Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 512383 5 pageshttpdxdoiorg1011552013512383

Research ArticleRotationally Symmetric Harmonic Diffeomorphismsbetween Surfaces

Li Chen1 Shi-Zhong Du2 and Xu-Qian Fan3

1 Faculty of Mathematics amp Computer Science Hubei University Wuhan 430062 China2The School of Natural Sciences and Humanities Shenzhen Graduate School The Harbin Institute of TechnologyShenzhen 518055 China

3Department of Mathematics Jinan University Guangzhou 510632 China

Correspondence should be addressed to Xu-Qian Fan txqfanjnueducn

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

Academic Editor Yuriy Rogovchenko

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

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

1 Introduction

The existence of harmonic diffeomorphisms between com-plete Riemannian manifolds has been extensively studiedplease see for example [1ndash34] In particular Heinz [17]proved that there is no harmonic diffeomorphism from theunit disc onto C with its flat metric On the other handSchoen [25] mentioned a question about the existenceor nonexistence of a harmonic diffeomorphism from thecomplex plane onto the hyperbolic 2-space At the presenttime many beautiful results about the asymptotic behavior ofharmonic embedding from C into the hyperbolic plane havebeen obtained please see for example [4 5 14 32] or thereview [33] byWan and the references therein In 2010 Collinand Rosenberg [10] constructed harmonic diffeomorphismsfrom C onto the hyperbolic plane In [7 24 28 29] theauthors therein studied the rotational symmetry case Oneof their results is the nonexistence of rotationally symmetricharmonic diffeomorphism fromC onto the hyperbolic plane

In this paper we will study the existence or nonexistenceof rotationally symmetric harmonic diffeomorphisms fromthe unit disk without the origin onto a punctured disc Forsimplicity let us denote

Dlowast

= D 0 119875 (119886) = D |119911| le 119890

minus119886

for 119886 gt 0 (1)

whereD is the unit disc and 119911 is the complex coordinate ofCWe will prove the following results

Theorem 1 For any 119886 gt 0 there is no rotationally symmetricharmonic diffeomorphism from Dlowast onto 119875(119886) with its hyper-bolic metric

And vice versa as shown below

Theorem 2 For any 119886 gt 0 there is no rotationally symmetricharmonic diffeomorphism from 119875(119886) onto Dlowast with its hyper-bolic metric

We will also consider the Euclidean case and will provethe following theorem

Theorem 3 For any 119886 gt 0 there is no rotationally symmetricharmonic diffeomorphism from Dlowast onto 119875(119886) with its Euclid-ean metric but on the other hand there are rotationallysymmetric harmonic diffeomorphisms from 119875(119886) ontoDlowast withits Euclidean metric

This paper is organized as follows In Section 2 we willprove Theorems 1 and 2 Theorem 3 will be proved inSection 3 At the last section we will give another proof forthe nonexistence of rotationally symmetric harmonic diffeo-morphism from C onto the hyperbolic disc

2 Abstract and Applied Analysis

2 Harmonic Maps from Dlowast to 119875(119886) withIts Hyperbolic Metric and Vice Versa

For convenience let us recall the definition about the har-monic maps between surfaces Let 119872 and119873 be two orientedsurfaces withmetrics 1205912|119889119911|2 and 120590

2

|119889119906|

2 respectively where119911 and 119906 are local complex coordinates of 119872 and 119873 respec-tively A 119862

2 map 119906 from 119872 to 119873 is harmonic if and only if 119906satisfies

119906119911119911

+

2120590119906

120590

119906119911119906119911= 0 (2)

Now let us proveTheorem 1

Proof of Theorem 1 First of all let us denote (119903 120579) as the polarcoordinates ofDlowast and 119906 as the complex coordinates of119875(119886) inC then the hyperbolic metric 120590

1119889|119906| on 119875(119886) can be written

as

minus120587 |119889119906|

119886 |119906| sin ((120587119886) ln |119906|)

(3)

Here |119906| is the norm of 119906with respect to the EuclideanmetricWewill prove this theorem by contradiction Suppose 119906 is

a rotationally symmetric harmonic diffeomorphism fromDlowast

onto 119875(119886) with the metric 1205901119889|119906| Because Dlowast 119875(119886) and the

metric 1205901119889|119906| are rotationally symmetric we can assume that

such a map 119906 has the form 119906 = 119891(119903)119890

119894120579 Substituting 119906 1205901to

(2) we can get

119891

10158401015840

+

119891

1015840

119903

minus

119891

119903

2

minus

sin ((120587119886) ln119891) + (120587119886) cos ((120587119886) ln119891)

119891 sin ((120587119886) ln119891)

times ((119891

1015840

)

2

minus

119891

2

119903

2

) = 0

(4)

for 1 gt 119903 gt 0 Since 119906 is a harmonic diffeomorphism fromDlowast

onto 119875(119886) we have

119891 (0) = 119890

minus119886

119891 (1) = 1

119891

1015840

(119903) gt 0 for 1 gt 119903 gt 0

(5)

or

119891 (0) = 1 119891 (1) = 119890

minus119886

(6)

119891

1015840

(119903) lt 0 for 1 gt 119903 gt 0 (7)

We will just deal with the case that (5) is satisfied the restcase is similar Let 119865 = ln119891 isin (minus119886 0) then we have

119891

1015840

=

119891

1015840

119891

gt 0 119865

10158401015840

=

119891

10158401015840

119891

minus (

119891

1015840

119891

)

2

(8)

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

119865

10158401015840

+

1

119903

119865

1015840

minus

120587

119886

ctg(

120587

119886

119865) (119865

1015840

)

2

+

1

119903

2

120587

119886

ctg(

120587

119886

119865)

= 0 for 1 gt 119903 gt 0

(9)

with 119865(0) = minus119886 119865(1) = 0 and 119865

1015840

(119903) gt 0 for 1 gt 119903 gt 0Regarding 119903 as a function of 119865 we have the following

relations

119865119903= 119903

minus1

119865 119865

119903119903= minus119903

minus3

119865119903119865119865

(10)

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

119903

10158401015840

119903

minus (

119903

1015840

119903

)

2

+

120587

119886

ctg(

120587

119886

119865)

119903

1015840

119903

minus (

119903

1015840

119903

)

3

120587

119886

ctg(

120587

119886

119865) = 0

(11)

for 0 gt 119865 gt minus119886 Let 119909 = (ln 119903)

1015840

(119865) from (11) we can get thefollowing equation

119909

1015840

+

120587

119886

ctg(

120587

119886

119865) sdot 119909 minus

120587

119886

ctg(

120587

119886

119865) sdot 119909

3

= 0 (12)

One can solve this Bernoulli equation to obtain

119909

minus2

= 1 + 1198880(sin(

120587

119886

119865))

2

(13)

Here 1198880is a constant depending on the choice of the function

119891 So

119909 =

1

radic1 + 1198880(sin ((120587119886) 119865))

2

(14)

Since 119909 = (ln 119903)

1015840

(119865) we can get

(ln 119903) (119865) = int

119865

0

119909 (119905) 119889119905 = int

119865

0

1

radic1 + 1198880(sin ((120587119886) 119905))

2

119889119905

(15)

Noting that 119909(119865) is continuous in (minus119886 0) and is equal to 1

as 119865 = minus119886 or 0 one can get 119909 is uniformly bounded for119865 isin [minus119886 0] So the right-hand side of (15) is uniformlybounded but the left-hand side will tend to minusinfin as 119865 rarr minus119886Hence we get a contradiction Therefore such 119891 does notexist Theorem 1 has been proved

We are going to proveTheorem 2

Proof of Theorem 2 First of all let us denote (119903 120579) as the polarcoordinates of 119875(119886) and 119906 as the complex coordinates of Dlowastin C then the hyperbolic metric 120590

2119889|119906| onDlowast can be written

as|119889119906|

|119906| ln (1 |119906|)

(16)

Here |119906| is the norm with respect to the Euclidean metric

Abstract and Applied Analysis 3

We will prove this theorem by contradiction The idea issimilar to the proof of Theorem 1 Suppose 120595 is a rotationallysymmetric harmonic diffeomorphism from 119875(119886) onto Dlowast

with the metric 1205902119889|119906| with the form 120595 = 119892(119903)119890

119894120579 thensubstituting 120595 120590

2to 119906 120590 in (2) respectively we can get

119892

10158401015840

+

119892

1015840

119903

minus

119892

119903

2

minus

1 + ln119892

119892 ln119892

((119892

1015840

)

2

minus

119892

2

119903

2

) = 0 (17)

for 1 gt 119903 gt 119890

minus119886 Since V is a harmonic diffeomorphism from119875(119886) onto Dlowast we have

119892 (119890

minus119886

) = 0 119892 (1) = 1

119892

1015840

(119903) gt 0 for 1 gt 119903 gt 119890

minus119886

(18)

or

119892 (119890

minus119886

) = 1 119892 (1) = 0 (19)

119892

1015840

(119903) lt 0 for 1 gt 119903 gt 119890

minus119886

(20)

