Spacelike mean curvature one surfaces
in de Sitter -space
The first author studied spacelike constant mean curvature one (CMC-) surfaces in de Sitter 3-space when the surfaces have no singularities except within some compact subset and are of finite total curvature on the complement of this compact subset. However, there are many CMC- surfaces whose singular sets are not compact. In fact, such examples have already appeared in the construction of trinoids given by Lee and the last author via hypergeometric functions.
In this paper, we improve the Osserman-type inequality given by the first author. Moreover, we shall develop a fundamental framework that allows the singular set to be non-compact, and then will use it to investigate the global behavior of CMC- surfaces.
2000 Mathematics Subject Classification:Primary 53A10; Secondary 53A35, 53C50.
A holomorphic map of a Riemann surface into the complex Lie group is called null if vanishes identically, where is a local complex coordinate of . We consider two projections, one into the hyperbolic -space
and the other into the de Sitter -space
where the definition of is in Appendix B. It is well-known that the projection of a holomorphic null immersion into by gives a conformal CMC- (constant mean curvature one) immersion (see [Br], [UY1], [CHR]). Moreover, conformal CMC- immersions are always given locally in such a manner.
On the other hand, spacelike CMC- surfaces given by the projection of holomorphic null immersions into by can have singularities, and are called CMC- faces. We work with this class of surfaces that is larger than the class of CMC- immersions. In fact, the class of CMC- immersions is too small, since there is only one, up to congruency, complete spacelike CMC- immersion [Ak, R], which we call an -horosphere. (We also give a simple proof of this here. See the last remark of Section 1.)
The relationship between CMC- surfaces in and CMC- faces in is analogous to that between minimal surfaces in Euclidean 3-space and spacelike maximal surfaces with singularities in Lorentz-Minkowski 3-space (called maxfaces [UY5]). Note that maximal surfaces also admit a Weierstrass-type representation formula ([K]). As in the case of maxfaces (see [UY5]), the first author [F] investigated the global behavior of CMC- faces in , in particular proving an Osserman-type inequality for complete CMC- faces of finite type whose ends are all elliptic, where a complete end of a CMC- face is called elliptic, parabolic, or hyperbolic if the monodromy matrix of the holomorphic lift is elliptic, parabolic, or hyperbolic, respectively (see Section 1). One of our main results is the following, which implies that the ellipticity or parabolicity of ends follows from completeness:
A complete end of a CMC- face in is never hyperbolic, so must be either elliptic or parabolic. Moreover, the total curvature over a neighborhood of such an end is finite.
We remark that there exist incomplete elliptic and parabolic ends.
It is remarkable that just completeness of an end is sufficient to conclude that it has finite total curvature. This is certainly not the case for CMC- surfaces in nor for minimal surfaces in , but is similar to the case of maximal surfaces in [UY5]. Although the asymptotic behavior of regular elliptic CMC- ends in is investigated in [F], there do also exist complete parabolic ends, and to describe them, a much deeper analysis is needed, which we will conduct in this article.
Suppose a CMC- face is complete. Then there exist a compact Riemann surface and a finite number of points such that is biholomorphic to , and
where is the hyperbolic Gauss map of and is the Euler characteristic of . Furthermore, equality holds if and only if each end is regular and properly embedded.
CMC- trinoids in were constructed by Lee and the last author using hypergeometric functions [LY], and those trinoids with elliptic ends are complete in the sense of [F], and attain equality in ( ‣ II). However, those having other types of ends are not complete, as their singular sets are not compact. For this reason, our goal is not only to prove the above two theorems, but also to extend the framework for CMC- surfaces to include a larger class of surfaces, relaxing the immersedness and completeness conditions. If is of finite topology, i.e. if is diffeomorphic to a compact Riemann surface with finitely many punctures , and if a CMC- face is weakly complete, whose precise definition will be given in Section 1, we say that is a weakly complete CMC- face of finite topology. We shall develop the framework under this more general notion, which includes all the trinoids in [LY].
In Section 1, we recall definitions and basic results. In Section 2, we investigate the monodromy of the hyperbolic metrics on a punctured disk around an end. As an application, we prove Theorem I in Section 3. In Section 4, we give a geometric interpretation of the hyperbolic Gauss map. In Section 5, we investigate the asymptotic behavior of regular parabolic ends, and prove Theorem II. In Appendix A, we prove meromorphicity of the Hopf differential for complete CMC- faces. In Appendix B, we explain the conjugacy classes of .
The representation formula
Let be the Lorentz-Minkowski space of dimension , with the Lorentz metric
Then de Sitter -space is
with metric induced from , which is a simply-connected Lorentzian -manifold with constant sectional curvature . We identify with the set of Hermitian matrices () by
and . Then is
with the metric
The projection mentioned in the introduction is written explicitly as . Note that the hyperbolic -space is given by and the projection is .
