Compact complete null curves in Complex 3-space
Antonio Alarcón and Francisco J. López
Abstract We prove that for any open orientable surface of finite topology, there exist a Riemann surface a relatively compact domain and a continuous map such that:
and are homeomorphic to and contain no relatively compact components in
is a complete null holomorphic curve, is an embedding and the Hausdorff dimension of is
Moreover, for any and compact null holomorphic curve with non-empty boundary there exist Riemann surfaces and homeomorphic to and a map in the above conditions such that where means Hausdorff distance in
Keywords Null holomorphic curves, Calabi-Yau problem, Plateau problem.
Mathematics Subject Classification (2010) 53C42, 32H02, 53A10.
Given an open Riemann surface a null (holomorphic) curve in is a holomorphic immersion satisfying that where is the complex differential. In this paper we deal with the existence of compact complete null curves in accordingly to the following
Let be a relatively compact domain in an open Riemann surface. A continuous map is said to be a compact complete null curve in if is a complete null curve.
The first approach to this matter was made by Martín and Nadirashvili in the context of simply connected minimal surfaces in [MN]. The corresponding finite topology case was considered in [AL1], see also [Na2, AN] for other related questions. Compact complete minimal surfaces manifest the interplay between two well studied topics on minimal surface theory: Plateau’s and Calabi-Yau’s problems. The first one consists of finding a compact minimal surface spanning a given closed curve in and it was independently solved by Douglas [Do] and Radó [Ra]. The second one deals with the construction of complete minimal surfaces in with bounded coordinate functions, and the most relevant examples were given by Jorge-Xavier [JX] and Nadirashvili [Na1].
In the very last few years, the study of the Calabi-Yau problem has evolved in the direction of null curves in Observe that a holomorphic map is a null curve if and only if both and are conformal minimal immersions. Furthermore, the Riemannian metric on induced by is twice the one induced by and Therefore, complete bounded null curves in provide complete bounded minimal surfaces in with well defined and bounded conjugate surface, and viceversa. Existence of a vast family of complete bounded null curves with arbitrary topology and other additional properties is known [MUY1, AL2].
If is a relatively compact domain in an open Riemann surface and is a continuous map such that is a conformal complete minimal immersion whose conjugate immersion is well defined, then does not necessarily extend as a continuous map to Therefore, it is natural to wonder whether there exist compact complete null curves in
In this paper we answer this question, proving considerably more.
Main Theorem. Let be an open orientable surface of finite topology.
Then there exist a Riemann surface a relatively compact domain and a continuous map such that:
and are homeomorphic to and contain no relatively compact components in
is a complete null curve, is an embedding and the Hausdorff dimension of is
Moreover, for any and for any compact null curve with non-empty boundary there exist Riemann surfaces and homeomorphic to and a map in the above conditions such that where means Hausdorff distance in
Unfortunately, the techniques we use do not give enough control over the topology of to assert that it consists, for instance, of a finite collection of Jordan curves. Concerning the second part of the theorem, it is worth mentioning that there exist Jordan curves in which are not spanned by any null curve. Main Theorem actually follows from a more general density result (see Theorem 5.1). Some other similar existence results for complex curves in null holomorphic curves in the complex Lie group Bryant surfaces in hyperbolic 3-space and minimal surfaces in can be also derived from it, see Corollary 5.3. See [MUY1, MUY2, AL2] for related results.
Part of this work was done when the first author was visiting the Institut de Mathématiques de Jussieu in Paris. He wish to thank this institution, and in particular Rabah Souam, for the kind invitation and hospitality, and for providing excellent working conditions.
We denote by and the Euclidean norm and metric in where or and for any compact topological space and continuous map we denote by the maximum norm of on We also denote by
2.1. Riemann surfaces
Given a Riemann surface with non empty boundary, we denote by the (possibly non-connected) 1-dimensional topological manifold determined by its boundary points. For any we denote by and the interior of in the closure of in and the topological frontier of in Open connected subsets of will be called domains, and those proper topological subspaces of being Riemann surfaces with boundary are said to be regions.
A Riemann surface is said to be a bordered Riemann surface if it is compact, and is smooth. A compact region of is said to be a bordered region if it is a bordered Riemann surface.
