generality. Bearing in mind these considerations, we focus on the following class of hypersurfaces.

Definition 1.1

An immersed, oriented hypersurface Σ in Rn+1 is an Hλ-hypersurface if its mean curvature function HΣ is given byHΣ(p)=Hλ(ηp)=ηp,v+λ,pΣ.

See that if Σ is an Hλ-hypersurface with Gauss map η, then Σ with the opposite orientation −η is trivially an Hλ-hypersurface. Thus, up to a change of the orientation, we assume λ>0.

The relevance of the class of Hλ-hypersurfaces lies in the fact that they satisfy some characterizations which are closely related to the theory of manifolds with density. Firstly, following Gromov [9], for an oriented hypersurface Σ in Rn+1 with respect to the density eϕC1(Rn+1), the weighted mean curvature Hϕ of Σ is defined byHϕ:=HΣη,ϕ, where ∇ is the gradient operator of Rn+1. Note that when the density is ϕv(x)=x,v, by using (1.2) and (1.3) it follows that Σ is an Hλ-hypersurface if and only if Hϕv=λ. In particular, as pointed out by Ilmanen [15], self-translating solitons are weighted minimal, i.e. Hϕv=0. On the other hand, although hypersurfaces of prescribed mean curvature do not come in general associated to a variational problem, the Hλ-hypersurfaces do. To be more specific, consider any measurable set ΩRn+1 having as boundary Σ=Ω and inward unit normal η along Σ. Then, the weighted area and volume of Ω with respect to the density ϕv are given respectively byAϕv(Σ):=ΣeϕvdΣ,Vϕv(Ω):=ΩeϕvdV, where dΣ and dV are the usual area and volume elements in Rn+1. So, in [2] it is proved that Σ has constant weighted mean curvature equal to λ if and only if Σ is a critical point under compactly supported variations of the functional Jϕv, whereJϕv:=AϕvλVϕv. Finally, observe that if f:ΣRn+1 is an Hλ-hypersurface, the family of translations of f in the v direction given by F(p,t)=f(p)+tv is the solution of the geometric flow(Ft)=(HΣλ)η, which corresponds to the mean curvature flow with a constant forcing term, that is, f is a self-translating soliton of the geometric flow (1.4). This flow already appeared when studying the volume preserving mean curvature flow, introduced by Huisken [13].

Throughout this work we focus our attention on Hλ-hypersurfaces which are invariant under the flow of an (n1)-group of translations and the isometric SO(n)-action of rotations that pointwise fixes a straight line. The first group of isometries generates cylindrical flat hypersurfaces, while the second one corresponds to rotational hypersurfaces. These isometries and the symmetries inherited by the invariant Hλ-hypersurfaces are induced to Equation (1.2) easing the treatment of its solutions. We must emphasize that, although the authors already defined the class of immersed Hλ-hypersurfaces in [5], the classification of neither cylindrical nor rotational Hλ-hypersurfaces in [6] was covered.

We next detail the organization of the paper. In Section 2 we study complete Hλ-hypersurfaces that have constant curvature. Firstly, we study in detail that, from classical theorems of Liebmann, Hilbert and Hartman-Nirenberg, any such Hλ-hypersurface must be flat, hence invariant by an (n1)-group of translations and described as the riemannian product α×Rn1, where α is a plane curve called the base curve. After that, we note that this product structure allows us to relate the condition of being an Hλ-hypersurface with the geometry of α. Indeed, we prove that the curvature κα is, essentially, the mean curvature of α×Rn1. Finally, in Theorem 2.2 we classify such Hλ-hypersurfaces by giving explicit parametrizations of the base curve α depending on the value of λ and the density vector.

Later, in Section 3 we introduce the phase plane for the study of rotational Hλ-hypersurfaces. In particular, we treat the ODE that the profile curve of a rotational Hλ-hypersurface satisfies as a non-linear autonomous system since the qualitative study of the solutions of this system will be carried out by a phase plane analysis, as the first author did jointly with Gálvez and Mira in [6].

