HARMONIC MAPPINGS BETWEEN RIEMANNIAN MANIFOLDS by Anand Arvind Joshi A Thesis ...

57 Pages·2006·0.29 MB·English

HARMONICMAPPINGS BETWEENRIEMANNIAN MANIFOLDS
by
AnandArvindJoshi
A ThesisPresented tothe
FACULTYOF THEGRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
InPartial Fulfillmentofthe
RequirementsfortheDegree
MASTEROFARTS
(MATHEMATICS)
August2006
Copyright 2006 AnandArvindJoshi
Dedication
Idedicatethisthesistomyparents Arvindand SuvarnaJoshi.
ii
Table of Contents
Dedication ii
Abstract iv
1 HarmonicMappings 1
1.1 SpaceofMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 ConnectionsintheSpaceofMaps . . . . . . . . . . . . . . . . . . . . 4
1.3 TheFirst VariationFormula . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 HarmonicMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The HeatFlowMethod 18
2.1 TheEells SampsonTheorem . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 TheHeat Flow Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Existence ofLocal and GlobalSolutions 31
3.1 ExistenceofLocalSolutions . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 ExistenceofGlobalTime-DependentSolutions . . . . . . . . . . . . . 43
Bibliography 48
Appendices 49
A Ho¨lderSpaces andSome Results onPDEs 49
A.1 Ho¨lderSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 SomeResults onPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 51
A.3 Fre´chet Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
iii
Abstract
HarmonicmappingsbetweentwoRiemannianmanifoldsisanobjectofextensivestudy,
due to their wide applications in mathematics, science and engineering. Proving the
existence of such mappings is challenging because of the non-linear nature of the cor-
responding partial differential equations. This thesis is an exposition of a theorem by
Eells and Sampson, which states that any given map from a Riemannian manifold to a
Riemannianmanifoldwithnon-positivesectionalcurvaturecan befreely homotopedto
aharmonicmap. Inparticular,thisprovestheexistenceofharmonicmapsbetweensuch
manifolds. Thetechniqueused fortheproofis theheat-flow method.
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Chapter 1
Harmonic Mappings
In this chapter we define and discuss harmonic mappings. Let (M,g) and (N,h) be m
and n dimensional Riemannian manifolds, and let u denote a smooth map from M to
N, i.e. u ∈ C∞(M,N). A natural questionto ask is: what is the‘least expanding’map
from M to N? In order to make precise what we mean by ‘least expanding’ map here,
weneedtoanalyzethespaceofmapsC∞(M,N). Inthesectionsthatfollow,wedothis
analysis and define an energy of maps on this space. A harmonic map will be a critical
pointofthisenergy as discussedlater.
1.1 Space of Maps
Let T M denote the tangent space of M and let T M∗ be the dual space of this tan-
x x
gent space. We know that u ∈ C∞(M,N) implies that du is a linear map from
x
T M to T N, i.e. du ∈ Hom(T M,T N). We want to find a metric on
x u(x) x x u(x)
Hom(T M,T N) so thatwecan define energy ofmaps. Wefirst provealemma.
x u(x)
Lemma 1.1.1. T M ∼= T M∗. The isomorphism is linear and induces a metric on
x x
T M∗.
x
1
Proof. The Riemannian metric g induces a natural linear isomorphism between T M
x
and its dual T M∗ defined as follows. Let X = m Xi(x)( ∂ ) ∈ T M and w =
x x i=0 ∂xi x x x
m w (x)(dxi) ∈ T M∗. Define ♭ : T M → TPM and♯ : T M∗ → T M
i=0 i x x x x x x
P
m m
X♭ = ( g (x)Xj(x))(dxi) , (1.1)
x ij x
i=1 i=1
X X
m m
∂
w♯ = ( gij(x)w (x))( ) . (1.2)
j ∂xi x
i=1 i=1
X X
Clearly ♯ and ♭ are linear. Also it can be verified that they are inverse of each other
resulting in a linear isomorphismbetween T M and T M∗. Now we define a metricg∗
x x x
on T M∗ by
x
g∗(w ,θ ) = g (w♯,θ♯) for w ,θ ∈ T M∗. (1.3)
x x x x x x x x x
Thisbilinearformg∗ isametricduetolinearityof♯. Wecanalsogetg∗((di) ,(dj) ) =
x x x x x x
gij where(gij) denotesthematrixinverseofg = (g ).
x ij
Proposition1.1.2. Hom(T M,T N) ∼= T M∗ ⊗T N.
x u(x) x u(x)
Proof. For every f ∈ Hom(T M,T N) we associate a bilinear map f† ∈ T M∗ ⊗
x u(x) x
T N by defining
u(x)
f†(V,w) = w(f(V)), ∀V ∈ T M,w ∈ T N∗. (1.4)
x u(x)
Wecanseethat,givensuchabilinearmap,wecanalsoassociatewithitalinearmapin
Hom(T M,T (x)).
