Complete Constant Mean Curvature Hypersurfaces in Euclidean space of dimension four or higher
In this article we provide a general construction when for immersed in Euclidean -space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC -hypersurfaces). More precisely our construction converts certain graphs in Euclidean -space to CMC -hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC -hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter . This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given and it allows us to realize infinitely many topological types as CMC -hypersurfaces in with ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when where the number of possible topological types for each is finite but tends to as .
MSC 53A05, 53C21.
Key words and phrases:Differential Geometry, constant mean curvature surfaces, partial differential equations, perturbation methods
Key words and phrases:Differential geometry and constant mean curvature surfaces and partial differential equations and perturbation methods
The general framework
Constant Mean Curvature (CMC) (hyper)surfaces in a Riemannian manifold can be described variationally as critical points of the induced intrinsic volume (or area in dimension two) functional, subject to an enclosed volume constraint. Alternatively they can be described as soap films (or fluid interfaces) in equilibrium under only the forces of surface tension and uniform enclosed pressure. In both cases the geometric condition is that the mean curvature of the hypersurface is constant as the name suggests.
Of particular interest are the complete CMC (hyper)surfaces of finite topological type smoothly immersed in Euclidean spaces and in particular in Euclidean three-space. The only classically known such examples were the round spheres, the cylinders, and more generally the rotationally invariant surfaces discovered by Delaunay in 1841 . Two major results were proved in the 1950’s characterizing the round two-spheres as the only closed CMC surface in Euclidean three-space, under the assumption of embeddedness (by Alexandrov ), or the assumption of zero genus (by Hopf ). These results and their methods of proof had a profound influence in Mathematics. They also led to the celebrated conjecture (or question according to some) by Hopf on whether the only immersed closed CMC surfaces in Euclidean three-space are round spheres. In 1986 Wente disproved the Hopf conjecture by constructing genus one closed immersed examples .
At that stage the only examples of finite topological type in Euclidean three-space were the classical ones and the Wente tori. Following a general gluing methodology developed by Schoen  and N.K. , and using the Delaunay surfaces as building blocks, most of the possible finite topological types were realized as immersed (or Alexandrov embedded) CMC surfaces for the first time [24, 25].  in particular settled the Hopf question for high genus closed surfaces by providing examples of any genus . In spite of its success the use of Delaunay pieces as building blocks has the limitation that it does not allow the construction of closed genus two CMC examples. In  a systematic and detailed refinement of the original gluing methodology made it possible to construct genus two (actually any genus ) closed examples with the Wente tori as building blocks. Since then, many other gluing problems have been successfully resolved by using this refined approach. These results include gluing constructions for special Lagrangian cones [12, 13, 11] and various gluing constructions for minimal surfaces [48, 32, 29, 28, 27, 30, 31].
It is worth pointing out that the constructions in [46, 24, 25] are quite general in two ways: First, in that each construction is reduced to finding graphs satisfying some rather general conditions. There is an abundance of such graphs and so a plethora of examples can be produced. Second, in that no symmetry is required—although it can be imposed in special cases—and indeed most examples constructed do not satisfy any symmetries. These constructions can serve then as a prototype for general constructions in other geometric settings—see [28, 29, 23].
We briefly mention that much progress has been made in the case of embedded, or more generally Alexandrov embedded, complete CMC surfaces of finite genus with ends. Meeks  proved that such (noncompact) surfaces have at least two ends and all their ends are cylindrically bounded. Motivated by [43, 24], Korevaar, Kusner, and Solomon  showed that each end converges exponentially fast to a Delaunay surface and if there are only two ends then the surface is Delaunay. Further progress in this direction was made in [34, 35] and also in understanding the moduli space of these surfaces as for example in . Moreover a significant success was that in some cases of genus zero, complete classification results were obtained with a satisfactory understanding of the surfaces involved [9, 10, 8].
We briefly also mention that various constructions extended the results of : Große-Brauckmann  used a conjugate surface construction to construct genus zero examples with ends under maximal (-fold dihedral) symmetry, including examples with large neck size for the first time. Various gluing constructions related to non-degeneracy [39, 45, 41, 42, 4, 22] were developed in certain cases, which allowed some new examples, in particular examples with asymptotically cylindrical ends , with noncatenoidal necks used as nodes instead of spheres , and a modified construction (end-to-end gluing) of the closed CMC examples [45, 22]. Recently the construction and estimates in  were refined in  by applying the improved methodology of . This way a large class of embedded examples was produced.  served also as in intermediate step for developing the high-dimensional constructions presented in this article.
