Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space

Shoichi Fujimori

Department of Mathematics, Kobe University, Kobe 657-8501, Japan

^{Correspondence}

ABSTRACT

We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere.

Key words: minimal surfaces, CMC 1 cousins, hyperbolic space.

RESUMO

Mostramos que as diferenciais de Hopf de um par de superfícies primas, a saber, uma superfície mínima em um espaço euclideano de dimensão 3 e uma superfície de curvatura média constante (CMC) um em um espaço hiperbólico de dimensão 3, se estendem holomorficamente em cada fim. Usando este resultado, obtemos condições para que o par seja um plano e uma horosfera.

Palavras-chave: superfícies mínimas, prismas de CMC 1, espaços hiperbólicos.

INTRODUCTION

As there is a way to deform simply-connected CMC 1 surfaces in hyperbolic 3-space to minimal surfaces in Euclidean 3-space (Umehara and Yamada 1992), one might expect that there exist cousins in these two spaces that are not simply-connected. However, although there are now many known examples of minimal surfaces in and also CMC 1 surfaces in (see, for example, Bryant 1987, Rossman et al. 1997, 2001, Sá Earp and Toubiana 2001, Yu 2001, Umehara and Yamada 1993), and although non-simply-connected cousins pairs are easily found, such a pair of surfaces with embedded ends is yet to be found. Our purpose is to investigate whether such a pair can exist. Toward this goal, we apply recent results in Collin et al. 2001 about the behavior of embedded CMC 1 ends in to give various conditions under which such a pair cannot exist.

RESULTS

Let D Ì be a simply-connected domain in the complex plane. Fix a point z₀Î D. Let g be a meromorphic function on D and w a holomorphic 1-form on D such that w has a zero of order 2k if and only if g has a pole of order k and so that w has no other zeros. Set

Then F₀: D ® is a minimal immersion with induced metric

Furthermore, g is stereographic projection of the Gauss map of F₀. This is the Weierstrass representation.

On the other hand, for Weierstrass data (g, w) on D, we can take F: D ® SL(2, ) such that

and set

Then F₁ : D ® { XÎ Herm(2) ; X Î SL(2, )} @ is a CMC 1 immersion with induced metric = , where = (-1) is the hyperbolic 3-space with sectional curvature -1. This is the Bryant representation (Bryant 1987, Umehara and Yamada 1993). F is unique up to the form A · F, A a constant in SL(2, ), so F₁ is unique up to rigid motions of (see Umehara and Yamada 1993).

This shows that given data (g, w) on D, we can locally construct a pair of isometric surfaces, a minimal surface F₀(D) in and a CMC 1 surface F₁(D) in (see Theorem 8 of Lawson 1970).

For both F₀ and F₁, the Hopf differential Q on D is defined by Q = wdg.

DEFINITION 1. Let M be a Riemann surface and F₀: M ® a conformal minimal immersion. Then a CMC 1 immersion F₁: M ® is a cousin surface of F₀ if

=

holds. We refer to any such pair of surfaces F₀ and F₁ as cousins.

The following lemma is immediately obtained from §177 of Nitsche 1989:

LEMMA 2. Let (g, w) be the Weierstrass data of a simply-connected CMC 1 surface F₁:D ® . Then any cousin minimal surface F₀ in

can be represented (up to a rigid motion) by the Weierstrass data (g,e^iqw) for some q Î [0, p).

Recall that a surface has finite topology if it is homeomorphic to a compact Riemann surface with a finite number of points {p₁,¼,p_k} removed, which we write as M = \{p₁,¼,p_k}. We have the following proposition, which follows directly from results in Collin et al. 2001 and Sá Earp and Toubiana 2001.

PROPOSITION 3. Let M =

\{p₁,¼,p_k} be a Riemann surface of finite topology, and let F₁ : M ® be a conformal CMC 1 immersion with properly embedded annular ends. Let F₀ : M ® be a minimal immersion with embedded ends, and assume that F₁ and F₀ are cousins. Then the Hopf differentials of F₁ and F₀ are holomorphic on

.