x u
We knowthat du ∈ Hom(T M,T N) isrepresented in localcoordinates by
x x u(x)
∂ m ∂uα ∂
du = (x) . (1.5)
x
∂xi ∂xi ∂yα
(cid:18)(cid:18) (cid:19)(cid:19) α=1(cid:18) (cid:19) (cid:18) (cid:19)u(x)
X
2
SincethebasisforT M∗ ⊗T N is givenby
x u(x)
∂
(dxi) ⊗( ) , (1.6)
x ∂yα u(x)
du isrepresented by
x
m n ∂uα ∂
du = (x)(dxi) ⊗ . (1.7)
x ∂xi x ∂yα
i=1 α=1(cid:18) (cid:19) (cid:18) (cid:19)u(x)
XX
We proved that gij is an inner product on T M∗. Also h is the induced inner
x u(x)
productinT . ThesetwoinnerproductsinduceaninnerproductonT M∗⊗T (x)N
u(x) x u
givenby
∂ ∂
(dxi) ⊗ ,(dxj) ⊗ = gijh (u(x)). (1.8)
x ∂yα x ∂yβ αβ
* (cid:18) (cid:19)u(x) (cid:18) (cid:19)u(x)+
Sincethisinnerproductisdefined everywhereonM, wedefinean innerproducton
sectionsbysetting
hσ,σ′i(x) = hσ(x),σ′(x)i , for x ∈ M;σ,σ′ ∈ Γ(T∗M ⊗u−1TN). (1.9)
x
Withthisinnerproduct,wedefineanorm ondu givenby
x
m n ∂uα ∂uβ
|du|2 = gijh (u) . (1.10)
αβ ∂xi ∂xj
i,j=1α,β=1 (cid:18) (cid:19)(cid:18) (cid:19)
X X
WiththisnormdefinedonHom(TM,TN),wenowdefinetheenergydensityofamap.
Definition 1.1.1. Givenu ∈ C∞(M,N), theenergydensityfunctionofu isdefined as
1
e(u)(x) = |du|2(x), x ∈ M. (1.11)
2
3
Definition 1.1.2. Let (M,g) be a compact Riemannian manifold. Given u ∈
C∞(M,N), theenergyorharmonicenergyofuis defined as
1
E(u) = e(u)dµ = |dµ |. (1.12)
g g
2
ZM ZM
The energy densityof u can be interpreted in a followingway. Let {e ,...,e }, and
1 m
{e′,...,e′ } be orthonormal bases with respect to g and h , for tangent spaces T M
1 n x u(x) x
and T N, respectively. Weexpressdu inthesebases as
u(x) x
n
du (e ) = λαe′ , i = 1,...,m. (1.13)
x i i α
α=1
X
Then, weget
m n
|du|2(x) = (λα)2. (1.14)
i
i=1 α=1
XX
Consequently,wecanregardtheenergydensityfunctionale(u)(x)asthe‘rateofexpan-
sion’ofthedifferentialdu : T M → T N ofuat x ∈ M. Thisiswhywecalle(u )
x x u(x) x
‘energy density’ofthemap.
Thus the energy E(u) is defined for each u ∈ C∞(M,N). The energy of maps E
can be regarded as a functional E : C∞(M,N) → R, and we want to find maps which
are criticalpointsofthisfunctionalE.
1.2 Connections in the Space of Maps
Having introduced an inner product and norm for u ∈ Hom(M,N) i.e. on Γ(TM∗ ⊗
u−1TN) we want to know what is the effect on energy if the map u is changed by a
small amount. In other words, we want to be able to take directional derivatives in the
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spaceΓ(TM∗⊗u−1TN). Todothat,wedevelopthenotionofconnectionforthisspace.
First, wedevelopconnectionsforTM∗ andu−1TN.
Let ∇ denotetheLevi-CivitaconnectionofM whichgivesusamap
∇ : Γ(TM) → Γ(TM∗ ⊗TN) (1.15)
which assigns a tensor field ∇Y ∈ Γ(TM∗ ⊗ TM) of type (1,1) to Y ∈ Γ(TM), a
(0,1)tensorfield. ∇Y isthecovariantdifferentialofY. Let♯and♭betheisomorphisms
betweenTM andTM∗ givenin(1.1)and(1.2). Wecandefineaconnection∇∗ inTM∗
by setting
∇∗ w(Y) = (∇ w♯)♭(Y), Y ∈ Γ(TM),w ∈ Γ(TM∗) (1.16)
X X
= g (∇ w♯,Y) (1.17)
x X
= X(g (w♯,Y))−g (w♯,∇ Y) bycompatibilityof∇ (1.18)
x x X
= Xw(Y)−w(∇ Y) (1.19)
X
whichcouldalsoserveasanalternatedefinitionfortheconnection∇∗ andalsoexplains
that the connection ∇∗ on TM∗ and connection ∇ on TM can be regarded as dual to
each other.
Because of the compatibility of ∇ with g , we can see that the connection ∇∗ is
ij
compatiblewiththemetricgij onTM∗ bythefollowingcomputation.
Lemma 1.2.1. Xg∗(w,θ) = g∗(∇∗ w,θ)+g∗(w,∇∗ ,θ).
X X
5
Proof.
RHS = g((∇∗ w)♯,θ♯)+g(w♯,(∇∗ θ)♯) (1.20)
X X
= g(∇ w♯,θ♯)+g(w♯,∇ θ♯) (1.21)
X X
= Xw(θ♯) = LHS (1.22)
Thus ∇∗ is a Riemannian connection. Now that we have introduced the connection
∇∗ inTM∗,whataretheconnectioncoefficients? Weuse(1.19)todothecomputation.
∂ ∂ ∂ ∂
(∇∗ dxk)( ) = dxk( )−dxk(∇ ) (1.23)
∂∂xj ∂xl ∂xj ∂xl (∂∂xj)∂xl
∂
= δk −Γk (1.24)
∂xj l jl
= −Γk (1.25)
jl
Thusweget an expressionfor∇∗ as
m
∇∗ dxk = − Γkdxj, 1 ≤ i,k ≤ m. (1.26)
∂ ij
∂xi
j=1
X
We notethat theconnection coefficients of∇∗ induced in TM∗ from ∇ are negativeof
theconnectioncoefficientsof∇.
Now consider the tangent bundle u−1TN ⊂ TM induced from TN by the map
u : M → N. At each pointx,
∂ ∂
◦u (x),..., ◦u (x) (1.27)
∂y1 ∂yn
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
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