We will only deal with the case that (18) is satisfied therest case is similar Let 119866 = ln119892 then (17) can be rewritten as

119866

10158401015840

+

1

119903

119866

1015840

minus

1

119866

(119866

1015840

)

2

+

1

119903

2119866

= 0 (21)

for 1 gt 119903 gt 119890

minus119886 with 119866(1) = 0 and lim119903rarr119890minus119886119866(119903) = minusinfin

Regarding 119903 as a function of 119866 using a similar formula of(10) from (21) we can get

119903

10158401015840

119903

minus (

119903

1015840

119903

)

2

+

119903

1015840

119903119866

minus

1

119866

(

119903

1015840

119903

)

3

= 0 119866 isin (minusinfin 0) (22)

Similar to solving (11) we can get the solution to (22) asfollows

(ln 119903)

1015840

(119866) =

1

radic1 + 1198881119866

2

119866 isin (minusinfin 0) (23)

for some nonnegative constant 1198881depending on the choice of

119892If 1198881is equal to 0 then 119892 = 119903 this is in contradiction to

(18)If 1198881is positive then taking integration on both sides of

(23) we can get

(ln 119903) (119866) = int

119866

0

1

radic1 + 1198881119905

2

119889119905

=

1

radic1198881

ln(radic1198881119866 + radic1 + 119888

1119866

2)

(24)

So

119903 = (radic1198881ln119892 +

radic1 + 1198881ln2119892)

1radic1198881

(25)

with lim119892rarr0+

119903(119892) = 0 On the other hand from (18) we have119903(0) = 119890

minus119886 Hence we get a contradiction Therefore such 119892

does not exist Theorem 2 has been proved

3 Harmonic Maps from Dlowast to 119875(119886) with ItsEuclidean Metric and Vice Versa

Now let us consider the case of that the target has theEuclidean metric

Proof of Theorem 3 Let us prove the first part of this theoremthat is show the nonexistence of rotationally symmetric har-monic diffeomorphism fromDlowast onto 119875(119886)with its Euclideanmetric The idea is similar to the proof of Theorem 1 so wejust sketch the proof here Suppose there is such a harmonicdiffeomorphism 120593 from Dlowast onto 119875(119886) with its Euclideanmetric with the form 120593 = ℎ(119903)119890

119894120579 and then we can get

ℎ

10158401015840

+

1

119903

ℎ

1015840

minus

1

119903

2

ℎ = 0 for 1 gt 119903 gt 0 (26)

withℎ (0) = 119890

minus119886

ℎ (1) = 1

ℎ

1015840

(119903) gt 0 for 1 gt 119903 gt 0

(27)

orℎ (0) = 1 ℎ (1) = 119890

minus119886

ℎ

1015840

(119903) lt 0 for 1 gt 119903 gt 0

(28)

We will just deal with the case that (27) is satisfied therest case is similar Let 119867 = (ln ℎ)

1015840 then we can get

119867

1015840

+ 119867

2

+

1

119903

119867 = 0 for 1 gt 119903 gt 0 (29)

Solving this equation we can get

119867 =

1

119903

+

1

1198883119903

3minus 119903

2

=

1

119903

+

119888

2

3

1198883119903 minus 1

minus

1198883

119903

minus

1

119903

2

(30)

Here 1198883is a constant depending on the choice of ℎ So

ln ℎ = ln 119903 + 1198883ln (1198883119903 minus 1) minus 119888

3ln 119903 +

1

119903

+ 1198884 (31)

Here 1198884is a constant depending on the choice of ℎ Hence

ℎ = 119903(1198883119903 minus 1)

1198883

119903

minus1198883

119890

1119903

119890

1198884

(32)

From (32) we can get lim119903rarr0

ℎ(119903) = infin On the other handfrom (27) ℎ(0) = 119890

minus119886 We get a contradiction Hence such afunction ℎ does not exist the first part of Theorem 3 holds

Now let us prove the second part of this theorem thatis show the existence of rotationally symmetric harmonicdiffeomorphisms from 119875(119886) onto Dlowast with its Euclideanmetric It suffices to find a map from 119875(119886) onto Dlowast with theform 119902(119903)119890

119894120579 such that

119902

10158401015840

+

1

119903

119902

1015840

minus

1

119903

2

119902 = 0 for 1 gt 119903 gt 119890

minus119886 (33)

with 119902(119890

minus119886

) = 0 119902(1) = 1 and 119902

1015840

gt 0 for 1 gt 119903 gt 119890

minus119886 Usingthe boundary condition and (32) we can get that

119902 = 119890

minus1

(119890

119886

minus 1)

minus119890119886

119903(119890

119886

119903 minus 1)

119890119886

119903

minus119890119886

119890

1119903 (34)

is a solution to (33)Therefore we finished the proof of Theorem 3

4 Abstract and Applied Analysis

4 Harmonic Maps from C tothe Hyperbolic Disc

In this section we will give another proof of the followingresult

Proposition 4 There is no rotationally symmetric harmonicdiffeomorphism from C onto the hyperbolic disc

Proof It is well-known that the hyperbolic metric on theunit disc is (2(1 minus 119911|

2

))|119889119911| We will also use the idea ofthe proof of Theorem 1 Suppose there is such a harmonicdiffeomorphism 120601 from C onto D with its hyperbolic metricwith the form 120601 = 119896(119903)119890

119894120579 and then we can get

119896

10158401015840

+

1

119903

119896

1015840

minus

1

119903

2

119896 +

2119896

1 minus 119896

2

[(119896

1015840

)

2

minus

119896

2

119903

2

]

= 0 for 119903 gt 0

(35)

with

119896 (0) = 0 119896

1015840

(119903) gt 0 for 119903 gt 0 (36)

Regarding 119903 as a function of 119896 setting V = (ln 119903)

1015840

(119896) (35) canbe rewritten as

(1 minus 119896

2

) V1015840

minus 2119896V + V3

(119896 + 119896

3

) = 0 (37)

That is

(Vminus2

)

1015840

+

4119896

1 minus 119896

2

Vminus2

=

2 (119896 + 119896

3

)

1 minus 119896

2

(38)

One can solve this equation to obtain

Vminus2

= 119896

2

+ 1198885(1 minus 119896

2

)

2 (39)

for some nonnegative constant 1198885depending on the choice of

the function 119896If 1198885= 0 then we can get 119903 = 119888

6119896 for some constant 119888

6 On

the other hand 120601 is a diffeomorphism so 119896 rarr 1 as 119903 rarr infinThis is a contradiction

If 1198885gt 0 then 119887

1ge 119896

2

+ 1198885(1 minus 119896

2

)

2

ge 1198872for some positive

constants 1198871and 1198872 So

10038161003816100381610038161003816

(ln 119903)

1015840

(119896)

10038161003816100381610038161003816

le

1

radic1198872

(40)

This is in contradiction to the assumption that 119903 rarr infin as119896 rarr 1

Therefore Proposition 4 holds

Acknowledgments

The author (Xu-Qian Fan) would like to thank ProfessorLuen-fai Tam for his very worthy advice The first author ispartially supported by the National Natural Science Foun-dation of China (11201131) the second author is partiallysupported by the National Natural Science Foundation ofChina (11101106)

References

[1] K Akutagawa ldquoHarmonic diffeomorphisms of the hyperbolicplanerdquo Transactions of the American Mathematical Society vol342 no 1 pp 325ndash342 1994

[2] K Akutagawa and S Nishikawa ldquoThe Gauss map and spacelikesurfaces with prescribed mean curvature in Minkowski 3-spacerdquoThe Tohoku Mathematical Journal vol 42 no 1 pp 67ndash82 1990

[3] T K-K Au and T Y-H Wan ldquoFrom local solutions to a globalsolution for the equationΔ119908 = 119890

2119908

minusΦ

2

119890

minus2119908rdquo inGeometry andGlobal Analysis pp 427ndash430 Tohoku University Sendai Japan1993

[4] T K K Au L-F Tam and T Y HWan ldquoHopf differentials andthe images of harmonicmapsrdquoCommunications in Analysis andGeometry vol 10 no 3 pp 515ndash573 2002

[5] T K K Au and T Y H Wan ldquoImages of harmonic maps withsymmetryrdquoThe Tohoku Mathematical Journal vol 57 no 3 pp321ndash333 2005

[6] Q Chen and J Eichhorn ldquoHarmonic diffeomorphisms betweencomplete Riemann surfaces of negative curvaturerdquo Asian Jour-nal of Mathematics vol 13 no 4 pp 473ndash533 2009

[7] L F Cheung and C K Law ldquoAn initial value approach to rota-tionally symmetric harmonic mapsrdquo Journal of MathematicalAnalysis and Applications vol 289 no 1 pp 1ndash13 2004

[8] H I Choi and A Treibergs ldquoNew examples of harmonic dif-feomorphisms of the hyperbolic plane onto itselfrdquoManuscriptaMathematica vol 62 no 2 pp 249ndash256 1988

[9] H I Choi and A Treibergs ldquoGauss maps of spacelike constantmean curvature hypersurfaces of Minkowski spacerdquo Journal ofDifferential Geometry vol 32 no 3 pp 775ndash817 1990

