The existence and smoothness of the solution is proved in [ES64] (Theorem 10.C p.154 and Proposition 6.B p.135). Note that Eells and Sampson prove these theorems under an additional assumption involving restrictions on a choice of isometric embeddingN →Rn. Hartman proved in [Har67, Assertion (A)] that this assumption is redundant. Finally, the convergence statement fort→ ∞is proved in [Har67, Assertion (B)].
We note that fort= 0 we haveψ0= id hence, by continuity,ψtis an isomorphism for anyt∈[0, ) if we take >0 small enough (after possibly shrinkingU).
BecauseM is compact, it can be covered by a finite set of adapted charts.
More precisely, there exists an >0, a finite set of charts {Ue1, . . . ,Uer} of M and charts{V1, . . . , Vr} of N such that F maps eachUep×[0, ) intoVp. Let us denote byψt,p:Et|
Uep→E0|
Uep the homomorphisms associated to each pair (Uep, Vp) of adapted charts.
Before we proceed to the proof of Proposition 5.3.1, we will first use our choice of adapted charts to defineCk norms on the spaces Γk(Et) which will be particularly well-adjusted to our arguments. Fix ap∈ {1, . . . , r}, let (xi)mi=1 be the coordinates of the chartUep ⊂ M and let (yα)nα=1 be the coordinates of the chartVp ⊂N. We set, as before,Eα=F∗∂y∂α. By shrinking the open setsUep slightly we can find precompact open subsetsUp ⊂Uep such that the sets{Up}rp=1 still coverM. A sections∈Γk(Et) can, locally onUep, be written ass=sαEα(·, t). Using this notation, we define, fork∈Nandt∈[0, ), the seminormsk·kΓk(Up;Et)on Γk(Et) as
kskΓk(Up;Et)= sup
∂|µ|
∂xµsα(x)
:x∈Up,1≤α≤n,|µ| ≤k
. Hereµ= (µ1, . . . , µm) is a multi-index and∂∂x|µ|µ = ∂x∂µµ11
1
· · ·∂x∂µmµm 1
. This expression is finite because Up is compact in Uep. We now define the normk·kΓk(Et) on Γk(Et) as
kskΓk(Et)= max
p=1,...,rkskΓk(Up;Et).
These norms induce the usual Banach space structure on the spaces Γk(Et).
For any of the setsUp⊂M, withp= 1, . . . , r, we will denote by Γk(Up;Et) the Banach space of sections ofEt overUp that extend tok-times differentiable sections over some open set containingUp. On this spacek·kΓk(Up;Et)defines a Banach norm.
By inspecting the definition of the (local) homomorphismsψt,p:Et|
Uep → E0|
Uep and the seminormsk·kΓk(Up;Et) we observe the following. For allk∈N andt∈[0, ), ifs∈Γk(Up;Et) is a section, then
kψt,p(s)kΓk(Up;E0)=kskΓk(Up;Et). (5.3) We will use this compatibility between the homomorphisms and seminorms in our proof of Proposition 5.3.1.
Proof of Proposition 5.3.1. Let the adjusted charts (Uep, Vp), associated homo-morphismsψt,p:Et|
Uep→E0|
Uep and choice of precompact opens Up⊂Uep be as above.
Let us denoteλ= lim inft→0λ1(Jt). There exists a sequence (tu)u∈N⊂[0, ) such thattu→0 asu→ ∞and
u→∞lim λ1(Jtu) =λ= lim inf
t→0 λ1(Jt).
It follows from Proposition 5.2.2 that for each u ∈ N there exists a smooth eigensectionsu∈Γ∞(Et) such thatJtusu=λ1(Jtu)·su. We normalise such thatksukΓ0(Et)= 1 for allu∈N.
Forp= 1, . . . , rwe denoteσu,p=ψtu,p(su|
Uep)∈Γ∞(Uep;E0). Our proof will rely on the following two lemmas.
Lemma 5.3.3. There exists a subsequence (uk)k∈N ⊂ N such that for each p= 1, . . . , rthe sequence(σuk,p)k∈Nconverges inΓ2(Up;E0)to a limiting section σp∈Γ2(Up;E0). At least one of these limiting sections is not the zero section.
Moreover, for allp, q= 1, . . . , r the sectionsσp andσq coincide onUp∩Uq. In the second lemma we consider the operatorJ0 restricted to the open sets Up. Since the Jacobi operatorsJt are ordinary differential operators, it follows that the value ofJtsat a point inM depends only on the germ of the section sat that point. Hence, we can applyJtalso to sections that are not globally defined.