An immersion into is called spacelike if the induced metric on the immersed surface is positive definite. The complex Lie group acts isometrically on , as well as , by
In fact, is isomorphic to the identity component of the isometry group of . Note that each element of corresponds to an orientation preserving and orthochronous (i.e., time orientation preserving) isometry. The group of orientation preserving isometries of is generated by and the map
Aiyama-Akutagawa [AA] gave a Weierstrass-type representation formula in terms of holomorphic data for spacelike CMC- immersions in . The first author [F] extended the notion of CMC- surfaces as follows, like as for the case of maximal surfaces in the Minkowski space [UY5].
Definition 1.1 ([F]).
Let be a -manifold. A -map is called a CMC- face if
there exists an open dense subset such that is a spacelike CMC- immersion,
for any singular point (that is, a point where the induced metric degenerates) , there exists a -differentiable function , defined on the intersection of neighborhood of with , such that extends to a -differentiable Riemannian metric on , where is the first fundamental form, i.e., the pull-back of the metric of by , and
for any .
A -map is called a frontal if lifts to a -map such that lies in the canonical contact plane-field on . Moreover, is called a wave front or a front if is an immersion, that is, is a Legendrian submanifold. If a frontal can lift up to a smooth map into , is called co-orientable, and otherwise it is called non-co-orientable. Wave fronts are a canonical class for investigating flat surfaces in the hyperbolic 3-space . In fact, like for CMC- faces (see Theorem II in the introduction), an Osserman-type inequality holds for flat fronts in (see [KUY2].) Although our CMC- faces belong to a special class of horospherical linear Weingarten surfaces (cf. [KU]), they may not be (wave) fronts in general, but are co-orientable frontals. In particular, there is a globally defined non-vanishing normal vector field on the whole of for a given CMC- face . It should be remarked that the limiting tangent plane at each singular point contains a lightlike direction, that is, a CMC- face is not spacelike on the singular set.
An oriented -manifold on which a CMC- face is defined always has a complex structure (see [F]). Since CMC- faces are all orientable and co-orientable (cf. [KU]), from now on, we will treat as a Riemann surface, and we can assume the existence of a globally defined non-vanishing normal vector field. The representation formula in [AA] can be extended for CMC- faces as follows:
Theorem 1.4 ([F, Theorem 1.9]).
Let be a simply connected Riemann surface. Let be a meromorphic function and a holomorphic -form on such that
is a Riemannian metric on and is not identically . Take a holomorphic immersion satisfying
Then defined by
is a CMC- face which is conformal away from its singularities. The induced metric on , the second fundamental form II, and the Hopf differential of are given as follows:
The singularities of the CMC- face occur at points where .
By definition, CMC- faces have dense regular sets. However, the projection of null holomorphic immersions might not have dense regular sets, in general. Such an example has been given in [F, Remark 1.8]. Fortunately, we can explicitly classify such degenerate examples, as follows: Let be a connected Riemann surface and be a null immersion. We assume that the set of singular points of the corresponding map
has an interior point. Then the secondary Gauss map is constant on and . Without loss of generality, we may assume . Since is an immersion, is positive definite. Then everywhere. Hence for each , one can take a complex coordinate such that . Then is a solution of
Without loss of generality, we may assume that . Then we have
and the corresponding map is computed as
whose image is a lightlike line in . Thus, we have shown that the image of any degenerate CMC- surface is a part of a lightlike line.
Theorem 1.4 is an analogue of the Bryant representation for CMC- surfaces in , which explains why CMC- surfaces in both and are characterized by the projections and . The CMC- surfaces in and are both typical examples in the class of linear Weingarten surfaces. A Bryant-type representation formula for linear Weingarten surfaces was recently given by J. Gálvez, A. Martínez and F. Milán [GMM].
Following the terminology of [UY1], is called a secondary Gauss map of . The pair is called Weierstrass data of , and is called a holomorphic null lift of .
Any maxface (see [UY5] for the definition) is locally obtained as the real part of some . Moreover, their first fundamental forms and Hopf differentials are given by (1.8). The meromorphic function can be identified with the Lorentzian Gauss map. In this case, we call the pair the Weierstrass data of the maxface.
The holomorphic null lift of a CMC- face is unique up to right-multiplication by matrices in , that is, for each , the projection of is also . Under the transformation , the secondary Gauss map changes by a Möbius transformation:
The conditions , , are invariant under this transformation.