REMARK 4. By Theorem 10 of Collin et al. 2001, all properly embedded annular CMC 1 ends in are conformal to a punctured disk, thus the assumption that F₁ is conformal is not actually a restriction on the possible choices of F₁. Because F₀ and F₁ are cousins, F₀ : is also conformal.

PROOF OF PROPOSITION 3. Let j₁ : ® be an arbitrary end of F₁, where = {z Î ; 0 < |z| < e} for some e > 0. As noted in Remark 4, we may assume that j₁ is conformal. Let j₀ : ® be the corresponding minimal end. By Theorem 10 of Collin et al. 2001, j₁ has finite total curvature and is regular. Then by Umehara and Yamada 1993, we can take the Weierstrass data associated with j₁ in the following form:

where , are nonzero holomorphic functions on D_e = {z Î ; |z| < e}, and m, n Î , m > 0, n < –1, m + n Î , m + n > –1.

By Lemma 2, there exists a q Î [0, p) such that (g, e^iqw) is the Weierstrass data associated with j₀. Because g is stereographic projection of the Gauss map of j₀, g is well-defined on , so m Î and hence – n Î .

The first and second coordinates of j₀ are

and j₀ is asymptotic to a catenoid or planar end, by Schoen 1983. Also g(0) = 0, and the limiting normal of the end j₀ must be vertical. Therefore, n must be –2 for the end to be embedded, and (0) must be 0 for the end j₀ to be well-defined on .

Lemma 2.4 of Sá Earp and Toubiana 2001 showed that 0 ¹ (0)(0) = (1 – m²)/4m. So m cannot be 1 because (0) ¹ 0 and (0) ¹ 0. Furthermore, Lemma 2.9 of Sá Earp and Toubiana 2001 showed that

So m cannot be 2. Therefore m > 3.

Thus the Hopf differentials wdg and e^iqwdg have order m + n – 1 > 0 at z = 0. Hence they are holomorphic at each end, as well as on M itself.

An end j₀ : ® (resp. j₁ : ® ) is said to be a planar end (resp. horosphere end) if m + n > 0. So we have the following corollary:

COROLLARY 5. Hypotheses being as in Proposition 3, then F₀ has only planar ends and F₁ has only horosphere ends.

COROLLARY 6. Let M =

\{p₁,¼,p_k} be a Riemann surface of finite topology so that

has genus zero. Let F₀ : M ® , F₁ : M ® be properly immersed cousin surfaces with embedded ends. Then F₀ is a plane and F₁ is a horosphere.

PROOF. Since there exists no nonzero holomorphic 2-differential on the sphere È {¥}, the Hopf differential is identically zero. So both F₀(M) and F₁(M) are totally umbilic. Therefore F₀ is a plane and F₁ is a horosphere.

COROLLARY 7. Let M =

\{p₁,¼,p_k} be a Riemann surface of finite topology so that

has genus g. Let F₀ : M ®

, F₁ : M ® be properly immersed cousin surfaces with embedded ends, and suppose they have total curvature more than –16p. Then F₀ is a plane and F₁ is a horosphere.

PROOF. Lopez 1992 showed that any minimal surface with total curvature –4p or –8p has a non-holomorphic Hopf differential Q on . Thus the only possibility (other than a plane) is that F₀ : M ® is a properly immersed minimal surface with embedded planar ends and total curvature –12p. By Theorem 4 of Jorge and Meeks 1983, each end of F₀ is embedded if and only if

holds, where K and dA are the Gaussian curvature and the area element of F₀. So k + g = 4. Since any complete minimal surface with finite total curvature and one embedded end is a plane, and since the only complete minimal surface in with finite total curvature and two embedded ends is the catenoid (Schoen 1983), F₀(M) is a torus with three embedded planar ends. But Theorem 26 of Kusner and Schmitt 1992 showed that such a surface does not exist, completing the proof.