[10] P Collin and H Rosenberg ldquoConstruction of harmonic diffeo-morphisms and minimal graphsrdquo Annals of Mathematics vol172 no 3 pp 1879ndash1906 2010

[11] J-M Coron and F Helein ldquoHarmonic diffeomorphisms min-imizing harmonic maps and rotational symmetryrdquo CompositioMathematica vol 69 no 2 pp 175ndash228 1989

[12] J A Galvez and H Rosenberg ldquoMinimal surfaces and har-monic diffeomorphisms from the complex plane onto certainHadamard surfacesrdquoAmerican Journal of Mathematics vol 132no 5 pp 1249ndash1273 2010

[13] Z-C Han ldquoRemarks on the geometric behavior of harmonicmaps between surfacesrdquo in Elliptic and Parabolic Methods inGeometry pp 57ndash66 A K Peters Wellesley Mass USA 1996

[14] Z-C Han L-F Tam A Treibergs and T Wan ldquoHarmonicmaps from the complex plane into surfaces with nonpositivecurvaturerdquo Communications in Analysis and Geometry vol 3no 1-2 pp 85ndash114 1995

[15] F Helein ldquoHarmonic diffeomorphisms between Riemannianmanifoldsrdquo in Variational Methods vol 4 of Progress in Nonlin-ear Differential Equations and Their Applications pp 309ndash318Birkhauser Boston Mass USA 1990

[16] F Helein ldquoHarmonic diffeomorphisms with rotational symme-tryrdquo Journal fur die Reine und AngewandteMathematik vol 414pp 45ndash49 1991

[17] E Heinz ldquoUber die Lusungen der Minimalflachengleichungrdquoin Nachrichten Der Akademie Der Wissenschaften in Got-tingen Mathematisch-physikalische klasse pp 51ndash56 1952Mathematisch-physikalisch-chemische abteilung

[18] J Jost and R Schoen ldquoOn the existence of harmonic diffeomor-phismsrdquo Inventiones Mathematicae vol 66 no 2 pp 353ndash3591982

Abstract and Applied Analysis 5

[19] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables Theory and Applicationvol 48 no 2 pp 175ndash187 2003

[20] P Li L-F Tam and J Wang ldquoHarmonic diffeomorphismsbetween Hadamard manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 9 pp 3645ndash3658 1995

[21] J Lohkamp ldquoHarmonic diffeomorphisms and Teichmullertheoryrdquo Manuscripta Mathematica vol 71 no 4 pp 339ndash3601991

[22] V Markovic ldquoHarmonic diffeomorphisms and conformal dis-tortion of Riemann surfacesrdquo Communications in Analysis andGeometry vol 10 no 4 pp 847ndash876 2002

[23] V Markovic ldquoHarmonic diffeomorphisms of noncompact sur-faces and Teichmuller spacesrdquo Journal of the LondonMathemat-ical Society vol 65 no 1 pp 103ndash114 2002

[24] A Ratto and M Rigoli ldquoOn the asymptotic behaviour ofrotationally symmetric harmonic mapsrdquo Journal of DifferentialEquations vol 101 no 1 pp 15ndash27 1993

[25] R M Schoen ldquoThe role of harmonic mappings in rigidityand deformation problemsrdquo in Complex Geometry vol 143 ofLecture Notes in Pure and Applied Mathematics pp 179ndash200Dekker New York NY USA 1993

[26] Y Shi ldquoOn the construction of some new harmonic maps fromR119898 to H119898rdquo Acta Mathematica Sinica vol 17 no 2 pp 301ndash3042001

[27] Y Shi and L-F Tam ldquoHarmonic maps from R119899 to H119898 withsymmetryrdquo Pacific Journal of Mathematics vol 202 no 1 pp227ndash256 2002

[28] A Tachikawa ldquoRotationally symmetric harmonic maps from aball into a warped product manifoldrdquo Manuscripta Mathemat-ica vol 53 no 3 pp 235ndash254 1985

[29] A Tachikawa ldquoA nonexistence result for harmonic mappingsfrom R119899 into H119899rdquo Tokyo Journal of Mathematics vol 11 no 2pp 311ndash316 1988

[30] L-F Tam and T Y-H Wan ldquoHarmonic diffeomorphisms intoCartan-Hadamard surfaces with prescribed Hopf differentialsrdquoCommunications in Analysis and Geometry vol 2 no 4 pp593ndash625 1994

[31] L-F Tam and T Y H Wan ldquoQuasi-conformal harmonicdiffeomorphism and the universal Teichmuller spacerdquo Journalof Differential Geometry vol 42 no 2 pp 368ndash410 1995

[32] T Y-H Wan ldquoConstant mean curvature surface harmonicmaps and universal Teichmuller spacerdquo Journal of DifferentialGeometry vol 35 no 3 pp 643ndash657 1992

[33] T Y H Wan ldquoReview on harmonic diffeomorphisms betweencomplete noncompact surfacesrdquo in Surveys in Geometric Anal-ysis and Relativity vol 20 of Advanced Lectures in Mathematicspp 509ndash516 International Press Somerville Mass USA 2011

[34] D Wu ldquoHarmonic diffeomorphisms between complete sur-facesrdquo Annals of Global Analysis and Geometry vol 15 no 2pp 133ndash139 1997

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

2 Harmonic Maps from Dlowast to 119875(119886) withIts Hyperbolic Metric and Vice Versa

For convenience let us recall the definition about the har-monic maps between surfaces Let 119872 and119873 be two orientedsurfaces withmetrics 1205912|119889119911|2 and 120590

2

|119889119906|

2 respectively where119911 and 119906 are local complex coordinates of 119872 and 119873 respec-tively A 119862

2 map 119906 from 119872 to 119873 is harmonic if and only if 119906satisfies

119906119911119911

+

2120590119906

120590

119906119911119906119911= 0 (2)

Now let us proveTheorem 1

Proof of Theorem 1 First of all let us denote (119903 120579) as the polarcoordinates ofDlowast and 119906 as the complex coordinates of119875(119886) inC then the hyperbolic metric 120590

1119889|119906| on 119875(119886) can be written

as

minus120587 |119889119906|

119886 |119906| sin ((120587119886) ln |119906|)

(3)

Here |119906| is the norm of 119906with respect to the EuclideanmetricWewill prove this theorem by contradiction Suppose 119906 is

a rotationally symmetric harmonic diffeomorphism fromDlowast

onto 119875(119886) with the metric 1205901119889|119906| Because Dlowast 119875(119886) and the

metric 1205901119889|119906| are rotationally symmetric we can assume that

such a map 119906 has the form 119906 = 119891(119903)119890

119894120579 Substituting 119906 1205901to

(2) we can get

119891

10158401015840

+

119891

1015840

119903

minus

119891

119903

2

minus

sin ((120587119886) ln119891) + (120587119886) cos ((120587119886) ln119891)

119891 sin ((120587119886) ln119891)

times ((119891

1015840

)

2

minus

119891

2

119903

2

) = 0

(4)

for 1 gt 119903 gt 0 Since 119906 is a harmonic diffeomorphism fromDlowast

onto 119875(119886) we have

119891 (0) = 119890

minus119886

119891 (1) = 1

119891

1015840

(119903) gt 0 for 1 gt 119903 gt 0

(5)

or

119891 (0) = 1 119891 (1) = 119890

minus119886

(6)

119891

1015840

(119903) lt 0 for 1 gt 119903 gt 0 (7)

We will just deal with the case that (5) is satisfied the restcase is similar Let 119865 = ln119891 isin (minus119886 0) then we have

119891

1015840

=

119891

1015840

119891

gt 0 119865

10158401015840

=

119891

10158401015840

119891

minus (

119891

1015840

119891

)

2

(8)

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

119865

10158401015840

+

1

119903

119865

1015840

minus

120587

119886

ctg(

120587

119886

119865) (119865

1015840

)

2

+

1

119903

2

120587

119886

ctg(

120587

119886

119865)

= 0 for 1 gt 119903 gt 0

(9)

with 119865(0) = minus119886 119865(1) = 0 and 119865

1015840

(119903) gt 0 for 1 gt 119903 gt 0Regarding 119903 as a function of 119865 we have the following

relations

119865119903= 119903

minus1

119865 119865

119903119903= minus119903

minus3

119865119903119865119865

(10)

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

119903

10158401015840

119903

minus (

119903

1015840

119903

)

2

+

120587

119886

ctg(

120587

119886

119865)

119903

1015840

119903

minus (

119903

1015840

119903

)

3

120587

119886

ctg(

120587

119886

119865) = 0

(11)

for 0 gt 119865 gt minus119886 Let 119909 = (ln 119903)

1015840

(119865) from (11) we can get thefollowing equation

119909

1015840

+

120587

119886

ctg(

120587

119886

119865) sdot 119909 minus

120587

119886

ctg(

120587

119886

119865) sdot 119909

3

= 0 (12)

