Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields

418 Pages·2013·3.133 MB·English

by  Yuan-Jen Chiang (auth.)

Frontiers in Mathematics
Yuan-Jen Chiang
Developments

of

Harmonic Maps,

Wave Maps and
Yang-Mills Fields

into Biharmonic Maps,
Biwave Maps
and
Bi-Yang-Mills Fields

Frontiers in Mathematics
Advisory Editorial Board
Leonid Bunimovich(Georgia Instituteof Technology, Atlanta)
Benoˆıt Perthame(Universite´ Pierre et Marie Curie, Paris)
Laurent Saloff-Coste(Cornell University, Ithaca)
Igor Shparlinski (Macquarie University, New South Wales)
Wolfgang Spro¨ssig (TU Bergakademie Freiberg)
Ce´dric Villani (Institut Henri Poincare´, Paris)
Yuan-Jen Chiang
Developments of Harmonic
Maps, Wave Maps and
Yang-Mills Fields into
Biharmonic Maps, Biwave
Maps and Bi-Yang-Mills
Fields
Yuan-JenChiang
DepartmentofMathematics
UniversityofMaryWashington
Fredericksburg,VA
USA
ISSN1660-8046 ISSN1660-8054(electronic)
ISBN978-3-0348-0533-9 ISBN978-3-0348-0534-6(eBook)
DOI10.1007/978-3-0348-0534-6
SpringerBaselHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013940741
MathematicsSubjectClassification(2010):58E20,58E15,58E12,81T13,53A10,53C07,53C12,
53C43,49Q05,35J47,35J48,35K05,35L70,35J10,32Q15
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InMemoryofProfessorJamesEells and
ProfessorJosephH.Sampson
Thenamesofthesetwopioneersofthetheory
ofharmonicmapswill beengravedinthe
mindsofall mathematicianswho workon
harmonicmaps,wave maps,andYang-Mills
fields,fortheirgreat andeverlasting
contributions.