COROLLARY 8. Let M =

\{p₁,¼,p_k} be a Riemann surface of finite topology so that

has genus one. Let F₀ : M ® , F₁ : M ® be properly immersed cousin surfaces with embedded ends. Then F₀ and F₁ each have at least 4 ends.

PROOF. By Theorem 4 of Jorge and Meeks 1983 again, the right hand side of (1) is –4kp. So k > 4, by Corollary 7.

REMARK 9. Theorem 3 of Miyaoka and Sato 1994 found examples of complete minimal surfaces of genus one with four embedded ends, but they all contain non-planar ends.

REMARK 10. Costa 1993 and Kusner and Schmitt 1992 found examples of complete minimal surfaces of genus one with four embedded planar ends. But none of them satisfies the condition that the Hopf differential extends holomorphically to the ends.

Defining annular ends to be those which are homeomorphic to punctured disks, Theorem 12 of Collin et al. 2001 showed that each end of a properly embedded non-totally-umbilic CMC 1 surface F₁ : M ® with annular ends is asymptotic to an end of a CMC 1 catenoid. In particular, such a surface does not have horosphere ends. We saw in the proof of Proposition 3, in conjunction with Remark 4, that any single embedded annular end asymptotic to a CMC 1 catenoid in cannot have a corresponding minimal cousin in with an embedded end. Hence, F₁ does not have a cousin F₀ : M ® with embedded ends. So we have the following corollary, in which we do not need to assume that M has finite topology, since finite topology was not assumed in Theorem 12 of Collin et al. 2001:

COROLLARY 11. Let M be a Riemann surface. Let F₁ : M ® be a conformal CMC 1 proper embedding with annular ends, and let F₀ : M ® be a minimal surface with embedded ends. Assume that F₁ and F₀ are cousins. Then F₀ is a plane and F₁ is a horosphere.

REMARK 12. Regarding Corollary 11:

(i) If the assumption that F₁ is embedded is removed, then the pair of cousin surfaces given by the Weierstrass data

is a counterexample. In fact, each end of F₁ in this example is an n-fold cover of an embedded end, and F₀ is an embedding.

(ii) If the assumption that F₁ is embedded is replaced with the weaker assumption that only the ends are embedded, then any possible counterexamples can not satisfy the conditions of Corollaries 6 or 7 or 8.

(iii) If the weaker assumption in (ii) is used, and the assumption that the ends of F₀ are embedded is removed, then the pair of cousin surfaces given by the Weierstrass data

is a counterexample to the corollary. In fact, in this example, each end of F₁ is embedded, and each end of F₀ is an n-fold cover of an embedded end.

(iv) If the assumption that F₁ is embedded is kept, but the assumption that each end of F₀ is embedded is removed, then the author does not know of any counterexamples to the corollary.

REMARK 13. Theorem 3.3 of Choi et al. 1990 showed that a properly embedded minimal surface in which has more than one end is minimally rigid. Corollary 3.4 of Umehara and Yamada 1992 showed that if cousin surfaces : D_e ® (–c²) (c > 0) associated with a minimal surface F₀ º r : D_e ® are well-defined on for all c, then all of the surfaces in the associate family of F₀ are well-defined on , where

is the projection. However, this cannot lead us to another proof of Corollary 11, because we only assume that the F_c is well-defined when c = 0, 1. Furthermore, we allow M to have positive genus, so we are not considering well-definedness merely on domains which are simply-connected or homeomorphic to .

ACKNOWLEDGMENTS

The author thanks Professors Wayne Rossman and Hitoshi Furuhata for their assistance with this work.

Manuscript received on February 4, 2003; accepted for publication on June 4, 2003; presented by MANFREDO DO CARMO

Mathematics Subject Classification: 53A10, 53C42.

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Publication Dates

• Publication in this collection
  26 Aug 2003
• Date of issue
  Sept 2003

History

• Received
  04 Feb 2003
• Accepted
  04 June 2003