Lemma 5.3.4. Consider the limiting sections σp ∈Γ2(Up;E0) as defined in Lemma 5.3.3. For all p= 1, . . . , r we have onUp that
J0σp=λ·σp.
We postpone the proof of these two lemmas and first finish proof of Proposi-tion 5.3.1.
It follows from the last statement of Lemma 5.3.3 that we can patch the limiting sections σp together to obtain a well-defined global limiting section σ ∈ Γ2(E0). More precisely, we let σ ∈ Γ2(E0) be the section that on each Up ⊂M is given by σ|U
p = σp. Note that the setsUp cover M and that by Lemma 5.3.3 the section is well-defined on intersectionsUp∩Uq. Because at least one of the limiting sectionsσpdoes not vanish, it follows thatσis not the zero section.
Now Lemma 5.3.4 implies thatσis an eigensection ofJ0. Namely, we have J0σ=λ·σ
because this holds on each subsetUp⊂M. It follows thatλis an eigenvalue of J0and hence that
λ1(J0)≤λ= lim inf
t→0 λ1(Jt).
We now prove Lemma 5.3.3 and Lemma 5.3.4. The proofs of these lemmas will rely on the fact that, in suitably chosen local coordinates, the coefficients of the differential operatorsJt depend continuously ont.
Let us first introduce the necessary notation. Let (Uep, Vp) be a pair of adapted charts as before, (xi)mi=1 the coordinates on Uep and (yα)nα=1 the coordinates
onVp. We put againEα=F∗∂y∂α. The Jacobi operatorsJt are second order differential operators. Hence, in local coordinates they can be written as
Jts(x) =
Aij,γα (x, t)∂2sα
∂xixj(x) +Bαi,γ(x, t)∂sα
∂xi(x) +Cαγ(x, t)sα(x)
Eγ(x, t), (5.4) whereAij,γα , Bαi,γ, Cαγ:Uep×[0, )→Rare suitable coefficient functions. Here we write any sectionsofEtoverUep ass=sαEα(·, t).
Our proofs of Lemma 5.3.3 and Lemma 5.3.4 are based on the following observation.
Lemma 5.3.5. LetU0⊂Uep be a precompact open subset. For alli, j= 1, . . . , m and α, γ = 1, . . . , n, we have that the maps t 7→ Aij,γα (·, t), t 7→ Bαi,γ(·, t) and t7→Cαγ(·, t)are continuous mappings from[0,1]intoC1(U0).
Proof. Denote bygij the coefficients of the inverse of the metric tensorg with respect to the coordinates (xi)mi=1 and byMΓkij the Christoffel symbols of the Levi-Civita connection of (M, g). The Jacobi operators are expressed locally as
Jts= ∆s−trgRN(s, df·)df·
=−gij
∇ ∂
∂xi
∇ ∂
∂xj
s−MΓkij∇ ∂
∂xk
s+RN
s, ∂f
∂xi ∂f
∂xj
,
withs∈Γ2(Up;Et). Recall that ∇ is the pullback connection on the bundle Et=ft∗T N. Let us denote byNΓγαβthe Christoffel symbols of the Levi-Civita connection of (N, h) on the chartVp. Then, for anys=sαEα(·, t)∈Γ1(Uep;Et), we can write the pullback connection as
∇ ∂
∂xi
s(x) =∂sα
∂xi(x)Eα(x, t) +sα(x)∂ftβ
∂xi(x)·NΓγαβ(ft(x))·Eγ(x, t).
The coefficient functionsAij,γα , Bαi,γ, Cαγ can be calculated by filling in this expres-sion for the connection∇into the local expression for the Jacobi operators. It follows that these functions can be expressed entirely in terms of the quantities
gij,∂ftβ
∂xi,MΓkij,(RN)δαβγ◦ftand NΓγαβ◦ft
and their first derivatives. Here (RN)δαβγ denote the coefficients of the Riemann curvature tensorRN in the coordinates onVp. As a result, in the expression for the coefficient functions only spatial derivatives of the functionsftup to second order appear. The statement of the lemma now follows immediately from our assumption that [0,1]→C3(M, N) :t7→ftis a continuous mapping.
We can now prove Lemma 5.3.3.
Proof of Lemma 5.3.3. Fix ap∈ {1, . . . , r}. Let us write su=sαuEα(·, t) onUep. Because eachsu is an eigensection of the Jacobi operatorJtu, we find that they satisfy
[Jtu−λ1(Jtu)]su= 0. (5.5) Hence, onUepthe coefficients (sαu)nα=1satisfy a second order linear elliptic system of differential equations. We will use Schauder estimates to obtain a uniform bound on theC2,µ-H¨older norm of these coefficients. To this end we will apply the results of [Mor54].