One can solve this Bernoulli equation to obtain

119909

minus2

= 1 + 1198880(sin(

120587

119886

119865))

2

(13)

Here 1198880is a constant depending on the choice of the function

119891 So

119909 =

1

radic1 + 1198880(sin ((120587119886) 119865))

2

(14)

Since 119909 = (ln 119903)

1015840

(119865) we can get

(ln 119903) (119865) = int

119865

0

119909 (119905) 119889119905 = int

119865

0

1

radic1 + 1198880(sin ((120587119886) 119905))

2

119889119905

(15)

Noting that 119909(119865) is continuous in (minus119886 0) and is equal to 1

as 119865 = minus119886 or 0 one can get 119909 is uniformly bounded for119865 isin [minus119886 0] So the right-hand side of (15) is uniformlybounded but the left-hand side will tend to minusinfin as 119865 rarr minus119886Hence we get a contradiction Therefore such 119891 does notexist Theorem 1 has been proved

We are going to proveTheorem 2

Proof of Theorem 2 First of all let us denote (119903 120579) as the polarcoordinates of 119875(119886) and 119906 as the complex coordinates of Dlowastin C then the hyperbolic metric 120590

2119889|119906| onDlowast can be written

as|119889119906|

|119906| ln (1 |119906|)

(16)

Here |119906| is the norm with respect to the Euclidean metric

Abstract and Applied Analysis 3

We will prove this theorem by contradiction The idea issimilar to the proof of Theorem 1 Suppose 120595 is a rotationallysymmetric harmonic diffeomorphism from 119875(119886) onto Dlowast

with the metric 1205902119889|119906| with the form 120595 = 119892(119903)119890

119894120579 thensubstituting 120595 120590

2to 119906 120590 in (2) respectively we can get

119892

10158401015840

+

119892

1015840

119903

minus

119892

119903

2

minus

1 + ln119892

119892 ln119892

((119892

1015840

)

2

minus

119892

2

119903

2

) = 0 (17)

for 1 gt 119903 gt 119890

minus119886 Since V is a harmonic diffeomorphism from119875(119886) onto Dlowast we have

119892 (119890

minus119886

) = 0 119892 (1) = 1

119892

1015840

(119903) gt 0 for 1 gt 119903 gt 119890

minus119886

(18)

or

119892 (119890

minus119886

) = 1 119892 (1) = 0 (19)

119892

1015840

(119903) lt 0 for 1 gt 119903 gt 119890

minus119886

(20)

We will only deal with the case that (18) is satisfied therest case is similar Let 119866 = ln119892 then (17) can be rewritten as

119866

10158401015840

+

1

119903

119866

1015840

minus

1

119866

(119866

1015840

)

2

+

1

119903

2119866

= 0 (21)

for 1 gt 119903 gt 119890

minus119886 with 119866(1) = 0 and lim119903rarr119890minus119886119866(119903) = minusinfin

Regarding 119903 as a function of 119866 using a similar formula of(10) from (21) we can get

119903

10158401015840

119903

minus (

119903

1015840

119903

)

2

+

119903

1015840

119903119866

minus

1

119866

(

119903

1015840

119903

)

3

= 0 119866 isin (minusinfin 0) (22)

Similar to solving (11) we can get the solution to (22) asfollows

(ln 119903)

1015840

(119866) =

1

radic1 + 1198881119866

2

119866 isin (minusinfin 0) (23)

for some nonnegative constant 1198881depending on the choice of

119892If 1198881is equal to 0 then 119892 = 119903 this is in contradiction to

(18)If 1198881is positive then taking integration on both sides of

(23) we can get

(ln 119903) (119866) = int

119866

0

1

radic1 + 1198881119905

2

119889119905

=

1

radic1198881

ln(radic1198881119866 + radic1 + 119888

1119866

2)

(24)

So

119903 = (radic1198881ln119892 +

radic1 + 1198881ln2119892)

1radic1198881

(25)

with lim119892rarr0+

119903(119892) = 0 On the other hand from (18) we have119903(0) = 119890

minus119886 Hence we get a contradiction Therefore such 119892

does not exist Theorem 2 has been proved

3 Harmonic Maps from Dlowast to 119875(119886) with ItsEuclidean Metric and Vice Versa

Now let us consider the case of that the target has theEuclidean metric

Proof of Theorem 3 Let us prove the first part of this theoremthat is show the nonexistence of rotationally symmetric har-monic diffeomorphism fromDlowast onto 119875(119886)with its Euclideanmetric The idea is similar to the proof of Theorem 1 so wejust sketch the proof here Suppose there is such a harmonicdiffeomorphism 120593 from Dlowast onto 119875(119886) with its Euclideanmetric with the form 120593 = ℎ(119903)119890

119894120579 and then we can get

ℎ

10158401015840

+

1

119903

ℎ

1015840

minus

1

119903

2

ℎ = 0 for 1 gt 119903 gt 0 (26)

withℎ (0) = 119890

minus119886

ℎ (1) = 1

ℎ

1015840

(119903) gt 0 for 1 gt 119903 gt 0

(27)

orℎ (0) = 1 ℎ (1) = 119890

minus119886

ℎ

1015840

(119903) lt 0 for 1 gt 119903 gt 0

(28)

We will just deal with the case that (27) is satisfied therest case is similar Let 119867 = (ln ℎ)

1015840 then we can get

119867

1015840

+ 119867

2

+

1

119903

119867 = 0 for 1 gt 119903 gt 0 (29)

Solving this equation we can get

119867 =

1

119903

+

1

1198883119903

3minus 119903

2

=

1

119903

+

119888

2

3

1198883119903 minus 1

minus

1198883

119903

minus

1

119903

2

(30)

Here 1198883is a constant depending on the choice of ℎ So

ln ℎ = ln 119903 + 1198883ln (1198883119903 minus 1) minus 119888

3ln 119903 +

1

119903

+ 1198884 (31)

Here 1198884is a constant depending on the choice of ℎ Hence

ℎ = 119903(1198883119903 minus 1)

1198883

119903

minus1198883

119890

1119903

119890

1198884

(32)

From (32) we can get lim119903rarr0

ℎ(119903) = infin On the other handfrom (27) ℎ(0) = 119890

minus119886 We get a contradiction Hence such afunction ℎ does not exist the first part of Theorem 3 holds

Now let us prove the second part of this theorem thatis show the existence of rotationally symmetric harmonicdiffeomorphisms from 119875(119886) onto Dlowast with its Euclideanmetric It suffices to find a map from 119875(119886) onto Dlowast with theform 119902(119903)119890

119894120579 such that

119902

10158401015840

+

1

119903

119902

1015840

minus

1

119903

2

119902 = 0 for 1 gt 119903 gt 119890

minus119886 (33)

with 119902(119890

minus119886

) = 0 119902(1) = 1 and 119902

1015840

gt 0 for 1 gt 119903 gt 119890

minus119886 Usingthe boundary condition and (32) we can get that

119902 = 119890

minus1

(119890

119886

minus 1)

minus119890119886

119903(119890

119886

119903 minus 1)

119890119886

119903

minus119890119886

119890

1119903 (34)

is a solution to (33)Therefore we finished the proof of Theorem 3

4 Abstract and Applied Analysis

4 Harmonic Maps from C tothe Hyperbolic Disc

In this section we will give another proof of the followingresult

Proposition 4 There is no rotationally symmetric harmonicdiffeomorphism from C onto the hyperbolic disc

Proof It is well-known that the hyperbolic metric on theunit disc is (2(1 minus 119911|

2

))|119889119911| We will also use the idea ofthe proof of Theorem 1 Suppose there is such a harmonicdiffeomorphism 120601 from C onto D with its hyperbolic metricwith the form 120601 = 119896(119903)119890

119894120579 and then we can get

119896

10158401015840

+

1

119903

119896

1015840

minus

1

119903

2

119896 +

2119896

1 minus 119896

2

[(119896

1015840

)

2

minus

119896

2

119903

2

]

= 0 for 119903 gt 0

(35)

with

119896 (0) = 0 119896

1015840

(119903) gt 0 for 119903 gt 0 (36)

Regarding 119903 as a function of 119896 setting V = (ln 119903)

1015840

(119896) (35) canbe rewritten as

(1 minus 119896

2

) V1015840

minus 2119896V + V3

(119896 + 119896

3

) = 0 (37)

That is

(Vminus2

)

1015840

+

4119896

1 minus 119896

2

Vminus2

=

2 (119896 + 119896

3

)

1 minus 119896

2

(38)

One can solve this equation to obtain

Vminus2

= 119896

2

+ 1198885(1 minus 119896

2

)

2 (39)

for some nonnegative constant 1198885depending on the choice of

the function 119896If 1198885= 0 then we can get 119903 = 119888

6119896 for some constant 119888

6 On

the other hand 120601 is a diffeomorphism so 119896 rarr 1 as 119903 rarr infinThis is a contradiction

If 1198885gt 0 then 119887

1ge 119896

2

+ 1198885(1 minus 119896

2

)

2

ge 1198872for some positive

constants 1198871and 1198872 So

10038161003816100381610038161003816

(ln 119903)

