A Delaunay-type classification result for prescribed mean curvature surfaces in $\mathbb{M}^2(κ)\times\mathbb{R}$
Antonio Bueno









A Delaunay-type classification result for prescribed mean curvature surfaces in 𝕄^2(κ)×
[
Mathematics Subject Classification: 53A10.

Keywords: Prescribed mean curvature, product space, rotational surface, existence of spheres, Delaunay-type classification.

The author was partially supported by MICINN-FEDER Grant No. MTM2016-80313-P, Junta de Andalucía Grant No. FQM325 and FPI-MINECO Grant No. BES-2014-067663.]


  Antonio Bueno






Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain. 
 
E-mail address: jabueno@ugr.es 



The purpose of this paper is to study immersed surfaces in the product spaces , whose mean curvature is given as a C^1 function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.




§ INTRODUCTION



Let $̋ be aC^1function defined on the 2-sphere of the Euclidean space^3. An immersed, oriented surfacein3̊is said to have prescribed mean curvature$̋ (for short,  is an $̋-surface) if its mean curvature functionH_satisfies

    H_(p)=(̋η_p),∀ p∈,

whereηdenotes the Gauss map of.  Obviously, when=̋H_0is chosen as a constant, the surfaces defined by Equation (<ref>) are just the surfaces with constant mean curvature equal toH_0. 

The definition of this class of immersed surfaces is motivated by a long standing conjecture due to Alexandrov <cit.> regarding the uniqueness of strictly convex spheres[By sphere we mean a closed (compact and without boundary) surface of genus zero.] with prescribed Weingarten curvature, i.e. in Equation (<ref>) the function$̋ is an arbitrary symmetric function of its principal curvatures and its Gauss map. This conjecture has been recently solved by Gálvez and Mira as a consequence of their outstanding work <cit.>, where the authors announced an extremely general Hopf-type theorem[In the literature, a Hopf-type theorem refers to a uniqueness result of immersed spheres in some class of immersed surfaces] for immersed surfaces governed by an elliptic PDE in an arbitrary oriented three-manifold.  For the particular but important case when we prescribe the mean curvature, that is when  is governed by Equation (<ref>), this uniqueness result states the following: let 𝒮 be a strictly convex sphere satisfying (<ref>). Then, any other immersed sphere governed by (<ref>) is a translation of 𝒮.

Besides the work of Guan and Guan <cit.>, where they proved the existence of $̋-spheres under symmetry conditions on$̋, the class of immersed surfaces in 3̊ defined by Equation (<ref>) was largely unexplored until we recently developed the global theory of prescribed mean curvature surfaces in ^n, in joint work with Gálvez and Mira <cit.>. In <cit.> we focused in the global theory of $̋-surfaces in3̊, obtaining a priori curvature and height estimates for, a structure theorem for properly embeddedwith finite topology, stability properties and a radius estimate for stable. In <cit.> we covered topics such as the analysis of rotational$̋-hypersurfaces obtaining Delaunay-type classification result, and exhibited examples of a vast variety of $̋-hypersurfaces for general choices of the prescribed function. 

A particular but important case is when∈̋C^1(^2)depends only on the height of the sphere. These functions are called rotationally symmetric for obvious reasons, and thus there exists a one dimensional function𝔥∈C^1([-1,1])such that(̋x)=𝔥(⟨x,e_3⟩), for everyx∈^2. For this particular case, Equation (<ref>) reads as

    H_(p)=𝔥(⟨η_p,e_3⟩)=𝔥(ν(p)),∀ p∈,

where

    ν:→,ν(p):=⟨η_p,e_3⟩,∀ p∈,

is the so called angle function.

Our aim in this paper is to extend this recently developed theory ofto the product spaces, where^2(κ)stands for the complete, simply connected surface of constant curvatureκ. We take as a starting point the natural mixture between the well-studied theory of constant mean curvature surfaces immersed in these product spaces and the theory ofin3̊.

The theory of immersed surfaces inhas experimented an extraordinary development since Abresch and Rosenberg <cit.> defined a holomorphic quadratic differential on every constant mean curvature surface, that vanishes on rotational examples. This quadratic differential, called in the literature the Abresch-Rosenberg differential, enabled the authors to extend the so called Hopf theorem: an immersion of a constant mean curvature sphere inis a rotational, embedded sphere. This milestone was the starting point for the growth of a fruitful theory of positive, constant mean curvature surfaces (CMC surfaces in the following) in; see <cit.> for a global picture of the development of this theory.