In particular, let be a CMC- face of a (not necessarily simply connected) Riemann surface . Then the holomorphic null lift is defined only on the universal cover of . Take a deck transformation in . Since , there exists a such that
The representation is called the monodromy representation, which induces a -representation satisfying
The action , , induces a rigid motion in , and the isometric motion as in (1.4) corresponds to
The secondary Gauss map of is .
Let be the Gaussian curvature of on the set of regular points of . Then
is a pseudometric of constant curvature , which degenerates at isolated umbilic points. We have
is induced from the canonical Hermitian metric of via . When the CMC- face is defined on , and are as well, so is well-defined on , and is called the lift metric. It is nothing but the dual metric of the CMC- surface in , see [UY3].
We now define two different notions of completeness for CMC- faces as follows:
We say a CMC- face is complete if there exists a symmetric -tensor field which vanishes outside a compact subset such that the sum is a complete Riemannian metric on .
We say that is weakly complete if it is congruent to an -horosphere or if the lift metric (1.17) is a complete Riemannian metric on .
Here, the -horosphere is the totally umbilic CMC- surface, which is also the only complete CMC- immersed surface (see Remark 1.21). It has the Weierstrass data and . The metric of an -horosphere cannot be defined by (1.17) as is constant and is identically , but can still be defined as the metric induced by , and is a complete flat metric on .
We say that is of finite type if there exists a compact set of such that the first fundamental form is positive definite and has finite total (absolute) curvature on .
Let be a CMC- face of finite topology, that is, is diffeomorphic to a compact Riemann surface with a finite number of points excluded. We can take a punctured neighborhood of which is biholomorphic to either the punctured unit disk or an annular domain, and is called a puncture-type end or an annular end, respectively.
Let be a CMC- face. If is complete, then
the singular set of is compact,
is weakly complete,
has finite topology and each end is of puncture-type.
(1) is obvious. If is totally umbilic, it is congruent to an -horosphere and the assertion is obvious. So we assume the Hopf differential does not vanish identically. Since the Gaussian curvature of is nonnegative, completeness implies (3) by the appendix of [UY5]. So we shall now prove that completeness implies weak completeness: Fix an end of . By an appropriate choice of a coordinate , the restriction of to a neighborhood of is . We denote by the induced metric of the corresponding CMC- surface into hyperbolic -space. Take a path such that as . Then by (1.5) and (1.8), holds, and hence completeness of implies that each lift of has infinite length with respect to . Here, and are the pull-backs of the Hermitian metric of by and , respectively. Yu [Y] showed that completeness of these two metrics are equivalent. Hence, has infinite length with respect to the metric . Since is well-defined on , also has infinite length with respect to , that is, the metric on is complete at . Thus, is a weakly complete end. ∎
Our definition of weak completeness of CMC- faces is somewhat more technical than that of maxfaces [UY5], but it is the correctly corresponding concept in : for data , weak completeness of the associated maxface in is equivalent to that of the CMC- face in .
The Hopf differential of a complete CMC- face is meromorphic on its compactification , even without assuming that all ends of are regular. See Appendix A. It should be remarked that for CMC- surfaces in hyperbolic -space, finiteness of total curvature is needed to show the meromorphicity of (see [Br]).
Monodromy of ends of CMC- faces
For any real number , we set
A matrix in is called
elliptic if it is conjugate to () in ,
parabolic if it is conjugate to () in , and
hyperbolic if it is conjugate to () in .
Any matrix in is of one of these three types, see Appendix B. Note that the parabolic matrices and are conjugate in if and only if . Though the set of conjugate classes of parabolic matrices is fully represented by , we may use various values of in this paper for the sake of simplicity.
Let be a weakly complete CMC- face of finite topology, where is diffeomorphic to a compact Riemann surface with finitely many punctures . Any puncture , or occasionally a small neighborhood of , is called an end of .
An end is called elliptic, parabolic or hyperbolic when the monodromy matrix is elliptic, parabolic or hyperbolic, respectively, where is as in Remark 1.10 and is the deck transformation corresponding to the counterclockwise loop about .
The Schwarzian derivative
Let be a local complex coordinate of a Riemann surface , and a meromorphic function on . Then
is the Schwarzian derivative of with respect to the coordinate .
If at , where denotes higher order terms, then the positive integer is called the (ramification) order of , and we have
We write , which we also call the Schwarzian derivative. The Schwarzian derivative depends on the choice of local coordinates, but the difference does not, that is, is a well-defined holomorphic -differential.
The Schwarzian derivative is invariant under Möbius transformations: holds for , where denotes the Möbius transformation as in (1.11). Conversely, if , there exists an such that .