1015840

(119896)

10038161003816100381610038161003816

le

1

radic1198872

(40)

This is in contradiction to the assumption that 119903 rarr infin as119896 rarr 1

Therefore Proposition 4 holds

Acknowledgments

The author (Xu-Qian Fan) would like to thank ProfessorLuen-fai Tam for his very worthy advice The first author ispartially supported by the National Natural Science Foun-dation of China (11201131) the second author is partiallysupported by the National Natural Science Foundation ofChina (11101106)

References

[1] K Akutagawa ldquoHarmonic diffeomorphisms of the hyperbolicplanerdquo Transactions of the American Mathematical Society vol342 no 1 pp 325ndash342 1994

[2] K Akutagawa and S Nishikawa ldquoThe Gauss map and spacelikesurfaces with prescribed mean curvature in Minkowski 3-spacerdquoThe Tohoku Mathematical Journal vol 42 no 1 pp 67ndash82 1990

[3] T K-K Au and T Y-H Wan ldquoFrom local solutions to a globalsolution for the equationΔ119908 = 119890

2119908

minusΦ

2

119890

minus2119908rdquo inGeometry andGlobal Analysis pp 427ndash430 Tohoku University Sendai Japan1993

[4] T K K Au L-F Tam and T Y HWan ldquoHopf differentials andthe images of harmonicmapsrdquoCommunications in Analysis andGeometry vol 10 no 3 pp 515ndash573 2002

[5] T K K Au and T Y H Wan ldquoImages of harmonic maps withsymmetryrdquoThe Tohoku Mathematical Journal vol 57 no 3 pp321ndash333 2005

[6] Q Chen and J Eichhorn ldquoHarmonic diffeomorphisms betweencomplete Riemann surfaces of negative curvaturerdquo Asian Jour-nal of Mathematics vol 13 no 4 pp 473ndash533 2009

[7] L F Cheung and C K Law ldquoAn initial value approach to rota-tionally symmetric harmonic mapsrdquo Journal of MathematicalAnalysis and Applications vol 289 no 1 pp 1ndash13 2004

[8] H I Choi and A Treibergs ldquoNew examples of harmonic dif-feomorphisms of the hyperbolic plane onto itselfrdquoManuscriptaMathematica vol 62 no 2 pp 249ndash256 1988

[9] H I Choi and A Treibergs ldquoGauss maps of spacelike constantmean curvature hypersurfaces of Minkowski spacerdquo Journal ofDifferential Geometry vol 32 no 3 pp 775ndash817 1990

[10] P Collin and H Rosenberg ldquoConstruction of harmonic diffeo-morphisms and minimal graphsrdquo Annals of Mathematics vol172 no 3 pp 1879ndash1906 2010

[11] J-M Coron and F Helein ldquoHarmonic diffeomorphisms min-imizing harmonic maps and rotational symmetryrdquo CompositioMathematica vol 69 no 2 pp 175ndash228 1989

[12] J A Galvez and H Rosenberg ldquoMinimal surfaces and har-monic diffeomorphisms from the complex plane onto certainHadamard surfacesrdquoAmerican Journal of Mathematics vol 132no 5 pp 1249ndash1273 2010

[13] Z-C Han ldquoRemarks on the geometric behavior of harmonicmaps between surfacesrdquo in Elliptic and Parabolic Methods inGeometry pp 57ndash66 A K Peters Wellesley Mass USA 1996

[14] Z-C Han L-F Tam A Treibergs and T Wan ldquoHarmonicmaps from the complex plane into surfaces with nonpositivecurvaturerdquo Communications in Analysis and Geometry vol 3no 1-2 pp 85ndash114 1995

[15] F Helein ldquoHarmonic diffeomorphisms between Riemannianmanifoldsrdquo in Variational Methods vol 4 of Progress in Nonlin-ear Differential Equations and Their Applications pp 309ndash318Birkhauser Boston Mass USA 1990

[16] F Helein ldquoHarmonic diffeomorphisms with rotational symme-tryrdquo Journal fur die Reine und AngewandteMathematik vol 414pp 45ndash49 1991

[17] E Heinz ldquoUber die Lusungen der Minimalflachengleichungrdquoin Nachrichten Der Akademie Der Wissenschaften in Got-tingen Mathematisch-physikalische klasse pp 51ndash56 1952Mathematisch-physikalisch-chemische abteilung

[18] J Jost and R Schoen ldquoOn the existence of harmonic diffeomor-phismsrdquo Inventiones Mathematicae vol 66 no 2 pp 353ndash3591982

Abstract and Applied Analysis 5

[19] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables Theory and Applicationvol 48 no 2 pp 175ndash187 2003

[20] P Li L-F Tam and J Wang ldquoHarmonic diffeomorphismsbetween Hadamard manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 9 pp 3645ndash3658 1995

[21] J Lohkamp ldquoHarmonic diffeomorphisms and Teichmullertheoryrdquo Manuscripta Mathematica vol 71 no 4 pp 339ndash3601991

[22] V Markovic ldquoHarmonic diffeomorphisms and conformal dis-tortion of Riemann surfacesrdquo Communications in Analysis andGeometry vol 10 no 4 pp 847ndash876 2002

[23] V Markovic ldquoHarmonic diffeomorphisms of noncompact sur-faces and Teichmuller spacesrdquo Journal of the LondonMathemat-ical Society vol 65 no 1 pp 103ndash114 2002

[24] A Ratto and M Rigoli ldquoOn the asymptotic behaviour ofrotationally symmetric harmonic mapsrdquo Journal of DifferentialEquations vol 101 no 1 pp 15ndash27 1993

[25] R M Schoen ldquoThe role of harmonic mappings in rigidityand deformation problemsrdquo in Complex Geometry vol 143 ofLecture Notes in Pure and Applied Mathematics pp 179ndash200Dekker New York NY USA 1993

[26] Y Shi ldquoOn the construction of some new harmonic maps fromR119898 to H119898rdquo Acta Mathematica Sinica vol 17 no 2 pp 301ndash3042001

[27] Y Shi and L-F Tam ldquoHarmonic maps from R119899 to H119898 withsymmetryrdquo Pacific Journal of Mathematics vol 202 no 1 pp227ndash256 2002

[28] A Tachikawa ldquoRotationally symmetric harmonic maps from aball into a warped product manifoldrdquo Manuscripta Mathemat-ica vol 53 no 3 pp 235ndash254 1985

[29] A Tachikawa ldquoA nonexistence result for harmonic mappingsfrom R119899 into H119899rdquo Tokyo Journal of Mathematics vol 11 no 2pp 311ndash316 1988

[30] L-F Tam and T Y-H Wan ldquoHarmonic diffeomorphisms intoCartan-Hadamard surfaces with prescribed Hopf differentialsrdquoCommunications in Analysis and Geometry vol 2 no 4 pp593ndash625 1994

[31] L-F Tam and T Y H Wan ldquoQuasi-conformal harmonicdiffeomorphism and the universal Teichmuller spacerdquo Journalof Differential Geometry vol 42 no 2 pp 368ndash410 1995

[32] T Y-H Wan ldquoConstant mean curvature surface harmonicmaps and universal Teichmuller spacerdquo Journal of DifferentialGeometry vol 35 no 3 pp 643ndash657 1992

[33] T Y H Wan ldquoReview on harmonic diffeomorphisms betweencomplete noncompact surfacesrdquo in Surveys in Geometric Anal-ysis and Relativity vol 20 of Advanced Lectures in Mathematicspp 509ndash516 International Press Somerville Mass USA 2011

[34] D Wu ldquoHarmonic diffeomorphisms between complete sur-facesrdquo Annals of Global Analysis and Geometry vol 15 no 2pp 133ndash139 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

We will prove this theorem by contradiction The idea issimilar to the proof of Theorem 1 Suppose 120595 is a rotationallysymmetric harmonic diffeomorphism from 119875(119886) onto Dlowast

with the metric 1205902119889|119906| with the form 120595 = 119892(119903)119890

119894120579 thensubstituting 120595 120590

2to 119906 120590 in (2) respectively we can get

119892

10158401015840

+

119892

1015840

119903

minus

119892

119903

2

minus

1 + ln119892

119892 ln119892

((119892

1015840

)

2

minus

119892

2

119903

2

) = 0 (17)

for 1 gt 119903 gt 119890

minus119886 Since V is a harmonic diffeomorphism from119875(119886) onto Dlowast we have

119892 (119890

minus119886

) = 0 119892 (1) = 1

119892

1015840

(119903) gt 0 for 1 gt 119903 gt 119890

minus119886

(18)

or

119892 (119890

minus119886

) = 1 119892 (1) = 0 (19)

119892

1015840

(119903) lt 0 for 1 gt 119903 gt 119890

minus119886

(20)

We will only deal with the case that (18) is satisfied therest case is similar Let 119866 = ln119892 then (17) can be rewritten as

119866

10158401015840

+

1

119903

119866

1015840

minus

1

119866

(119866

1015840

)