When trying to extend Equation (<ref>) to the product spaces, we find out two major difficulties: the spaces^2(κ)×do not carry a Lie group structure, and thus we are not able to define a left-invariant Gauss map in order to prescribe some function on a fixed sphere, just like in Equation (<ref>). The difficulty when trying to extend this theory to the spacesℍ^2(κ)×comes from the fact that they have two non-isometric Lie group structures: one unimodular and other non-unimodular, see <cit.> for details.

Nonetheless, inwe have a notion of angle function as well, defined by measuring the projection of a unit normal vector fieldηofonto the vertical Killing vector field∂_z, just like in Equation (<ref>). Bearing this in mind, we can define the following class of immersed surfaces in:

Let be ∈̋C^1([-1,1]). An immersed, oriented surface  in ^2(κ)× has prescribed mean curvature$̋ if its mean curvatureH_satisfies

    H_(p)=(̋ν(p)),∀ p∈,

whereν(p):=⟨η_p,∂_z⟩is the angle function ofandηis a unit normal vector field on.

In analogy with the Euclidean case, we will simply say thatis an$̋-surface.

Besides the trivial choice of $̋ as a constant, there are other prescribed functions that generate some known classes of immersed surfaces in. Indeed, if we consider the function(̋x)=x, ∀ x∈ [-1,1], the class of immersed surfaces described by Equation (<ref>) are the self-translating solitons of the mean curvature flow (MCF for short); see <cit.> for a recent development of this theory. These particular, almost trivial, choices of the prescribed function$̋ and the classes of immersed surfaces generated by them, show the richness and wideness of the family of .

The rest of the introduction is devoted to detail the organization of the paper, and highlight some of the main results.

In Section <ref> we will exhibit some basic properties of immersed  in the product spaces. We will show that  obey the geometric maximum principle, as they are locally governed by an elliptic, second order, quasilinear PDE. We make special emphasis in the ambient isometries that preserve the class of ; any such isometry must keep invariant the angle function in order to preserve Equation (<ref>). Specifically, in Lemma <ref> we will show that, for an arbitrary prescribed function $̋, almost all the isometries of the spacesare induced as isometries for the class of.

When studying the properties of CMC surfaces in the product spaces, one of the key tools is the existence of a sphere with the same mean curvature. This is the motivation for the contents of Section <ref>, where we take advantage of the symmetries of Equation (<ref>) to study rotationalimmersed in. We should emphasize that the arbitrariness of the prescribed function$̋ prevents us from obtaining a first integral to study these rotational examples. 

In the same fashion as in Section 2 in <cit.>, we approach this study by means of a phase plane analysis of the resulting ODE that the profile curve of a rotational  satisfies. In Section <ref> we obtain some necessary conditions on $̋ in order to ensure the existence of an$̋-sphere. We also prove in Theorem <ref> that a rotational $̋-sphere has monotonous angle function, and thus is unique among all immersed$̋-spheres due to Gálvez and Mira uniqueness theorem <cit.>. 

Finally, in Section <ref> we give a classification of complete, rotational , provided that ; see Equation (<ref>) for a definition of the space ℭ^1_κ. In Theorem <ref> we prove the existence of a rotational, embedded $̋-sphere, and in Theorem <ref> we announce a Delaunay-type classification result for. In the same fashion as in the CMC case, every complete rotationalis either an$̋-sphere, a vertical circular cylinder,  a properly embedded surface of unduloid type or a properly immersed (non-embedded) surface of nodoid type. Moreover, in analogy to the CMC case, in the space 2r there also exist rotational, embedded $̋-surfaces diffeomorphic to^1×^1, i.e. rotational, embedded $̋-tori.





