Abstract and Applied Analysis

Volume 2013 (2013), Article ID 512383, 5 pages

http://dx.doi.org/10.1155/2013/512383

Research Article

Rotationally Symmetric Harmonic Diffeomorphisms between Surfaces

Li Chen,¹ Shi-Zhong Du,² and Xu-Qian Fan³

¹Faculty of Mathematics & Computer Science, Hubei University, Wuhan 430062, China

²The School of Natural Sciences and Humanities, Shenzhen Graduate School, The Harbin Institute of Technology, Shenzhen 518055, China

³Department of Mathematics, Jinan University, Guangzhou 510632, China

Received 12 February 2013; Revised 22 April 2013; Accepted 22 April 2013

Academic Editor: Yuriy Rogovchenko

Copyright © 2013 Li Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We show the nonexistence of rotationally symmetric harmonic diffeomorphism between the unit disk without the origin and a punctured disc with hyperbolic metric on the target.

1. Introduction

The existence of harmonic diffeomorphisms between complete Riemannian manifolds has been extensively studied, please see, for example, [1–34]. In particular, Heinz [17] proved that there is no harmonic diffeomorphism from the unit disc onto with its flat metric. On the other hand, Schoen [25] mentioned a question about the existence, or nonexistence, of a harmonic diffeomorphism from the complex plane onto the hyperbolic 2-space. At the present time, many beautiful results about the asymptotic behavior of harmonic embedding from into the hyperbolic plane have been obtained, please see, for example, [4, 5, 14, 32] or the review [33] by Wan and the references therein. In 2010, Collin and Rosenberg [10] constructed harmonic diffeomorphisms from onto the hyperbolic plane. In [7, 24, 28, 29], the authors therein studied the rotational symmetry case. One of their results is the nonexistence of rotationally symmetric harmonic diffeomorphism from onto the hyperbolic plane.

In this paper, we will study the existence, or nonexistence, of rotationally symmetric harmonic diffeomorphisms from the unit disk without the origin onto a punctured disc. For simplicity, let us denote

where is the unit disc and is the complex coordinate of . We will prove the following results.

Theorem 1. For any , there is no rotationally symmetric harmonic diffeomorphism from onto with its hyperbolic metric.

And vice versa as shown below.

Theorem 2. For any , there is no rotationally symmetric harmonic diffeomorphism from onto with its hyperbolic metric.

We will also consider the Euclidean case and will prove the following theorem.

Theorem 3. For any , there is no rotationally symmetric harmonic diffeomorphism from onto with its Euclidean metric; but on the other hand, there are rotationally symmetric harmonic diffeomorphisms from onto with its Euclidean metric.

This paper is organized as follows. In Section 2, we will prove Theorems 1 and 2. Theorem 3 will be proved in Section 3. At the last section, we will give another proof for the nonexistence of rotationally symmetric harmonic diffeomorphism from onto the hyperbolic disc.

2. Harmonic Maps from to with Its Hyperbolic Metric and Vice Versa

For convenience, let us recall the definition about the harmonic maps between surfaces. Let and be two oriented surfaces with metrics and , respectively, where and are local complex coordinates of and , respectively. A map from to is harmonic if and only if satisfies

Now let us prove Theorem 1.

Proof of Theorem 1. First of all, let us denote as the polar coordinates of and as the complex coordinates of in ; then the hyperbolic metric on can be written as

Here is the norm of with respect to the Euclidean metric.

We will prove this theorem by contradiction. Suppose is a rotationally symmetric harmonic diffeomorphism from onto , with the metric . Because , and the metric are rotationally symmetric, we can assume that such a map has the form . Substituting to (2), we can get

for . Since is a harmonic diffeomorphism from onto , we have

or

We will just deal with the case that (5) is satisfied; the rest case is similar. Let , then we have

Using this fact, we can get from (4) the following equation:

with , , and for .

Regarding as a function of , we have the following relations:

Using these facts, we can get from (9) the following equation:

for . Let ; from (11) we can get the following equation:

One can solve this Bernoulli equation to obtain

Here is a constant depending on the choice of the function . So

Since , we can get

Noting that is continuous in and is equal to as , or , one can get is uniformly bounded for . So the right-hand side of (15) is uniformly bounded, but the left-hand side will tend to as . Hence, we get a contradiction. Therefore, such does not exist, Theorem 1 has been proved.

We are going to prove Theorem 2.

Proof of Theorem 2. First of all, let us denote as the polar coordinates of and as the complex coordinates of in ; then the hyperbolic metric on can be written as

Here is the norm with respect to the Euclidean metric.