Contrary to the case of Euclidean three-space very little is currently known in the case of higher-dimensional Euclidean spaces: Rotationally invariant CMC hypersurfaces analogous to the ones found by Delaunay have been constructed . In 1982 Hsiang  demonstrated that the theorem of Hopf does not extend to higher dimensions by constructing immersed CMC hyperspheres that are not round. Jleli has studied moduli spaces  and has developed an end-to-end gluing construction  which will provide new symmetric closed examples  when  appears. He also constructed examples bifurcating from the Delaunay-like ones .
Finally we briefly mention that constructions of CMC hypersurfaces have also been carried out in compact ambient manifolds under certain metric restrictions. Ye  provided the first such example, proving that there exists a foliation by CMC hyperspheres in a neighborhood of a non-degenerate critical point of the scalar curvature. Pacard and Xu  partially extended Ye’s result. Mazzeo and Pacard extended Ye’s result to geodesic tubes . Further constructions of CMC surfaces (two-dimensional) condensing around geodesic intervals or rays were provided in , and for CMC hypersurfaces condensing around higher dimensional submanifolds in .
Brief discussion of the results
In this article we extend the results of  to higher dimensions, that is to the construction of CMC -dimensional hypersurfaces in Euclidean -space for . Note that although the present proof and construction work for with small appropriate modifications, we restrict our attention to to simplify the presentation. For the same reason we restrict our attention to the construction of CMC hypersurfaces of finite topological type.
Our constructions as in [24, 2] are based on a suitable family of graphs which consists of small perturbations of a central graph (see 2.14). Our graphs have vertices, edges, rays, and nonzero weights assigned to the edges and the rays (see 2.1). is balanced in the sense that the resultant forces exerted on the vertices by the edges and rays vanish (see 2.6 and 2.9), and moreover its edges have even integer lengths. The other graphs in have approximately prescribed resultant forces (unbalancing condition) and prescribed small changes of the lengths of the edges (flexibility condition).
Given and a small nonzero a family of initial immersions is constructed, where the image of each such immersion is built around a properly chosen , and consists of unit spheres (with small geodesic balls removed) centered at the vertices of , and appropriately perturbed Delaunay pieces of parameter times the corresponding weight of . We have then the following.
Theorem 1.1 (Main Theorem).
Given a family of graphs , there exists such that for all , there exists a and an immersion built around as outlined above which admits a small graphical perturbation which has mean curvature . Moreover the immersion is an embedding if the central graph satisfies certain conditions (see 2.10) and .
Note that the conditions in 2.10 are the expected ones, that is they ensure that the various pieces stay away from each other and that the Delaunay pieces are embedded. It is easy then to realize infinitely many topological types as immersed complete CMC surfaces with ends, where any can be given in advance. These constructions (when no symmetries are imposed) have continuous parameters, reflecting thus the asymptotics of the Delaunay ends. Moreover there is further great variety in the immersions of a given number of ends and topological type reflected by the central graphs we can choose.
We can restrict our attention to embedded examples. In this case we could find examples with and then we have only finitely many topological types for each , with the number of topological types for each tending to as .
Outline of the approach
The construction in this article is an extension to high dimensions of the constructions in [24, 2]
with  serving also as an intermediate step in the development of this article.
The main difficulties and their resolution in extending to high dimensions are the following:
(1). A careful understanding of the geometry and analysis of the Delaunay hypersurfaces in high dimensions is needed, which to the best of our knowledge is new at least at this level of detail. In particular understanding their periods requires some work and is similar to work for special Legendrian submanifolds [14, 11].
(2). The conformal covariance of the Laplacian in dimension two is not available anymore. Moreover the linearized operator in dimension two can be formulated with respect to a conformal metric which compactifies the catenoidal necks in the limit and actually converts the catenoidal necks of the Delaunay surfaces into spherical regions isometric to the actual spherical regions, introducing thus new symmetries which did not exist in the induced metric ; all of this is unavailable in high dimensions. We resolved this difficulty by understanding the linearized equation on the catenoidal necks using Fourier decompositions on the meridians and some estimates. This is a simpler version of the approach in the analysis of the linearized equation on the (complicated and only approximately rotationally invariant) necks in . Note also that since we cannot compactify the necks we use appropriate weighted estimates.