Introduction
We present an overview of the developmentsof harmonic maps, wave maps, and
Yang-Mills fields into biharmonic maps, biwave maps, and bi-Yang-Mills fields.
ThetheoryofharmonicmapsbetweenRiemannianmanifoldswasfirstestablished
by Eells and Sampson (Chiang’s Ph.D. adviser) [129] in 1964. Wave maps are
harmonic maps on Minkowski spaces and were studied in the early 1990s. In the
last two decades, there were many new developments in wave maps achieved by
a numberofmathematicians.Yang-Millsfieldsarethe criticalpointsofthe Yang-
Millsfunctionalsofconnectionswhosecurvaturetensorsareharmonic.Theywere
firstexploredbyanumberofphysicistsinthe1950s,andsincethenthereweremany
new developments in this subject. Biharmonic maps, which generalize harmonic
maps,werefirststudiedbyJiang[196–198]in1986.Inrecentyears,therehasbeen
progressinbiharmonicmaps,accomplishedbyquiteafewmathematicians.Biwave
maps are biharmonic maps on Minkowski spaces which generalize wave maps,
and they were first studied by Chiang [75,76] in 2009 and with Wolak [84] later.
Bi-Yang-Millsfields,whichgeneralizeYang-Millsfields,werefirstinvestigatedby
Ichiyama,Inoguchi,andUrakawa[191,192]in2009.Moreover,exponentiallyhar-
monicmapswerefirstintroducedbyEellsandLemaire[125]in1990.Exponential
wavemapsareexponentiallyharmonicmapsonMinkowskispaces,whichwerefirst
studiedbyChiangandYang[88]in2007.ExponentialYang-Millsconnectionswere
firstexploredbyMatsuuraandUrakawa[260]in1995.Sincethisbookcoversbroad
topicsinterveningharmonicmaps,wavemaps,Yang-Millsfields,biharmonicmaps,
biwavemaps,bi-Yang-Millsfields,exponentiallyharmonicmaps,exponentialwave
maps,andexponentialYang-Millsconnections,it isimpossibleto describedetails
completelyandextensively.However,wetrytopresentthemostcrucialingredients
oftherecentdevelopmentsofthetopics.
Harmonic maps were first introduced by Sampson in the hope of obtaining a
homotopyversionof the highly successfulHodge theory of cohomologyin 1952.
Notlongafterthathisthencolleague,JohnNash(oneofthethreeNobellaureates
in Economicsin 1994) proposed a quite differentbut equivalentdefinition – both
of them were Moore Instructors at MIT at the time. Fuller [150] also came upon
harmonicmapsin 1954.The definition,whetherin termsof the energyfunctional
vii
viii Introduction
or the Euler-Lagrangeequations,seemsverynaturalto us today,butit was notso
obvioushalfacenturyago.
Eells and Sampson [129] collaborated on the first paper on harmonic maps of
Riemannian manifolds at the Institute for Advanced Study at Princeton in 1964.
This paper is usually considered as the pioneering work in harmonic maps. They
alsopublishedasecondandthirdjointpaper[130,131]afterwards.
Withaneyetowardthephysicalconceptofkineticenergy.mv2/,aharmonicmap
2
f W.Mm; g /!.Nn; h /fromanm-dimensionalRiemannianmanifoldintoan
ij ˛ˇ
n-dimensional Riemannian manifold is defined as a critical point of the energy
functional
Z Z
1 1
E.f/D jdfj2dvD h f˛fˇgijdv; (1)
2 2 ˛ˇ i j
M M
wheredvisthevolumeformofM determinedbythemetricg. Inordertoderive
the associated Euler-Lagrange equations, we consider a one-parameter family of
mapsff g2C1.M (cid:2)Œ0; 1(cid:2); N/fromacompactmanifoldM (withoutboundary)
t
into a Riemannian manifold N such that f is the endpointof a segment starting