§  BASIC PROPERTIES OF $̋-SURFACES IN THE PRODUCT SPACES


LET^2(Κ)BE THE COMPLETE, SIMPLY CONNECTED SURFACE WITH CONSTANT CURVATUREΚ. THEN, UP TO ISOMETRIES,^2(Κ)IS ONE OF THE FOLLOWING SURFACES: IFΚ=0WE GET THE USUAL FLAT PLANE^2; IFΚ<0WE OBTAIN THE HYPERBOLIC PLANEℍ^2(1/√(-Κ)); AND IFΚ>0WE HAVE THE 2-SPHERE^2(1/√(Κ)). WE DROP OUT THE CASEΚ=0, SINCE THE THEORY OF IMMERSEDIN^3HAS BEEN WIDELY STUDIED IN <CIT.>. WHENΚ≠ 0, THESE NON-FLAT SURFACES CAN BE REGARDED AS ISOMETRIC IMMERSIONS IN THE SPACE^3_Κ, WHERE^3_ΚSTANDS FOR THE USUAL EUCLIDEAN SPACE IFΚ>0, OR FOR THE LORENTZ-MINKOWSKI SPACE𝕃^3(THAT IS,3̊ENDOWED WITH THE METRIC WITH SIGNATURE+,+,-) IFΚ<0. INDEED, THE SURFACE^2(Κ)CAN BE DEFINED AS THE QUADRIC

    ^2(Κ)={(X_1,X_2,X_3)∈^3_Κ; X_1^2+X_2^2+Κ X_3^2=1/Κ, (1-Κ)X_3≥ 0}.


UP TO AN HOMOTHETIC CHANGE OF THE METRIC, WE WILL SUPPOSE THATΚ=± 1. HENCEFORTH, WE WILL DROP THE DEPENDENCE ONΚAND JUST WRITE^2.

THE PRODUCT SPACES2RARE DEFINED AS THE RIEMANNIAN PRODUCT OF THE SURFACE^2WITH THE REAL LINE, ENDOWED WITH THE USUAL PRODUCT METRIC. IN2RWE HAVE THE TWO USUAL PROJECTIONSΠ_1:2R→^2ANDΠ_2:2R→. THE HEIGHT FUNCTION IS DEFINED TO BE THE SECOND PROJECTION, AND IS COMMONLY DENOTED BYZ(P):=Π_2(P)FOR ALLP∈2R. THE GRADIENT OF THE HEIGHT FUNCTION IS A UNITARY KILLING VECTOR FIELD, WHICH IS USUALLY DENOTED IN THE LITERATURE BY∂_Z, AND THE DIRECTION GENERATED BY∂_ZIS CALLED THE VERTICAL DIRECTION. THE PROJECTIONΠ_1IS A RIEMANNIAN SUBMERSION, WHOSE FIBERS ARE THE VERTICAL LINES{P}×:=Π_1^-1(P), FORP∈^2. THE VERTICAL PLANES ARE THE SURFACES GIVEN BYΓ×:=Π_1^-1(Γ), WHEREΓ⊂^2IS A GEODESIC, WHICH ARE TOTALLY GEODESIC SURFACES ISOMETRIC TO^2. THE HORIZONTAL PLANES ARE THE SURFACES GIVEN BY^2×{T_0}:=Π_2^-1(T_0), T_0∈, WHICH ARE TOTALLY GEODESIC SURFACES ISOMETRIC TO^2.

LETBE ANANDΗA UNIT NORMAL VECTOR FIELD ON, AND TAKE SOMEP∈. SUPPOSE THATΗ_PIS NOT AN HORIZONTAL VECTOR. THUS, THE IMPLICIT FUNCTION THEOREM ENSURES US THAT A NEIGHBORHOOD OFPINCAN BE EXPRESSED AS A VERTICAL GRAPH(X,U(X))OF A FUNCTIONU:Ω⊂^2→. IN THIS SITUATION, EQUATION (<REF>) HAS THE FOLLOWING DIVERGENCE EXPRESSIONDIV_^2(∇^^2 U/√(1+|∇^^2 U|^2)) = 2(1/√(1+|∇^^2U|^2)),WHERE DIV_^2AND∇^^2ARE THE DIVERGENCE AND GRADIENT OPERATORS, BOTH COMPUTED W.R.T. THE METRIC OF^2. IFΗ_PIS HORIZONTAL, THENCAN BE EXPRESSED AS A HORIZONTAL GRAPH WHICH ALSO SATISFIES A DIVERGENCE-TYPE EQUATION, SEE E.G. SECTION 5 IN <CIT.>. IN PARTICULAR,SATISFY THE HOPF MAXIMUM PRINCIPLE IN ITS INTERIOR AND BOUNDARY VERSIONS, A PROPERTY THAT HAS THE FOLLOWING GEOMETRIC IMPLICATION: