Research Article Rotationally Symmetric Harmonic...
Transcript of Research Article Rotationally Symmetric Harmonic...
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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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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