Let be a CMC- face with the hyperbolic Gauss map , a secondary Gauss map and the Hopf differential . Then
Here we give a proof that the only complete CMC- immersion is the totally umbilic one, that is, the -horosphere, which is simpler than the original proofs in [Ak, R]. (The proof is essentially the same as for the case of maximal surfaces in given in [UY5, Remark 1.2].) Let be a complete CMC- immersion. Without loss of generality, we may assume that is both connected and simply connected. Then the Weierstrass data as in Theorem 1.4 is single-valued on . Since has no singular points, we may assume that holds on . Since , the metric is a complete flat metric on . Then the uniformization theorem yields that is bi-holomorphic to , and must be a constant function, which implies that the image of must be totally umbilic.
2. Monodromy of punctured hyperbolic metrics
By Remark 1.10, the monodromy of a holomorphic null immersion is elliptic, parabolic or hyperbolic if and only if the monodromy of its secondary Gauss map is elliptic, parabolic or hyperbolic, respectively. In this section, in an abstract setting, we give results needed for investigating the behavior of at a puncture-type end, in terms of the monodromy of .
Lifts of -projective connections on a punctured disk.
be the punctured unit disk and a holomorphic -differential on . Then there exists a holomorphic developing map such that , where is the universal cover of . For any other holomorphic function such that , there exists an so that . Thus there exists a matrix such that
where is the generator of corresponding to a counterclockwise loop about the origin. We call the monodromy matrix of . If there exists a so that , is called a -projective connection on and is called a -lift of . A -projective connection on has a removable singularity, a pole or an essential singularity at , and is said to have a regular singularity at if it has at most a pole of order 2 at . (The general definition of projective connections is given in [T] and [UY2]. There exist holomorphic -differentials on which are not -projective connections.) When , it is conjugate to one of the matrices in (1.18). The -projective connection is then called elliptic, parabolic or hyperbolic when is elliptic, parabolic or hyperbolic, respectively. This terminology is independent of the choice of .
By the property (1.13), the Schwarzian derivative of the secondary Gauss map of a CMC- face is an example of a -projective connection.
Note that a -lift has the ambiguity for . The property that (resp. ) is independent of this ambiguity.
Let be a -lift of a -projective connection . Then
is also a -lift of , because for any . However, , and one can show that there is no matrix such that , that is, is not -equivalent to .
In the rest of this article, as well as in the following proposition, we use
which is motivated by an isomorphism between and . See Appendix B.
Let be a -projective connection on . Then the following assertions hold:
Suppose that is elliptic. Then,
there exist a real number and a single-valued meromorphic function on such that
is a -lift of .
has a regular singularity at if and only if has at most a pole at .
Suppose that is parabolic and take an arbitrary positive number . Then,
for each , there exists a single-valued meromorphic function on such that
is a -lift of .
The function has at most a pole at if and only if has a pole of order exactly at .
is holomorphic at if and only if has at most a pole of order at .
When is holomorphic at , (resp. ) holds for sufficiently small if and only if (resp. ).
Suppose that is hyperbolic. Then,
there exist a positive number and a single-valued meromorphic function on such that
is a -lift of .
has at most a pole at if and only if has a pole of order exactly at .
In the statements of Proposition 2.2, the function is defined by
where is considered as a function defined on the universal cover of .
To prove this, we consider the following ordinary differential equation
If we assume has a regular singularity at , then for some and (2.3) has the fundamental system of solutions
where are holomorphic functions on such that . The constant is called the log-term coefficient and are the solutions of the indicial equation
In the above setting, if .
Proof of Proposition 2.2.
Take the matrix as in (2.1).
We first prove the elliptic case. Since is elliptic, there exist a and an such that . So , and is single-valued on , proving the first part of (1). If the origin is at most a pole of , a direct calculation shows that has a regular singularity. To show the converse, we set , with as in (2.4). Then by Lemma 2.4 we have . The monodromy matrix of is conjugate to
Since is elliptic, the log-term coefficient and . Thus
Since , there exists a so that . Then
so . If , then is meromorphic, proving (1). Otherwise,
for some , and (1) follows from
Next, we assume is parabolic and take a positive number and . Then by Theorem B.1 and Remark B.3 in Appendix B, there exists a matrix such that is one of , , , . Note that and are not conjugate in . Replacing with if (see Remark 2.1), we can choose a -lift such that
Then, . Here the -ambiguity of does not affect the -action. Thus,
Hence is a single-valued meromorphic function on , proving the first part of (2). If has at most a pole at , then a direct computation shows that has a pole of order exactly . Therefore, it suffices to show that has at most a pole at when has a regular singularity. We now show this:
We set . Since is parabolic, (2.6) yields that the log-term coefficient and is non-positive. Hence
Here is single-valued on and has at most a pole at . Take a matrix such that . Then