2

+

1

119903

2119866

= 0 (21)

for 1 gt 119903 gt 119890

minus119886 with 119866(1) = 0 and lim119903rarr119890minus119886119866(119903) = minusinfin

Regarding 119903 as a function of 119866 using a similar formula of(10) from (21) we can get

119903

10158401015840

119903

minus (

119903

1015840

119903

)

2

+

119903

1015840

119903119866

minus

1

119866

(

119903

1015840

119903

)

3

= 0 119866 isin (minusinfin 0) (22)

Similar to solving (11) we can get the solution to (22) asfollows

(ln 119903)

1015840

(119866) =

1

radic1 + 1198881119866

2

119866 isin (minusinfin 0) (23)

for some nonnegative constant 1198881depending on the choice of

119892If 1198881is equal to 0 then 119892 = 119903 this is in contradiction to

(18)If 1198881is positive then taking integration on both sides of

(23) we can get

(ln 119903) (119866) = int

119866

0

1

radic1 + 1198881119905

2

119889119905

=

1

radic1198881

ln(radic1198881119866 + radic1 + 119888

1119866

2)

(24)

So

119903 = (radic1198881ln119892 +

radic1 + 1198881ln2119892)

1radic1198881

(25)

with lim119892rarr0+

119903(119892) = 0 On the other hand from (18) we have119903(0) = 119890

minus119886 Hence we get a contradiction Therefore such 119892

does not exist Theorem 2 has been proved

3 Harmonic Maps from Dlowast to 119875(119886) with ItsEuclidean Metric and Vice Versa

Now let us consider the case of that the target has theEuclidean metric

Proof of Theorem 3 Let us prove the first part of this theoremthat is show the nonexistence of rotationally symmetric har-monic diffeomorphism fromDlowast onto 119875(119886)with its Euclideanmetric The idea is similar to the proof of Theorem 1 so wejust sketch the proof here Suppose there is such a harmonicdiffeomorphism 120593 from Dlowast onto 119875(119886) with its Euclideanmetric with the form 120593 = ℎ(119903)119890

119894120579 and then we can get

ℎ

10158401015840

+

1

119903

ℎ

1015840

minus

1

119903

2

ℎ = 0 for 1 gt 119903 gt 0 (26)

withℎ (0) = 119890

minus119886

ℎ (1) = 1

ℎ

1015840

(119903) gt 0 for 1 gt 119903 gt 0

(27)

orℎ (0) = 1 ℎ (1) = 119890

minus119886

ℎ

1015840

(119903) lt 0 for 1 gt 119903 gt 0

(28)

We will just deal with the case that (27) is satisfied therest case is similar Let 119867 = (ln ℎ)

1015840 then we can get

119867

1015840

+ 119867

2

+

1

119903

119867 = 0 for 1 gt 119903 gt 0 (29)

Solving this equation we can get

119867 =

1

119903

+

1

1198883119903

3minus 119903

2

=

1

119903

+

119888

2

3

1198883119903 minus 1

minus

1198883

119903

minus

1

119903

2

(30)

Here 1198883is a constant depending on the choice of ℎ So

ln ℎ = ln 119903 + 1198883ln (1198883119903 minus 1) minus 119888

3ln 119903 +

1

119903

+ 1198884 (31)

Here 1198884is a constant depending on the choice of ℎ Hence

ℎ = 119903(1198883119903 minus 1)

1198883

119903

minus1198883

119890

1119903

119890

1198884

(32)

From (32) we can get lim119903rarr0

ℎ(119903) = infin On the other handfrom (27) ℎ(0) = 119890

minus119886 We get a contradiction Hence such afunction ℎ does not exist the first part of Theorem 3 holds

Now let us prove the second part of this theorem thatis show the existence of rotationally symmetric harmonicdiffeomorphisms from 119875(119886) onto Dlowast with its Euclideanmetric It suffices to find a map from 119875(119886) onto Dlowast with theform 119902(119903)119890

119894120579 such that

119902

10158401015840

+

1

119903

119902

1015840

minus

1

119903

2

119902 = 0 for 1 gt 119903 gt 119890

minus119886 (33)

with 119902(119890

minus119886

) = 0 119902(1) = 1 and 119902

1015840

gt 0 for 1 gt 119903 gt 119890

minus119886 Usingthe boundary condition and (32) we can get that

119902 = 119890

minus1

(119890

119886

minus 1)

minus119890119886

119903(119890

119886

119903 minus 1)

119890119886

119903

minus119890119886

119890

1119903 (34)

is a solution to (33)Therefore we finished the proof of Theorem 3

4 Abstract and Applied Analysis

4 Harmonic Maps from C tothe Hyperbolic Disc

In this section we will give another proof of the followingresult

Proposition 4 There is no rotationally symmetric harmonicdiffeomorphism from C onto the hyperbolic disc

Proof It is well-known that the hyperbolic metric on theunit disc is (2(1 minus 119911|

2

))|119889119911| We will also use the idea ofthe proof of Theorem 1 Suppose there is such a harmonicdiffeomorphism 120601 from C onto D with its hyperbolic metricwith the form 120601 = 119896(119903)119890

119894120579 and then we can get

119896

10158401015840

+

1

119903

119896

1015840

minus

1

119903

2

119896 +

2119896

1 minus 119896

2

[(119896

1015840

)

2

minus

119896

2

119903

2

]

= 0 for 119903 gt 0

(35)

with

119896 (0) = 0 119896

1015840

(119903) gt 0 for 119903 gt 0 (36)

Regarding 119903 as a function of 119896 setting V = (ln 119903)

1015840

(119896) (35) canbe rewritten as

(1 minus 119896

2

) V1015840

minus 2119896V + V3

(119896 + 119896

3

) = 0 (37)

That is

(Vminus2

)

1015840

+

4119896

1 minus 119896

2

Vminus2

=

2 (119896 + 119896

3

)

1 minus 119896

2

(38)

One can solve this equation to obtain

Vminus2

= 119896

2

+ 1198885(1 minus 119896

2

)

2 (39)

for some nonnegative constant 1198885depending on the choice of

the function 119896If 1198885= 0 then we can get 119903 = 119888

6119896 for some constant 119888

6 On

the other hand 120601 is a diffeomorphism so 119896 rarr 1 as 119903 rarr infinThis is a contradiction

If 1198885gt 0 then 119887

1ge 119896

2

+ 1198885(1 minus 119896

2

)

2

ge 1198872for some positive

constants 1198871and 1198872 So

10038161003816100381610038161003816

(ln 119903)

1015840

(119896)

10038161003816100381610038161003816

le

1

radic1198872

(40)

This is in contradiction to the assumption that 119903 rarr infin as119896 rarr 1

Therefore Proposition 4 holds

Acknowledgments

The author (Xu-Qian Fan) would like to thank ProfessorLuen-fai Tam for his very worthy advice The first author ispartially supported by the National Natural Science Foun-dation of China (11201131) the second author is partiallysupported by the National Natural Science Foundation ofChina (11101106)

References

[1] K Akutagawa ldquoHarmonic diffeomorphisms of the hyperbolicplanerdquo Transactions of the American Mathematical Society vol342 no 1 pp 325ndash342 1994

[2] K Akutagawa and S Nishikawa ldquoThe Gauss map and spacelikesurfaces with prescribed mean curvature in Minkowski 3-spacerdquoThe Tohoku Mathematical Journal vol 42 no 1 pp 67ndash82 1990

[3] T K-K Au and T Y-H Wan ldquoFrom local solutions to a globalsolution for the equationΔ119908 = 119890

2119908

minusΦ

2

119890

minus2119908rdquo inGeometry andGlobal Analysis pp 427ndash430 Tohoku University Sendai Japan1993

[4] T K K Au L-F Tam and T Y HWan ldquoHopf differentials andthe images of harmonicmapsrdquoCommunications in Analysis andGeometry vol 10 no 3 pp 515ndash573 2002

[5] T K K Au and T Y H Wan ldquoImages of harmonic maps withsymmetryrdquoThe Tohoku Mathematical Journal vol 57 no 3 pp321ndash333 2005

[6] Q Chen and J Eichhorn ldquoHarmonic diffeomorphisms betweencomplete Riemann surfaces of negative curvaturerdquo Asian Jour-nal of Mathematics vol 13 no 4 pp 473ndash533 2009

[7] L F Cheung and C K Law ldquoAn initial value approach to rota-tionally symmetric harmonic mapsrdquo Journal of MathematicalAnalysis and Applications vol 289 no 1 pp 1ndash13 2004

[8] H I Choi and A Treibergs ldquoNew examples of harmonic dif-feomorphisms of the hyperbolic plane onto itselfrdquoManuscriptaMathematica vol 62 no 2 pp 249ndash256 1988

[9] H I Choi and A Treibergs ldquoGauss maps of spacelike constantmean curvature hypersurfaces of Minkowski spacerdquo Journal ofDifferential Geometry vol 32 no 3 pp 775ndash817 1990

[10] P Collin and H Rosenberg ldquoConstruction of harmonic diffeo-morphisms and minimal graphsrdquo Annals of Mathematics vol172 no 3 pp 1879ndash1906 2010

