Invariant hypersurfaces with linear prescribed mean curvature
Introduction
Let us consider an oriented hypersurface Σ immersed into whose mean curvature is denoted by and its Gauss map by . Following [5], given a function , Σ is said to be a hypersurface of prescribed mean curvature if for every point . Observe that when the prescribed function is constant, Σ is a hypersurface of constant mean curvature (CMC).
It is a classical problem in Differential Geometry the study of hypersurfaces in whose principal curvatures and its Gauss map η satisfy a prescribed relation of the form . 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 , i.e. with prescribed Gauss-Kronecker curvature, while the Christoffel problem focuses on the prescribed relation . Note that for the function , 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 has been unexplored for general choices of 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 . The same authors have also studied rotational hypersurfaces in , 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 , 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 a linear function, that is, for every , where and v is a unit vector called the density vector. Note that if we are studying hypersurfaces with constant mean curvature equal to λ. Moreover, if , 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 in , we can get 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 is an -hypersurface if its mean curvature function is given by
The relevance of the class of 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 with respect to the density , the weighted mean curvature of Σ is defined by where ∇ is the gradient operator of . Note that when the density is , by using (1.2) and (1.3) it follows that Σ is an if and only if . In particular, as pointed out by Ilmanen [15], self-translating solitons are weighted minimal, i.e. . On the other hand, although hypersurfaces of prescribed mean curvature do not come in general associated to a variational problem, the do. To be more specific, consider any measurable set having as boundary and inward unit normal η along Σ. Then, the weighted area and volume of Ω with respect to the density are given respectively by where dΣ and dV are the usual area and volume elements in . 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 , where Finally, observe that if is an , the family of translations of f in the v direction given by is the solution of the geometric flow 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 which are invariant under the flow of an -group of translations and the isometric -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 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 in [5], the classification of neither cylindrical nor rotational in [6] was covered.
We next detail the organization of the paper. In Section 2 we study complete that have constant curvature. Firstly, we study in detail that, from classical theorems of Liebmann, Hilbert and Hartman-Nirenberg, any such must be flat, hence invariant by an -group of translations and described as the riemannian product , 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 with the geometry of α. Indeed, we prove that the curvature is, essentially, the mean curvature of . Finally, in Theorem 2.2 we classify such -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 . In particular, we treat the ODE that the profile curve of a rotational 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 . Firstly, if these 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 and the complete, rotational intersecting the rotation axis with upwards and downwards orientation respectively. Then: For any , is properly embedded, simply connected and converges to the CMC cylinder of radius . Moreover: If , intersects infinitely many times. If , intersects a finite number of times and is a graph outside a compact set. If , is a proper graph over the ball of radius .
For , is properly immersed (with infinitely-many self-intersections), simply connected and has unbounded distance to the rotation axis.
For , is a horizontal hyperplane.
For , is a strictly convex, entire graph.
Secondly, for staying away from the axis of rotation, we state that: Theorem 1.3 Let Σ be a complete, rotational non-intersecting the rotation axis. Then: either Σ is the CMC cylinder of radius , or Σ is properly immersed and diffeomorphic to . One end converges to with the same asymptotic behavior as in item 1 in Theorem 1.2, and: If , the other end has infinitely-many self-intersections and unbounded distance to the rotation axis. If , the other end is a graph outside a compact set.
For the very particular case that , both results agree with the ones obtained in [17].
Section snippets
Constant curvature -hypersurfaces
The aim of this section is to obtain a classification result for complete with constant curvature . Let us observe that not every value of the curvature is admissible. Indeed, by Theorem 47 in [22] we see that , and so no of negative constant curvature exist in , even locally. This result is a generalization of Hilbert's celebrated theorem [12]. If , then the is totally umbilic and so it is a round sphere. This result generalizes
The phase plane of rotational -hypersurfaces
This section is devoted to review the main features of the phase plane for the study of rotational . 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 , where there was no a priori relation between the density vector and the ruling directions, for a rotational 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 . In particular, the linearized of (3.3) at is given by whose eigenvalues are
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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