We will prove this theorem by contradiction. The idea is similar to the proof of Theorem 1. Suppose is a rotationally symmetric harmonic diffeomorphism from onto with the metric , with the form , then substituting to in (2), respectively, we can get

for . Since is a harmonic diffeomorphism from onto , we have

or

We will only deal with the case that (18) is satisfied; the rest case is similar. Let , then (17) can be rewritten as

for , with and .

Regarding as a function of , using a similar formula of (10), from (21) we can get

Similar to solving (11), we can get the solution to (22) as follows:

for some nonnegative constant depending on the choice of .

If is equal to , then ; this is in contradiction to (18).

If is positive, then taking integration on both sides of (23), we can get

So

with . On the other hand, from (18), we have . Hence, we get a contradiction. Therefore, such does not exist, Theorem 2 has been proved.

3. Harmonic Maps from to with Its Euclidean Metric and Vice Versa

Now let us consider the case of that the target has the Euclidean metric.

Proof of Theorem 3. Let us prove the first part of this theorem, that is, show the nonexistence of rotationally symmetric harmonic diffeomorphism from onto with its Euclidean metric. The idea is similar to the proof of Theorem 1, so we just sketch the proof here. Suppose there is such a harmonic diffeomorphism from onto with its Euclidean metric with the form , and then we can get

with

or

We will just deal with the case that (27) is satisfied; the rest case is similar. Let ; then we can get

Solving this equation, we can get

Here is a constant depending on the choice of . So

Here is a constant depending on the choice of . Hence

From (32), we can get . On the other hand, from (27), . We get a contradiction. Hence such a function does not exist; the first part of Theorem 3 holds.

Now let us prove the second part of this theorem, that is, show the existence of rotationally symmetric harmonic diffeomorphisms from onto with its Euclidean metric. It suffices to find a map from onto with the form such that

with , and for . Using the boundary condition and (32), we can get that

is a solution to (33).

Therefore, we finished the proof of Theorem 3.

4. Harmonic Maps from to the Hyperbolic Disc

In this section, we will give another proof of the following result.

Proposition 4. There is no rotationally symmetric harmonic diffeomorphism from onto the hyperbolic disc.

Proof. It is well-known that the hyperbolic metric on the unit disc is . We will also use the idea of the proof of Theorem 1. Suppose there is such a harmonic diffeomorphism from onto with its hyperbolic metric with the form , and then we can get

with

Regarding as a function of , setting , (35) can be rewritten as

That is,

One can solve this equation to obtain

for some nonnegative constant depending on the choice of the function .

If , then we can get for some constant . On the other hand, is a diffeomorphism, so as . This is a contradiction.

If , then for some positive constants and . So

This is in contradiction to the assumption that as .

Therefore, Proposition 4 holds.

Acknowledgments

The author (Xu-Qian Fan) would like to thank Professor Luen-fai Tam for his very worthy advice. The first author is partially supported by the National Natural Science Foundation of China (11201131); the second author is partially supported by the National Natural Science Foundation of China (11101106).

References

K. Akutagawa, “Harmonic diffeomorphisms of the hyperbolic plane,” Transactions of the American Mathematical Society, vol. 342, no. 1, pp. 325–342, 1994.

K. Akutagawa and S. Nishikawa, “The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space,” The Tohoku Mathematical Journal, vol. 42, no. 1, pp. 67–82, 1990.

T. K.-K. Au and T. Y.-H. Wan, “From local solutions to a global solution for the equation ,” in Geometry and Global Analysis, pp. 427–430, Tohoku University, Sendai, Japan, 1993.

T. K. K. Au, L.-F. Tam, and T. Y. H. Wan, “Hopf differentials and the images of harmonic maps,” Communications in Analysis and Geometry, vol. 10, no. 3, pp. 515–573, 2002.

T. K. K. Au and T. Y. H. Wan, “Images of harmonic maps with symmetry,” The Tohoku Mathematical Journal, vol. 57, no. 3, pp. 321–333, 2005.

Q. Chen and J. Eichhorn, “Harmonic diffeomorphisms between complete Riemann surfaces of negative curvature,” Asian Journal of Mathematics, vol. 13, no. 4, pp. 473–533, 2009.

L. F. Cheung and C. K. Law, “An initial value approach to rotationally symmetric harmonic maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 1–13, 2004.

H. I. Choi and A. Treibergs, “New examples of harmonic diffeomorphisms of the hyperbolic plane onto itself,” Manuscripta Mathematica, vol. 62, no. 2, pp. 249–256, 1988.

H. I. Choi and A. Treibergs, “Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space,” Journal of Differential Geometry, vol. 32, no. 3, pp. 775–817, 1990.

P. Collin and H. Rosenberg, “Construction of harmonic diffeomorphisms and minimal graphs,” Annals of Mathematics, vol. 172, no. 3, pp. 1879–1906, 2010.