[11] J-M Coron and F Helein ldquoHarmonic diffeomorphisms min-imizing harmonic maps and rotational symmetryrdquo CompositioMathematica vol 69 no 2 pp 175ndash228 1989

[12] J A Galvez and H Rosenberg ldquoMinimal surfaces and har-monic diffeomorphisms from the complex plane onto certainHadamard surfacesrdquoAmerican Journal of Mathematics vol 132no 5 pp 1249ndash1273 2010

[13] Z-C Han ldquoRemarks on the geometric behavior of harmonicmaps between surfacesrdquo in Elliptic and Parabolic Methods inGeometry pp 57ndash66 A K Peters Wellesley Mass USA 1996

[14] Z-C Han L-F Tam A Treibergs and T Wan ldquoHarmonicmaps from the complex plane into surfaces with nonpositivecurvaturerdquo Communications in Analysis and Geometry vol 3no 1-2 pp 85ndash114 1995

[15] F Helein ldquoHarmonic diffeomorphisms between Riemannianmanifoldsrdquo in Variational Methods vol 4 of Progress in Nonlin-ear Differential Equations and Their Applications pp 309ndash318Birkhauser Boston Mass USA 1990

[16] F Helein ldquoHarmonic diffeomorphisms with rotational symme-tryrdquo Journal fur die Reine und AngewandteMathematik vol 414pp 45ndash49 1991

[17] E Heinz ldquoUber die Lusungen der Minimalflachengleichungrdquoin Nachrichten Der Akademie Der Wissenschaften in Got-tingen Mathematisch-physikalische klasse pp 51ndash56 1952Mathematisch-physikalisch-chemische abteilung

[18] J Jost and R Schoen ldquoOn the existence of harmonic diffeomor-phismsrdquo Inventiones Mathematicae vol 66 no 2 pp 353ndash3591982

Abstract and Applied Analysis 5

[19] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables Theory and Applicationvol 48 no 2 pp 175ndash187 2003

[20] P Li L-F Tam and J Wang ldquoHarmonic diffeomorphismsbetween Hadamard manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 9 pp 3645ndash3658 1995

[21] J Lohkamp ldquoHarmonic diffeomorphisms and Teichmullertheoryrdquo Manuscripta Mathematica vol 71 no 4 pp 339ndash3601991

[22] V Markovic ldquoHarmonic diffeomorphisms and conformal dis-tortion of Riemann surfacesrdquo Communications in Analysis andGeometry vol 10 no 4 pp 847ndash876 2002

[23] V Markovic ldquoHarmonic diffeomorphisms of noncompact sur-faces and Teichmuller spacesrdquo Journal of the LondonMathemat-ical Society vol 65 no 1 pp 103ndash114 2002

[24] A Ratto and M Rigoli ldquoOn the asymptotic behaviour ofrotationally symmetric harmonic mapsrdquo Journal of DifferentialEquations vol 101 no 1 pp 15ndash27 1993

[25] R M Schoen ldquoThe role of harmonic mappings in rigidityand deformation problemsrdquo in Complex Geometry vol 143 ofLecture Notes in Pure and Applied Mathematics pp 179ndash200Dekker New York NY USA 1993

[26] Y Shi ldquoOn the construction of some new harmonic maps fromR119898 to H119898rdquo Acta Mathematica Sinica vol 17 no 2 pp 301ndash3042001

[27] Y Shi and L-F Tam ldquoHarmonic maps from R119899 to H119898 withsymmetryrdquo Pacific Journal of Mathematics vol 202 no 1 pp227ndash256 2002

[28] A Tachikawa ldquoRotationally symmetric harmonic maps from aball into a warped product manifoldrdquo Manuscripta Mathemat-ica vol 53 no 3 pp 235ndash254 1985

[29] A Tachikawa ldquoA nonexistence result for harmonic mappingsfrom R119899 into H119899rdquo Tokyo Journal of Mathematics vol 11 no 2pp 311ndash316 1988

[30] L-F Tam and T Y-H Wan ldquoHarmonic diffeomorphisms intoCartan-Hadamard surfaces with prescribed Hopf differentialsrdquoCommunications in Analysis and Geometry vol 2 no 4 pp593ndash625 1994

[31] L-F Tam and T Y H Wan ldquoQuasi-conformal harmonicdiffeomorphism and the universal Teichmuller spacerdquo Journalof Differential Geometry vol 42 no 2 pp 368ndash410 1995

[32] T Y-H Wan ldquoConstant mean curvature surface harmonicmaps and universal Teichmuller spacerdquo Journal of DifferentialGeometry vol 35 no 3 pp 643ndash657 1992

[33] T Y H Wan ldquoReview on harmonic diffeomorphisms betweencomplete noncompact surfacesrdquo in Surveys in Geometric Anal-ysis and Relativity vol 20 of Advanced Lectures in Mathematicspp 509ndash516 International Press Somerville Mass USA 2011

[34] D Wu ldquoHarmonic diffeomorphisms between complete sur-facesrdquo Annals of Global Analysis and Geometry vol 15 no 2pp 133ndash139 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

4 Harmonic Maps from C tothe Hyperbolic Disc

In this section we will give another proof of the followingresult

Proposition 4 There is no rotationally symmetric harmonicdiffeomorphism from C onto the hyperbolic disc

Proof It is well-known that the hyperbolic metric on theunit disc is (2(1 minus 119911|

2

))|119889119911| We will also use the idea ofthe proof of Theorem 1 Suppose there is such a harmonicdiffeomorphism 120601 from C onto D with its hyperbolic metricwith the form 120601 = 119896(119903)119890

119894120579 and then we can get

119896

10158401015840

+

1

119903

119896

1015840

minus

1

119903

2

119896 +

2119896

1 minus 119896

2

[(119896

1015840

)

2

minus

119896

2

119903

2

]

= 0 for 119903 gt 0

(35)

with

119896 (0) = 0 119896

1015840

(119903) gt 0 for 119903 gt 0 (36)

Regarding 119903 as a function of 119896 setting V = (ln 119903)

1015840

(119896) (35) canbe rewritten as

(1 minus 119896

2

) V1015840

minus 2119896V + V3

(119896 + 119896

3

) = 0 (37)

That is

(Vminus2

)

1015840

+

4119896

1 minus 119896

2

Vminus2

=

2 (119896 + 119896

3

)

1 minus 119896

2

(38)

One can solve this equation to obtain

Vminus2

= 119896

2

+ 1198885(1 minus 119896

2

)

2 (39)

for some nonnegative constant 1198885depending on the choice of

the function 119896If 1198885= 0 then we can get 119903 = 119888

6119896 for some constant 119888

6 On

the other hand 120601 is a diffeomorphism so 119896 rarr 1 as 119903 rarr infinThis is a contradiction

If 1198885gt 0 then 119887

1ge 119896

2

+ 1198885(1 minus 119896

2

)

2

ge 1198872for some positive

constants 1198871and 1198872 So

10038161003816100381610038161003816

(ln 119903)

1015840

(119896)

10038161003816100381610038161003816

le

1

radic1198872

(40)

This is in contradiction to the assumption that 119903 rarr infin as119896 rarr 1

Therefore Proposition 4 holds

Acknowledgments

The author (Xu-Qian Fan) would like to thank ProfessorLuen-fai Tam for his very worthy advice The first author ispartially supported by the National Natural Science Foun-dation of China (11201131) the second author is partiallysupported by the National Natural Science Foundation ofChina (11101106)

References

[1] K Akutagawa ldquoHarmonic diffeomorphisms of the hyperbolicplanerdquo Transactions of the American Mathematical Society vol342 no 1 pp 325ndash342 1994

[2] K Akutagawa and S Nishikawa ldquoThe Gauss map and spacelikesurfaces with prescribed mean curvature in Minkowski 3-spacerdquoThe Tohoku Mathematical Journal vol 42 no 1 pp 67ndash82 1990

[3] T K-K Au and T Y-H Wan ldquoFrom local solutions to a globalsolution for the equationΔ119908 = 119890

2119908

minusΦ

2

119890

minus2119908rdquo inGeometry andGlobal Analysis pp 427ndash430 Tohoku University Sendai Japan1993

[4] T K K Au L-F Tam and T Y HWan ldquoHopf differentials andthe images of harmonicmapsrdquoCommunications in Analysis andGeometry vol 10 no 3 pp 515ndash573 2002

[5] T K K Au and T Y H Wan ldquoImages of harmonic maps withsymmetryrdquoThe Tohoku Mathematical Journal vol 57 no 3 pp321ndash333 2005

[6] Q Chen and J Eichhorn ldquoHarmonic diffeomorphisms betweencomplete Riemann surfaces of negative curvaturerdquo Asian Jour-nal of Mathematics vol 13 no 4 pp 473ndash533 2009

[7] L F Cheung and C K Law ldquoAn initial value approach to rota-tionally symmetric harmonic mapsrdquo Journal of MathematicalAnalysis and Applications vol 289 no 1 pp 1ndash13 2004