(3). Since we do not use the end-to-end gluing idea which simplifies at the expense of limiting the scope of the construction, we still have to use the ideas of , modified for the high dimensions, to understand the linearized equation on the central—where the fusion with the Delaunay pieces occurs—spherical regions. We also use semi-localization, that is studying the linearized equation on the extended standard regions and combining the results.
(4). Because of the generality of the construction the whole scheme is quite involved. We tried to carefully organize the various steps so the whole structure of the proof is conceptually clear and easy to follow.
(5). We remark also that motivated by the geometric principle we achieve much faster decay away from the central spherical regions (compared to ), by introducing simple dislocations between the central spherical regions and the Delaunay pieces attached.
(6). Finally we remark that instead of monitoring the use of the extended substitute kernel at each step we chose to use a balancing formula  on the final hypersurface to estimate the unbalancing error because this seems to provide better control.
Organization of the presentation
Appendix A contains a thorough treatment of the essential information about the geometry of Delaunay surfaces. Appendix B provides standard background on the quadratic error estimates. Finally, in Appendix C we study the Dirichlet problem on a flat annulus.
Section 2 contains a description of the family of graphs which provides the structure for the immersion of the initial surfaces. We discuss the unbalancing and flexibility conditions and we associate to each graph in the family two parameters which give quantitative meaning to these conditions. In Section 3, we describe the building blocks of the construction, spheres with balls removed and Delaunay pieces with perturbations near their boundaries. The Delaunay building blocks are not necessarily CMC near their boundaries; the estimates are controlled by the parameters describing the perturbation. We are careful to describe these building blocks independent of any reference to a family of graphs. The building blocks depend only on general parameters and not on the structure of a graph. In Section 4 we study the linear operator on compact pieces of Delaunay surfaces. At this stage we choose a fixed large constant and a small depending on . For any , we consider regions on a Delaunay immersion with parameter . The size of the regions considered depend upon and and the choice of along with our understanding of the geometry of Delaunay surfaces provide good geometric estimates. Again, the statements and proofs of this section do not reference or rely on a graph or family of graphs.
In Section 5 we construct a family of initial surfaces which depend upon a parameter and a pair of parameters . We presume a given family of graphs . The parameter satisfies where depends upon and the graph but not on the structure of . The parameters and determine and thus a graph in the family . We build the initial surface by positioning and fusing building blocks at designated locations given by the structure of . The parameters describing the building blocks are encoded in , and the graph (but not the structure) of .
In Section 6 we study the linearized operator on the family of initial surfaces. We define the extended substitute kernel and solve the modified linear problem. Section 7 contains the prescribing of substitute and extended substitute kernel. We prove the Main Theorem in Section 8 using a fixed point theorem.
For , , a domain in a Riemannian manifold, , and we define the norm
Here is a geodesic ball centered at with radius in the metric . For simplicity, when or we may omit them from the notation.
Note from the definition that
If and , then we write
if and .
Throughout this paper we make extensive use of cut-off functions, and thus we adopt the following notation: Let be a smooth function such that
on and on
is an odd function.
For with , let be defined by where is a linear function with . Then has the following properties:
is weakly monotone.
on a neighborhood of and on a neighborhood of .
For a subset of a Riemannian manifold we write for the distance function from in . For we define a tubular neighborhood of by
In both cases we may omit or if understood from the context and if is finite we may just enumerate its points.
CB was supported in part by National Science Foundation grants DMS-1308420 and DMS-1609198. This material is also based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while CB was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester. NK would like to thank the Mathematics Department and the MRC at Stanford University for providing a stimulating mathematical environment and generous financial support during Fall 2011, Winter 2012 and Spring 2016. NK was also partially supported by NSF grants DMS-1105371 and DMS-1405537.