t
at f.x/.D f .x// determinedin length and direction by the vector field fP.x/. If
0
M is a non-closedmanifold,we assume that fP.x/ hascompactsupport,which is
contained in the interior of M. We now compute the first variation of the energy
functional:
ˇ Z ˇ Z
d ˇ ˇ
EP.f/ D E.f /ˇ D .df ; D df /ˇ dv D .df;DfP/dv
Zdt t tD0 Z M t t t tD0Z M
D div.w/dv(cid:3) .(cid:3)f;fP/dv D (cid:3) .(cid:3)f;fP/dv; 8fP; (2)
M M M
by the divergencetheorem,where(cid:3)˛.f/ D trace .Ddf/; D is the connectionon
g
T(cid:2)M˝f(cid:3)1TNinducedbytheLevi-CivitaconnectionsonM andN,div.w/Dwj ;
jj
andwj D h f˛fPˇgij isavectorfieldonM.Themapf W M ! N isharmonic
˛ˇ i
ifthetensionfield
(cid:3)˛.f/ Dtrace .Ddf/Dgijf˛ Dgij.f˛ C(cid:4)0˛ fˇf(cid:5)/
g ijj i;j ˇ(cid:5) i j
Dgij.f˛(cid:3)(cid:4)kf˛C(cid:4)0˛ fˇf(cid:5)/ (3)
ij ij k ˇ(cid:5) i j
vanishesidentically,wheref˛ D @f˛; f˛ D @2f˛ ; f˛ D f˛ (cid:3)(cid:4)kf˛,and(cid:4)k
i @xi ij @xi@xj i;j ij ij k ij
and(cid:4)0˛ aretheChristoffelsymbolsoftheLevi-CivitaconnectionsonM andN,
ˇ(cid:5)
respectively.
Assumethatf.x/ D f .x/isharmonicandthat(cid:6) D @ft.Wenextcomputethe
0 @t
secondvariationoftheenergyfrom(2):
Introduction ix
ˇ Z ˇ
d2 ˇ d ˇ
ER.f/ D E.f /ˇ D (cid:3) .(cid:3)f ; (cid:6)/dvˇ
dtZ2 t tD0 M dt t ˇ tD0
ˇ
D(cid:3) Œ.D (cid:3)f ; (cid:6)/C.(cid:3)f ; D (cid:6)/(cid:2)ˇ dv: (4)
t t t t
M tD0
Att D0,f.x/isharmonicandtheabovesecondvariationbecomes
Z ˇ
ˇ
ER.f/D(cid:3) .D (cid:3)f; (cid:6)/ˇ dv:
t
M tD0
The components of D (cid:3)f are f˛ D @fi˛jj C (cid:4)0˛ f(cid:7)(cid:6)(cid:8): By Eisenhart [137],
t ijjjt @t (cid:7)(cid:8) ijj
f˛ (cid:3)f˛ D(cid:3)Rk f˛CR0˛ fˇf(cid:5)(cid:6)(cid:7)andusingthecurvatureformulaonM(cid:2)
ijjjt ijtjj ijt k ˇ(cid:5)(cid:7) i j
Œ0; 1(cid:2) ! N;thefirstcurvaturetermvanishesandf˛ D f˛ CR0˛ fˇf(cid:5)(cid:6)(cid:7),
ijjjt ijtjj ˇ(cid:5)(cid:7) i j
whereR0 isthe RiemanniancurvatureofN:But f˛ Df˛ D(cid:6)˛; andso f˛ D
ijt tji ji ijjjt
(cid:6)˛ CR0˛ fˇf(cid:5)(cid:6)(cid:7):Therefore,D (cid:3)f hascomponents
jijj ˇ(cid:5)(cid:7) i j t
gij(cid:6)˛ CgijR0˛ fˇf(cid:5)(cid:6)(cid:7): (5)
jijj ˇ(cid:5)(cid:7) i j
Denotethefirsttermby.4(cid:6)/˛:Thuswearriveat
Z
ER.f/ D (cid:3) Œ.4(cid:6); (cid:6)/CgijR0 (cid:6)˛fˇf(cid:5)(cid:6)(cid:7)(cid:2)dv: (6)
˛ˇ(cid:5)(cid:7) i j
M
Usingtheintegrationbypartsgivesd.D(cid:6); (cid:6)/D.4(cid:6); (cid:6)/C.D(cid:6); D(cid:6)/,andthenthe
divergencetheoremrecasts(6)into
Z h i
ER.f/D jD(cid:6)j2(cid:3)gijR0 (cid:6)˛fˇf(cid:5)(cid:6)(cid:7) dv: (7)
˛ˇ(cid:5)(cid:7) i j
M
(Recallthat(cid:6) isasectionoff(cid:3)1.TN/;i.e.,avectorfieldalongf:Forgiven(cid:6),we
obtainasuitablevariationoff bysettingf .x/Dexp t(cid:6).x/; a 2M:)IfN has
t f.a/
negativesectionalcurvature,i.e.,R0 (cid:9)˛(cid:10)ˇ(cid:9)(cid:5)(cid:10)(cid:7) (cid:4) 0forarbitraryvectorfields(cid:9)
˛ˇ(cid:5)(cid:7)
and (cid:10), then it follows from (7) that ER.f/ (cid:5) 0; so every harmonic map is a local
minimumfortheenergyE.
Observe that if all the f are harmonic for t near 0, then (cid:3)f D 0; and also
t t
D (cid:3)f D0:By(4)–(6),thisistheJacobiequation(att D0),namely,
t t
J .(cid:6)/D4(cid:6)CR0.df;df/(cid:6) Dgij(cid:6)˛ CgijR0˛ fˇf(cid:5)(cid:6)(cid:7) D0: (8)
f jijj ˇ(cid:5)(cid:7) i j
Itisalinearequationfor(cid:6).Solutionsof(8)arecalledJacobifieldsalongf.