J.-M. Coron and F. Hélein, “Harmonic diffeomorphisms, minimizing harmonic maps and rotational symmetry,” Compositio Mathematica, vol. 69, no. 2, pp. 175–228, 1989.

J. A. Gálvez and H. Rosenberg, “Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces,” American Journal of Mathematics, vol. 132, no. 5, pp. 1249–1273, 2010.

Z.-C. Han, “Remarks on the geometric behavior of harmonic maps between surfaces,” in Elliptic and Parabolic Methods in Geometry, pp. 57–66, A K Peters, Wellesley, Mass, USA, 1996.

Z.-C. Han, L.-F. Tam, A. Treibergs, and T. Wan, “Harmonic maps from the complex plane into surfaces with nonpositive curvature,” Communications in Analysis and Geometry, vol. 3, no. 1-2, pp. 85–114, 1995.

F. Hélein, “Harmonic diffeomorphisms between Riemannian manifolds,” in Variational Methods, vol. 4 of Progress in Nonlinear Differential Equations and Their Applications, pp. 309–318, Birkhäuser, Boston, Mass, USA, 1990.

F. Hélein, “Harmonic diffeomorphisms with rotational symmetry,” Journal für die Reine und Angewandte Mathematik, vol. 414, pp. 45–49, 1991.

E. Heinz, “Uber die Lusungen der Minimalflachengleichung,” in Nachrichten Der Akademie Der Wissenschaften in Gottingen, Mathematisch-physikalische klasse, pp. 51–56, 1952, Mathematisch-physikalisch-chemische abteilung.

J. Jost and R. Schoen, “On the existence of harmonic diffeomorphisms,” Inventiones Mathematicae, vol. 66, no. 2, pp. 353–359, 1982.

D. Kalaj, “On harmonic diffeomorphisms of the unit disc onto a convex domain,” Complex Variables. Theory and Application, vol. 48, no. 2, pp. 175–187, 2003.

P. Li, L.-F. Tam, and J. Wang, “Harmonic diffeomorphisms between Hadamard manifolds,” Transactions of the American Mathematical Society, vol. 347, no. 9, pp. 3645–3658, 1995.

J. Lohkamp, “Harmonic diffeomorphisms and Teichmüller theory,” Manuscripta Mathematica, vol. 71, no. 4, pp. 339–360, 1991.

V. Markovic, “Harmonic diffeomorphisms and conformal distortion of Riemann surfaces,” Communications in Analysis and Geometry, vol. 10, no. 4, pp. 847–876, 2002.

V. Markovic, “Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces,” Journal of the London Mathematical Society, vol. 65, no. 1, pp. 103–114, 2002.

A. Ratto and M. Rigoli, “On the asymptotic behaviour of rotationally symmetric harmonic maps,” Journal of Differential Equations, vol. 101, no. 1, pp. 15–27, 1993.

R. M. Schoen, “The role of harmonic mappings in rigidity and deformation problems,” in Complex Geometry, vol. 143 of Lecture Notes in Pure and Applied Mathematics, pp. 179–200, Dekker, New York, NY, USA, 1993.

Y. Shi, “On the construction of some new harmonic maps from to ,” Acta Mathematica Sinica, vol. 17, no. 2, pp. 301–304, 2001.

Y. Shi and L.-F. Tam, “Harmonic maps from to with symmetry,” Pacific Journal of Mathematics, vol. 202, no. 1, pp. 227–256, 2002.

A. Tachikawa, “Rotationally symmetric harmonic maps from a ball into a warped product manifold,” Manuscripta Mathematica, vol. 53, no. 3, pp. 235–254, 1985.

A. Tachikawa, “A nonexistence result for harmonic mappings from into ,” Tokyo Journal of Mathematics, vol. 11, no. 2, pp. 311–316, 1988.

L.-F. Tam and T. Y.-H. Wan, “Harmonic diffeomorphisms into Cartan-Hadamard surfaces with prescribed Hopf differentials,” Communications in Analysis and Geometry, vol. 2, no. 4, pp. 593–625, 1994.

L.-F. Tam and T. Y. H. Wan, “Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space,” Journal of Differential Geometry, vol. 42, no. 2, pp. 368–410, 1995.

T. Y.-H. Wan, “Constant mean curvature surface, harmonic maps, and universal Teichmüller space,” Journal of Differential Geometry, vol. 35, no. 3, pp. 643–657, 1992.

T. Y. H. Wan, “Review on harmonic diffeomorphisms between complete noncompact surfaces,” in Surveys in Geometric Analysis and Relativity, vol. 20 of Advanced Lectures in Mathematics, pp. 509–516, International Press, Somerville, Mass, USA, 2011.

D. Wu, “Harmonic diffeomorphisms between complete surfaces,” Annals of Global Analysis and Geometry, vol. 15, no. 2, pp. 133–139, 1997.