Throughout this paper, and will denote a fixed but arbitrary bordered Riemann surface and a smooth conformal metric on repectively. We call the open Riemann surface and write for the number of ends of (or equivalently, for the number of connected components of ).
A subset is said to be Runge if is injective, where is the inclusion map. If is a compact region in this simply means that has no relatively compact components in
In the remainder of this subsection we introduce the necessary notations for a precise statement of the approximation result by null curves in Lemma 2.12, which is the starting point of the paper.
A 1-form on a subset is said to be of type if for any conformal chart in for some function
Assume that is compact. We say that
a function can be uniformly approximated on by holomorphic functions in if and only if there exists a sequence of holomorphic functions such that uniformly on and
a 1-form on can be uniformly approximated on by holomorphic 1-forms in if and only if there exists a sequence of holomoprhic 1-forms on such that uniformly on for any closed conformal disc on
The following definition is crucial in our arguments.
Definition 2.3 (Admissible set).
A compact subset is said to be admissible if and only if
is a finite collection of pairwise disjoint compact regions in with boundary,
consists of a finite collection of pairwise disjoint analytical Jordan arcs, and
any component of with an endpoint admits an analytical extension in such that the unique component of with endpoint lies in
The next one deals with the notion of smoothness of functions and 1-forms on admissible subsets.
Assume that is an admissible subset of
A function is said to be smooth if and only if admits a smooth extension to an open domain in containing and for any component of and any open analytical Jordan arc in containing admits a smooth extension to satisfying that
A 1-form on is said to be smooth if and only if is a smooth function for any closed conformal disk on such that is an admissible set.
Given a smooth function on an admissible we denote by the 1-form of type (1,0) given by and where is any conformal chart on satisfying that Then is well defined and smooth. The -norm of on is given by
A sequence of smooth functions on is said to converge in the -topology to a smooth function on if If in addition is (the restriction to of) a holomorphic function on for all we also say that can be uniformly -approximated on by holomorphic functions on
Likewise one can define the notions of smoothness, (vectorial) differential, -norm and uniform -approximation for maps admissible.
Let be an admissible subset of and let be a Runge open subset of whose closure in is a compact region and is an isomorphism, where is the inclusion map. is said to be a tubular neighborhood of if is an isomorphism and where is the inclusion map and means Euler characteristic. In particular, (respectively, ) consists of a family of pairwise disjoint open (respectively, compact) annuli.
For instance, assume that is a finite family of pairwise disjoint smooth Jordan curves in take and set where means Riemannian distance in If is small enough, the exponential map is a diffeomorphism and where is a normal field along in In this case, is said to be a metric tubular neighborhood of (of radious ). Furthermore, if denotes the projection we denote by the natural orthogonal projection.
We denote by the family of Runge bordered regions such that is a tubular neighborhood of Given we say that if and only if
2.2. Null curves in
Throughout this paper we adopt column notation for both vectors and matrices of linear transformations in and make the convention
Let us start this subsection by introducing some operators which are strongly related to the geometry of and null curves.
Let We denote by
the usual Hermitian inner product in
the Euclidean scalar product of
Notice that for all and the equality holds if and only if .
A vector is said to be null if and only if We denote by
A basis of is said to be -conjugate if and only if
We denote by the complex orthogonal group or in other words, the group of matrices whose column vectors determine a -conjugate basis of We also denote by the complex linear transformation induced by
It is clear that for all
Let be an open Riemann surface.
A holomorphic map is said to be a null curve if and only if and never vanishes on
Conversely, given an exact holomorphic vectorial 1-form on satisfying that and never vanishes on then the map defines a null curve in
If is a null curve then the pull back metric on is determined by the expresion Given two subsets we denote by the distance between and in the Riemannian surface
Let be a null curve and Then is a null curve as well and where
Given a subset we denote by the space of maps extending as a null curve to an open neighborhood of in
Let be an admissible subset. A smooth map is said to be a generalized null curve in if and only if on and never vanishes on
The following technical lemma is a key tool in this paper.