2. Finite Graphs
The gluing construction carried out in this article uses round spheres and pieces of Delaunay surfaces to build initial hypersurfaces which are then perturbed to become CMC hypersurfaces. The parameters of the Delaunay pieces and the positioning of the spheres and the Delaunay pieces are naturally encoded by graphs. In this article for simplicity we restrict ourselves to finite graphs which we discuss in this section. The initial graph we use should satisfy all of the relations one expects for a singular CMC surface and thus we impose a balancing restriction on each vertex and a restriction on the length of each edge. We first define the kind of graphs we will be using:
Definition 2.1 (Graphs).
We define a finite graph in for some to be a collection such that
is a finite collection of vertices.
is a finite collection of edges in , each with its two endpoints in .
is a finite collection of rays in , each with its one endpoint in .
is a function.
Given a finite graph , the input of a function or vector valued function of will be given by .
Definition 2.3 (Edge and Vertex Relations).
Let denote the collection of edges and rays that have as an endpoint. We have then
We also define the set of attachments
Finally for each we denote the unit vector pointing away from and in the direction of by .
For a graph , let denote the space of functions from to , let denote the space of functions from to , and let denote the space of functions from to . Equip each of these spaces with the maximum norm.
We define such that
measures the deviation from balancing at the vertex . Here and denotes as usual the -dimensional volume of .
We let such that for , equals the length of .
The constant will arise because of various normalizations throughout the argument. Absorbing it into the definition of will be convenient later.
Our construction will be based on a family of graphs that are perturbations of some fixed graph which we will call the central graph (see 2.9). The idea of the construction is to replace each edge or ray of by a Delaunay piece of parameter , where is a sufficiently small global parameter. (See Section 3 for a description of the Delaunay pieces.) The construction of the initial surfaces requires appropriate small perturbations of depending on and on other parameters. The central graph will be the limit of the graphs employed as . In this limit our surfaces will tend to tangentially touching unit spheres. Correspondingly, the period of the Delaunay surfaces will tend to . Therefore has to satisfy the condition that its edges have even integer length. Moreover the balancing conditions satisfied by CMC surfaces (see 7.1, (A.4), (A.3)) imply the vanishing of on . These considerations motivate the following definition.
Let be a finite graph. If for all , we say is a balanced graph. We call a central graph if is balanced and for all .
Finally, we define central graphs that guarantee that our construction produces an embedded CMC hypersurface:
Definition 2.10 (Pre-embedded graphs).
We say is pre-embedded if it is a central graph with and
For all and all , , where measures the angle between the two vectors .
For all that do not share any common endpoints, the Euclidean distance between is greater than .
For any two rays , .
For a pre-embedded and sufficiently small , each of the initial surfaces constructed from one of the possible perturbations of is embedded. In the singular setting, when , the angle condition between edges and rays about a fixed vertex allows for a singular surface with unit spheres touching tangentially. We do not require a strict inequality for this condition since the change in the period for small (on the order ) dominates both the radius change and the changes we allow via unbalancing and dislocation (on the order ). The second item requires a strict inequality as the maximum radius of an embedded Delaunay surface is on the order but we allow for the edges to move with order where can be quite large. The final condition also requires a strict inequality. Indeed if the central graph has two parallel rays pointing into the same half-plane, then the family of graphs on which we base our initial surfaces may include graphs with intersecting rays.
Deforming the graphs
Given a central graph , we will consider perturbations of this graph subject to parameters . We need the perturbations to be smoothly dependent on the parameters and are thus interested in graphs which can be deformed in this way.
Definition 2.11 (Isomorphic graphs).
We define two graphs as isomorphic if there exists a one-to-one correspondence between the vertices, edges, and rays, such that corresponding rays and edges emanate from the corresponding vertices. For convenience we will often use the same letter to denote corresponding objects for isomorphic graphs. Using this correspondence, for isomorphic to , we identify with respectively.
We proceed to define the function , which quantifies the length change of each edge for a perturbation of .
Given a graph isomorphic to a central graph , we define such that (following 2.11) for all ,
and therefore the length of the edge of corresponding to is
Definition 2.14 (Families of graphs).
We define a family of graphs to be a collection of graphs parametrized by such that the following hold:
Note that by the above definition each with is a modification of the central graph that is unbalanced as prescribed by while the lengths of the edges remain unchanged. Perturbing to is achieved by changing the lengths of the edges as prescribed by . Note that by 2.14.5 is unmodified under this perturbation. However, is not necessarily equal to , as the edges may rotate to accommodate the changes in edge length.