To finish, in Section 4 we prove our two main results in which we get a classification of complete rotational Hλ-hypersurfaces. Firstly, if these Hλ-hypersurfaces intersect the axis of rotation, we see that they must do it orthogonally and we get the first classification result:

Theorem 1.2

Let be Σ+ and Σ the complete, rotational Hλ-hypersurfaces intersecting the rotation axis with upwards and downwards orientation respectively. Then:

  • 1.

    For any λ>0, Σ+ is properly embedded, simply connected and converges to the CMC cylinder C(n1λn) of radius n1λn. Moreover:

    • 1.1.

      If λ>n1/2, Σ+ intersects C(n1λn) infinitely many times.

    • 1.2.

      If λ=n1/2, Σ+ intersects C(n1λn) a finite number of times and is a graph outside a compact set.

    • 1.3.

      If λ<n1/2, Σ+ is a proper graph over the ball of radius n1λn.

  • 2.

    For λ>1, Σ is properly immersed (with infinitely-many self-intersections), simply connected and has unbounded distance to the rotation axis.

  • 3.

    For λ=1, Σ is a horizontal hyperplane.

  • 4.

    For λ<1, Σ is a strictly convex, entire graph.

Secondly, for Hλ-hypersurfaces staying away from the axis of rotation, we state that:

Theorem 1.3

Let Σ be a complete, rotational Hλ-hypersurface non-intersecting the rotation axis. Then:

  • 1.

    either Σ is the CMC cylinder C(n1λn) of radius n1λn, or

  • 2.

    Σ is properly immersed and diffeomorphic to Sn1×R. One end converges to C(n1λn) with the same asymptotic behavior as in item 1 in Theorem 1.2, and:

    • 2.1.

      If λ>1, the other end has infinitely-many self-intersections and unbounded distance to the rotation axis.

    • 2.2.

      If λ1, the other end is a graph outside a compact set.

Moreover, if λ<1 and the unit normal of Σ at the points with horizontal tangent hyperplane is en+1, then Σ is embedded.

For the very particular case that n=2, both results agree with the ones obtained in [17].

Section snippets

Constant curvature Hλ-hypersurfaces

The aim of this section is to obtain a classification result for complete Hλ-hypersurfaces with constant curvature K0. Let us observe that not every value of the curvature is admissible. Indeed, by Theorem 47 in [22] we see that K00, and so no Hλ-hypersurfaces of negative constant curvature exist in Rn+1, even locally. This result is a generalization of Hilbert's celebrated theorem [12]. If K0>0, then the Hλ-hypersurface is totally umbilic and so it is a round sphere. This result generalizes

The phase plane of rotational Hλ-hypersurfaces

This section is devoted to review the main features of the phase plane for the study of rotational Hλ-hypersurfaces. To do so we follow [6], where the phase plane was used to study rotational hypersurfaces of prescribed mean curvature given by Equation (1.1).

Let us fix the notation. Firstly, observe that in contrast with cylindrical Hλ-hypersurfaces, where there was no a priori relation between the density vector and the ruling directions, for a rotational Hλ-hypersurface the density vector and

Proofs of Theorems 1.2 and 1.3

This section is devoted to prove Theorem 1.2, Theorem 1.3 at the same time.

We begin by analyzing the qualitative properties of system (3.3), most of them already studied in the previous section. First, it is useful to study its linearized system at the unique equilibrium e0=(n1λn,0). In particular, the linearized of (3.3) at e0 is given by(01n2λ2n1n), whose eigenvalues areμ1=n+n14λ2n12,andμ2=nn14λ2n12.

Standard theory of non-linear autonomous systems enables us to summarize the

Acknowledgments

The first author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P. For the second author, this research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. Irene Ortiz was partially supported by MICINN/FEDER project PGC2018-097046-B-I00 and Fundación Séneca project 19901/GERM/15, Spain.

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