Invariant hypersurfaces with linear prescribed mean curvature Antonio Buenoy, Irene Ortizz yDepartamento de Geometr´ıay Topolog´ıa,Universidad de Granada, E-18071 Granada, Spain. E-mail address: [email protected] zDepartamento de Ciencias e Inform´atica,Centro Universitario de la Defensa de San Javier, E-30729 Santiago de la Ribera, Spain. E-mail address: [email protected] 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. Contents 1: Introduction1 2: Constant curvature λ-hypersurfaces3 H 3: The phase plane of rotational λ-hypersurfaces7 H 4: Classification of rotational λ-hypersurfaces8 H 5: References 21 1 Introduction n Let us consider an oriented hypersurface Σ immersed into R +1 whose mean curvature is denoted n n+1 1 n by HΣ and its Gauss map by η :Σ S R . Following [BGM1], given a function C (S ), ! ⊂ H 2 Σ is said to be a hypersurface of prescribed mean curvature if H arXiv:1908.07378v1 [math.DG] 20 Aug 2019 H (p) = (ηp); (1.1) Σ H for every point p Σ. Observe that when the prescribed function is constant, Σ is a hypersurface 2 H of constant mean curvature (CMC). It is a classical problem in the Differential Geometry the study of hypersurfaces which are defined by means of a prescribed curvature function in terms of the Gauss map, being remarkable the Minkowski and Christoffel problems for ovaloids ([Min, Chr]). In particular, when such prescribed function is the mean curvature, 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 Mathematics Subject Classification: 53A10, 53C42, 34C05, 34C40. Keywords: Prescribed mean curvature hypersurface, weighted mean curvature, non-linear autonomous system. 1 in the '50s, [Ale, Pog], and more recently by Guan and Guan in [GuGu]. Nevertheless, the global n geometry of complete, non-compact hypersurfaces of prescribed mean curvature in R +1 has been unexplored for general choices of until recently. In this framework, the first author jointly with H G´alvez and Mira have started to develop the global theory of hypersurfaces with prescribed mean curvature in [BGM1], taking as a starting point the well-studied global theory of CMC hypersurfaces n n in R +1. The same authors have also studied rotational hypersurfaces in R +1, getting a Delaunay- type classification result and several examples of rotational hypersurfaces with further symmetries and topological properties (see [BGM2]). For prescribed mean curvature surfaces in R3, see [Bue1] for the resolution of the Bj¨orlingproblem and [Bue2] for the obtention of half-space theorems 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 n consider C1(S ) a linear function, that is, H 2 (x) = a x; v + λ H h i n for every x S , where a; λ R and v is a unit vector called the density vector. Note that if a = 0 2 2 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. [CSS, HuSi, Ilm, MSHS, SpXi] and references therein). Therefore, we will assume that a and λ are not null in order to avoid the trivial cases. Furthermore, after a homothety of factor n 1=a in R +1, we can get a = 1 without loss of generality. Bearing in mind these considerations, we focus on the following class of hypersurfaces. n+1 Definition 1.1 An immersed, oriented hypersurface Σ in R is an λ-hypersurface if its mean H curvature function HΣ is given by H (p) = λ(ηp) = ηp; v + λ, p Σ: (1.2) Σ H h i 8 2 See that if Σ is an λ-hypersurface with Gauss map η, then Σ with the opposite orientation η is H − trivially a λ-hypersurface. Thus, up to a change of the orientation, we assume λ > 0. H− The relevance of the class of λ-hypersurfaces lies in the fact that they satisfy some characteri- H zations which are closely related to the theory of manifolds with density. Firstly, following Gromov n φ n [Gro], for an oriented hypersurface Σ in R +1 with respect to the density e C1(R +1), the 2 weighted mean curvature Hφ of Σ is defined by Hφ := H η; φ ; (1.3) Σ − h r i where is the gradient operator of M. Note that when the density is φv(x) = x; v , by using (1.2) r h i and (1.3) it follows that Σ is an λ-hypersurface if and only if Hφ = λ. In particular, as pointed H v out by Ilmanen [Ilm], 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 λ-hypersurfaces do. To be more specific, consider any measurable set n H Ω R +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 by Z Z φv φv Aφv (Σ) := e dΣ;Vφv (Ω) := e dV; Σ Ω 2 n where dΣ and dV are the usual area and volume elements in R +1. So, in [BCMR] 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 , where Jφ := Aφ λVφ : v v − v n+1 Finally, observe that if f :Σ R is an λ-hypersurface, the family of translations of f in the v ! H direction given by F (p; t) = f(p) + tv is the solution of the geometric flow @F ? = (H λ)η; (1.4) @t Σ − 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 in the context of studying the volume preserving mean curvature flow, introduced by Huisken [Hui]. Throughout this work we focus our attention on λ-hypersurfaces which are invariant under the H flow of an (n 1)-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 in- herited by the invariant λ-hypersurfaces are induced to Equation (1.2) easing the treatment of H its solutions. We must emphasize that, although the authors already defined the class of immersed λ-hypersurfaces in [BGM1], the classification of neither cylindrical nor rotational λ-hypersurfaces H H in [BGM2] was covered. We next detail the organization of the paper. In Section2 we study complete λ-hypersurfaces H that have constant curvature. By classical theorems of Liebmann, Hilbert and Hartman-Nirenberg, any such λ-hypersurface must be flat, hence invariant by an (n 1)-group of translations and H n 1 − described as the riemannian product α R − , where α is a plane curve called the base curve. This × product structure allows us to relate the condition of being an λ-hypersurface with the geometry H n 1 of α. Indeed, the curvature κα is, essentially, the mean curvature of α R − . In Theorem 2.1 we × classify such λ-hypersurfaces by giving explicit parametrizations of the base curve. H Later, in Section3 we introduce the phase plane for the study of rotational λ-hypersurfaces. H In particular, we treat the ODE that the profile curve of a rotational λ-hypersurface satisfies as H 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 G´alvez and Mira in [BGM2]. Finally, in Section4 we give a complete classification of rotational λ-hypersurfaces intersecting the H axis of rotation in Theorem 4.3 and non-intersecting such axis in Theorem 4.4. To get such results we develop along this section a discussion depending on the value of λ, namely λ > 1; λ = 1 and λ < 1. 2 Constant curvature λ-hypersurfaces H The aim of this section is to obtain a classification result for complete λ-hypersurfaces with H constant curvature. By classical theorems of Liebmann, Hilbert, and Hartman-Nirenberg, any such λ-hypersurface must be flat, hence invariant by an (n 1)-parameter group of translations H n+1 − a1;:::;an−1 = Ft1;:::;tn−1 ; ti R where ai R with i = 1; :::; n 1, are linearly independent G f 2 g 2 − 3 Pn 1 n+1 and Ft1;:::;tn−1 (p) = p + − tiai, for every p R . Any λ-hypersurface invariant by such a i=1 2 H group is called a cylindrical flat λ-hypersurface, and the directions a1; :::; an 1 are known as ruling H − directions.