Throughout the paper, let denote the standard orthonormal basis of .
We now choose a frame associated to each edge in the graph and use this frame to determine a frame on each edge for any graph in .
For we choose once for all one of its endpoints to call . We call then its other endpoint and we define . For we choose once and for all an ordered, positively oriented orthonormal frame , such that , where is the endpoint of if and if . We have therefore when and
Given two unit vectors such that , let denote the unique rotation defined in the following manner.
If , take to be the identity.
If , set and . We define to be the rotation in the plane given by that rotates to , that is in closed form
The rotation depends smoothly on and .
Simplifying the expression, using the definition of , we observe that for ,
This expression is clearly smooth in . ∎
For and the corresponding attachment on an isomorphic graph, let
We use the rotation defined above to describe an orthonormal frame on the edges and rays of any graph in the family . By the smooth dependence on , and the presumed smallness of their norms, for and a corresponding edge or ray on any graph in the family. It follows that the rotation we need will always be well-defined.
For with as in 2.14, given we define an orthonormal frame uniquely by requiring the following:
depends smoothly on .
3. The Building Blocks
The initial hypersurfaces we construct will be built out of appropriately fused pieces of spheres and perturbed Delaunay hypersurfaces. The positioning of these pieces and the parameter of each Delaunay piece is determined by the graphs of and the parameters . The building blocks however can be described independently of any reference to the graphs of . To highlight this fact, we first develop the immersions of the building blocks to depend upon other general parameters not related to any graph. In Section 5 we use these immersions to produce a family of hypersurfaces from a family of graphs , where each hypersurface will depend on the central graph of as well as the parameters .
Spherical building blocks
Let be as in A.6. Immediately we see that
We determine now sphere diffeomorphisms that will be used to guarantee that the immersion is well-defined. First we define a rotation which maps to for a given orthonormal frame and a perturbation of .
Let be two orthonormal frames of with the same orientation. We define to be the unique rotation such that
We now define a map on that consists of local frame transformations and smoothly transits to the identity map away from these transformations. In application, the first vector in each frame will describe the positioning of an edge on a graph .
Definition 3.3 (Spherical Building Blocks).
We assume given two sets of positively oriented ordered orthonormal frames and , where
That is, the first vectors in each frame of are not close, while the first vectors in each pair of frames are close. We define then a family of diffeomorphisms , smoothly dependent on , by
Delaunay building blocks
We now describe a general immersion of an appropriately perturbed Delaunay piece. For a description of Delaunay immersions, see Section A. Throughout this subsection, let be the value defined in 3.1, let , and let and be as in A.10 so that is the domain period and the translational period of a Delaunay hypersurface of parameter . We presume throughout that is a constant chosen sufficiently small to guarantee that all immersions are smooth and well-defined and that all error estimates will hold as stated. Finally, we let denote a possibly large constant that is independent of .
Let be cutoff functions such that:
Given with and with , we define two smooth immersions and such that, for ,
To aid the reader, we describe the geometry of the immersion in some detail. For , the image is a geodesic hyperannulus sitting on a unit sphere with the sphere centered at . The annulus is centered at with inner radius . When , the immersion smoothly interpolates between the annular region on the dislocated sphere and an annular region centered at on a unit sphere centered at the origin. For , the immersion remains on the unit sphere centered at the origin, while for , the immersion smoothly transits between this sphere and a Delaunay piece with parameter . The same procedure happens toward the other end. First, the Delaunay piece transits back to a unit sphere centered at . This position represents the location of the end of a Delaunay piece with parameter and periods, with initial end at the origin. Finally, this sphere transits to a unit sphere centered at , a dislocation of from the previously described sphere.
Of course, the immersion has the same behavior as near the origin. The only difference is that the Delaunay immersion continues out to infinity and there is no transiting back to a sphere.
Let or as the situation dictates. For a fixed, large constant ,
On the region where , the only difference between the immersions comes from the cutoff function applied to , where the is appropriate for the domain. Thus the estimates on these regions are immediate.
Let denote the mean curvature of the immersion .
From these definitions and 3.6 we immediately bound the error on the mean curvature.
For as in 3.6, denoting the unit normal of the immersion , and ,