[8] H I Choi and A Treibergs ldquoNew examples of harmonic dif-feomorphisms of the hyperbolic plane onto itselfrdquoManuscriptaMathematica vol 62 no 2 pp 249ndash256 1988

[9] H I Choi and A Treibergs ldquoGauss maps of spacelike constantmean curvature hypersurfaces of Minkowski spacerdquo Journal ofDifferential Geometry vol 32 no 3 pp 775ndash817 1990

[10] P Collin and H Rosenberg ldquoConstruction of harmonic diffeo-morphisms and minimal graphsrdquo Annals of Mathematics vol172 no 3 pp 1879ndash1906 2010

[11] J-M Coron and F Helein ldquoHarmonic diffeomorphisms min-imizing harmonic maps and rotational symmetryrdquo CompositioMathematica vol 69 no 2 pp 175ndash228 1989

[12] J A Galvez and H Rosenberg ldquoMinimal surfaces and har-monic diffeomorphisms from the complex plane onto certainHadamard surfacesrdquoAmerican Journal of Mathematics vol 132no 5 pp 1249ndash1273 2010

[13] Z-C Han ldquoRemarks on the geometric behavior of harmonicmaps between surfacesrdquo in Elliptic and Parabolic Methods inGeometry pp 57ndash66 A K Peters Wellesley Mass USA 1996

[14] Z-C Han L-F Tam A Treibergs and T Wan ldquoHarmonicmaps from the complex plane into surfaces with nonpositivecurvaturerdquo Communications in Analysis and Geometry vol 3no 1-2 pp 85ndash114 1995

[15] F Helein ldquoHarmonic diffeomorphisms between Riemannianmanifoldsrdquo in Variational Methods vol 4 of Progress in Nonlin-ear Differential Equations and Their Applications pp 309ndash318Birkhauser Boston Mass USA 1990

[16] F Helein ldquoHarmonic diffeomorphisms with rotational symme-tryrdquo Journal fur die Reine und AngewandteMathematik vol 414pp 45ndash49 1991

[17] E Heinz ldquoUber die Lusungen der Minimalflachengleichungrdquoin Nachrichten Der Akademie Der Wissenschaften in Got-tingen Mathematisch-physikalische klasse pp 51ndash56 1952Mathematisch-physikalisch-chemische abteilung

[18] J Jost and R Schoen ldquoOn the existence of harmonic diffeomor-phismsrdquo Inventiones Mathematicae vol 66 no 2 pp 353ndash3591982

Abstract and Applied Analysis 5

[19] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables Theory and Applicationvol 48 no 2 pp 175ndash187 2003

[20] P Li L-F Tam and J Wang ldquoHarmonic diffeomorphismsbetween Hadamard manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 9 pp 3645ndash3658 1995

[21] J Lohkamp ldquoHarmonic diffeomorphisms and Teichmullertheoryrdquo Manuscripta Mathematica vol 71 no 4 pp 339ndash3601991

[22] V Markovic ldquoHarmonic diffeomorphisms and conformal dis-tortion of Riemann surfacesrdquo Communications in Analysis andGeometry vol 10 no 4 pp 847ndash876 2002

[23] V Markovic ldquoHarmonic diffeomorphisms of noncompact sur-faces and Teichmuller spacesrdquo Journal of the LondonMathemat-ical Society vol 65 no 1 pp 103ndash114 2002

[24] A Ratto and M Rigoli ldquoOn the asymptotic behaviour ofrotationally symmetric harmonic mapsrdquo Journal of DifferentialEquations vol 101 no 1 pp 15ndash27 1993

[25] R M Schoen ldquoThe role of harmonic mappings in rigidityand deformation problemsrdquo in Complex Geometry vol 143 ofLecture Notes in Pure and Applied Mathematics pp 179ndash200Dekker New York NY USA 1993

[26] Y Shi ldquoOn the construction of some new harmonic maps fromR119898 to H119898rdquo Acta Mathematica Sinica vol 17 no 2 pp 301ndash3042001

[27] Y Shi and L-F Tam ldquoHarmonic maps from R119899 to H119898 withsymmetryrdquo Pacific Journal of Mathematics vol 202 no 1 pp227ndash256 2002

[28] A Tachikawa ldquoRotationally symmetric harmonic maps from aball into a warped product manifoldrdquo Manuscripta Mathemat-ica vol 53 no 3 pp 235ndash254 1985

[29] A Tachikawa ldquoA nonexistence result for harmonic mappingsfrom R119899 into H119899rdquo Tokyo Journal of Mathematics vol 11 no 2pp 311ndash316 1988

[30] L-F Tam and T Y-H Wan ldquoHarmonic diffeomorphisms intoCartan-Hadamard surfaces with prescribed Hopf differentialsrdquoCommunications in Analysis and Geometry vol 2 no 4 pp593ndash625 1994

[31] L-F Tam and T Y H Wan ldquoQuasi-conformal harmonicdiffeomorphism and the universal Teichmuller spacerdquo Journalof Differential Geometry vol 42 no 2 pp 368ndash410 1995

[32] T Y-H Wan ldquoConstant mean curvature surface harmonicmaps and universal Teichmuller spacerdquo Journal of DifferentialGeometry vol 35 no 3 pp 643ndash657 1992

[33] T Y H Wan ldquoReview on harmonic diffeomorphisms betweencomplete noncompact surfacesrdquo in Surveys in Geometric Anal-ysis and Relativity vol 20 of Advanced Lectures in Mathematicspp 509ndash516 International Press Somerville Mass USA 2011

[34] D Wu ldquoHarmonic diffeomorphisms between complete sur-facesrdquo Annals of Global Analysis and Geometry vol 15 no 2pp 133ndash139 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

[19] D Kalaj ldquoOn harmonic diffeomorphisms of the unit disc ontoa convex domainrdquo Complex Variables Theory and Applicationvol 48 no 2 pp 175ndash187 2003

[20] P Li L-F Tam and J Wang ldquoHarmonic diffeomorphismsbetween Hadamard manifoldsrdquo Transactions of the AmericanMathematical Society vol 347 no 9 pp 3645ndash3658 1995

[21] J Lohkamp ldquoHarmonic diffeomorphisms and Teichmullertheoryrdquo Manuscripta Mathematica vol 71 no 4 pp 339ndash3601991

[22] V Markovic ldquoHarmonic diffeomorphisms and conformal dis-tortion of Riemann surfacesrdquo Communications in Analysis andGeometry vol 10 no 4 pp 847ndash876 2002

[23] V Markovic ldquoHarmonic diffeomorphisms of noncompact sur-faces and Teichmuller spacesrdquo Journal of the LondonMathemat-ical Society vol 65 no 1 pp 103ndash114 2002

[24] A Ratto and M Rigoli ldquoOn the asymptotic behaviour ofrotationally symmetric harmonic mapsrdquo Journal of DifferentialEquations vol 101 no 1 pp 15ndash27 1993

[25] R M Schoen ldquoThe role of harmonic mappings in rigidityand deformation problemsrdquo in Complex Geometry vol 143 ofLecture Notes in Pure and Applied Mathematics pp 179ndash200Dekker New York NY USA 1993

[26] Y Shi ldquoOn the construction of some new harmonic maps fromR119898 to H119898rdquo Acta Mathematica Sinica vol 17 no 2 pp 301ndash3042001

[27] Y Shi and L-F Tam ldquoHarmonic maps from R119899 to H119898 withsymmetryrdquo Pacific Journal of Mathematics vol 202 no 1 pp227ndash256 2002

[28] A Tachikawa ldquoRotationally symmetric harmonic maps from aball into a warped product manifoldrdquo Manuscripta Mathemat-ica vol 53 no 3 pp 235ndash254 1985

[29] A Tachikawa ldquoA nonexistence result for harmonic mappingsfrom R119899 into H119899rdquo Tokyo Journal of Mathematics vol 11 no 2pp 311ndash316 1988

[30] L-F Tam and T Y-H Wan ldquoHarmonic diffeomorphisms intoCartan-Hadamard surfaces with prescribed Hopf differentialsrdquoCommunications in Analysis and Geometry vol 2 no 4 pp593ndash625 1994

[31] L-F Tam and T Y H Wan ldquoQuasi-conformal harmonicdiffeomorphism and the universal Teichmuller spacerdquo Journalof Differential Geometry vol 42 no 2 pp 368ndash410 1995

[32] T Y-H Wan ldquoConstant mean curvature surface harmonicmaps and universal Teichmuller spacerdquo Journal of DifferentialGeometry vol 35 no 3 pp 643ndash657 1992

[33] T Y H Wan ldquoReview on harmonic diffeomorphisms betweencomplete noncompact surfacesrdquo in Surveys in Geometric Anal-ysis and Relativity vol 20 of Advanced Lectures in Mathematicspp 509ndash516 International Press Somerville Mass USA 2011

[34] D Wu ldquoHarmonic diffeomorphisms between complete sur-facesrdquo Annals of Global Analysis and Geometry vol 15 no 2pp 133ndash139 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of