3. Completeness Lemma
In this section we state and prove the technical result which is the kernel of this paper (Lemma 3.1 below). Its proof requires of the approximation result by null curves in Lemma 2.12. Lemma 3.1 has more strength than similar results in [MN, AL1], and its proof presents some differences.
Let with let be a metric tubular neighborhood of in and denote by the orthogonal projection. Consider an analytical map and so that
Then, for any and there exist and such that
is an embedding,
Roughly speaking, the above Lemma asserts that a compact null curve in can be perturbed into another compact null curve with larger intrinsic radious. This deformation hardly modifies the null curve in a prescribed compact set and, in addition, the effect of the deformation in the boundaries of the null curves is quite controlled. The bounds and in (3.1) and Items (3.1.c) and (3.1.e) follow the spirit of Nadirashvili’s original construction [Na1] (see (4.1) below).
The null curve which proves the lemma will be obtained from after two different perturbation processes. In the first one, which is enclosed in Claim 3.2 below, the effect of the deformation is strong over a family of Jordan arcs in (see the definition of the arcs in Items (C.1), (C.2) and (C.3)) and slight on a compact region containing In the second stage, see Claim 3.3, the deformation mainly acts on a family of compact discs (see the definition of in (E.2)) and hardly works on a compact region containing
3.1. Proof of Lemma 3.1
Take to be specified later.
By (3.1) and a continuity argument, for any there exists a simply connected open neighborhood of in such that
Set and observe that is an open covering of Take satisfying that
and note that is an open covering of as well. One has that where are pairwise disjoint compact annuli. Write and for the two components of For each let denote the additive cyclic group of integers modulus Since is an open covering of then there exist and a collection of compact Jordan arcs satisfying that
and have a common endpoint and are otherwise disjoint for all and
For any fix and such that
where is the orthogonal projection. This choice is possible since is dense in Label
for all and notice that
Since then we can take such that is a -conjugate basis of Denote by the complex orthogonal matrix for all and
Let be a family of pairwise disjoint analytical compact Jordan arcs in such that
(see (A.1), (B.2) and (B.3)),
has initial point final point and it is otherwise disjoint from and
As we announced above, the null curve will be obtained from after two deformation procedures. The first one strongly works in (see property (D.2) below) and is enclosed in the following
There exists such that, for any
if is a Borel measurable subset, then
where means third (complex) coordinate and Euclidean length in and
Let be a smooth parameterization of with and From (A.2) and (C.1) there exists a non-empty open subset satisfying that
Then (B.3), (3.5) and (C.1) give that Consider such that and
Set Take large enough so that
if and is odd, and
if and is even.
Notice that the curves are continuous, weakly differentiable and satisfy that Up to replacing for for all items (D.1), (D.2) and (D.3) formally hold. To finish, approximate by a smooth curve matching smoothly with at and so that the map given by for all and is a generalized null curve satisfying all the above items. Indeed, if is chosen close enough to (D.1) follows from (3.6) and (D.2) from (3.7). Apply Lemma 2.12 to and we are done. ∎
Denote by the closed disc in bounded by and a piece, named of connecting and Since is simply connected, then (A.1), (B.3) and (C.1) give that
Consider simply-connected compact neighborhoods and in of and respectively, satisfying the following properties:
is a compact disc and is a Jordan arc disjoint from the set (see Figure 3.1),
if is an arc connecting and and then
The existence of such compact discs follows from a continuity argument. In order to achieve properties (E.3) and (E.4) take into account Claim 3.2.
Let be the bijection where means integer part.
The second deformation process in the proof of Lemma 3.1 mainly acts in and is contained in the following claim.
There exists a sequence satisfying the following properties:
if is an arc connecting and and then
Properties (F.1), (F.4) and (F.6) follow from (E.3), (E.4) and (D.3), whereas (F.2), (F.3) and (F.5) make no sense. We reason by induction. Assume that we have defined satisfying the desired properties and let us construct
Label and write
Let denote a Jordan arc in disjoint from with initial point and final point and otherwise disjoint from Choose so that never vanishes. Denote and without loss of generality assume that is admissible.
Let be a smooth parameterization of with Label and consider the parameterization Write and notice that for all
From the definition of one has