The system of differential equations in Equation (5.5) is elliptic because the Jacobi operators are elliptic differential operators. The bounds on the H¨older norms of solutions to this equation that are provided by Morrey’s results depend on a uniform ellipticity constant which in Morrey’s paper is denotedM (defined in [Mor54, Equation 1.6]). This constant depends only on the coefficients of the second order part of the system in Equation (5.5). That is, it depends only on the coefficientsAij,γα . Because, by Lemma 5.3.5, these coefficient functions depend continuously ont, it follows that the constantM can be taken uniformly overu∈N.
Take a precompact open U0 ⊂ Uep such that Up ⊂ U0 ⊂ U0 ⊂ Uep. The coefficients of the system of differential equations in Equation (5.5) are a combi-nation of the coefficients ofJtu and the constant termλ1(Jtu). It follows from Lemma 5.3.5 that theC0,µ-H¨older norms (even C1 norms) of the coefficients of Jtu can be bounded uniformly in u. The constant term λ1(Jtu) can also be bounded uniformly inu, since the sequence (λ1(Jtu))u∈Nis convergent. So the coefficients of the system of differential equations in Equation (5.5) have uniformly (inu) boundedC0,µ-H¨older norms. Moreover, because we normalised the sectionssu such thatksukΓ0(Et)= 1, it follows that theC0norm (and hence also theL2norm) of the coefficients sαu is also bounded uniformly in u. We now apply [Mor54, Theorem 4.7] (withG =U0, G1 = Up, in the notation of that paper) to conclude that onUp theC2,µ-H¨older norms of the coefficientssαu are uniformly bounded inu.
We recall the notation σu,p = ψtu,p(su|
Uep). It follows from the definition of the homomorphismsψt,p thatsu andσu,phave the same coefficients on Uep. Namely, if we writeσu,p=σu,pα Eα(·,0), thensαu =σαu,pforα= 1, . . . , n. Hence, also theC2,µ-H¨older norms of the coefficients σu,pα are uniformly bounded. It now follows from the Arzel`a-Ascoli theorem that there exists a subsequence of (σu,p)u∈N that converges in Γ2(Up;E0) to a limiting section. We denote this limiting section byσp. By choosing subsequent refinements of the subsequence we can arrange for this to hold for eachp= 1, . . . , r. We denote the indices of this subsequence by (uk)k∈N⊂N.
We now prove that is it not possible that all limiting sections σp vanish identically. If this was the case, and all sectionsσpvanish, then this would imply kσuk,pkΓ0(Up;E0)→0 ask→ ∞for allp= 1, . . . , r. However, this contracts that for allu∈Nwe have, by Equation (5.3), that
max
p=1,...,rkσu,pkΓ0(Up;E0)= max
p=1,...,rksukΓ0(Up;Et)=ksukΓ0(Et)= 1.
Finally, we prove the last statement of the lemma. Let (Uep, Vp) and (Ueq, Vq) be two pairs of adapted charts with corresponding local homomorphismsψt,p
andψt,q. Recall that the mapsψt,p:Et|
Uep→E0|
Uepare isomorphisms fortsmall enough. It can be easily seen from the definition of these homomorphisms that, on the compact setUp∩Uq, the maps
ψt,q◦ψ−1t,p:E0|U
p∩Uq →E0|U
p∩Uq
converge uniformly to the identity map ast→0. It follows that σp|U
p∩Uq = lim
k→∞ψtuk,p(suk|U
p∩Uq)
= lim
k→∞ψtuk,q◦ψ−1t
uk,p◦ψtuk,p(suk|U
p∩Uq)
= lim
k→∞ψtuk,q(suk|U
p∩Uq)
=σq|U
p∩Uq,
where the limits are taken in Γ0(Up∩Uq;E0).
We finish this section with the proof of Lemma 5.3.4.
Proof of Lemma 5.3.4. Fix a p∈ {1, . . . , p}. Let (Uep, Vp) be a pair of adapted charts and let the homomorphismsψt,p and the frame (Eα)nα=1 be as before.
We claim that
kψt,p◦ Jt− J0◦ψt,pkop→0 ast→0. (5.6) Here,k·kopis the operator norm on the space of bounded linear operators from Γ2(Up;Et) to Γ0(Up;E0) (equipped with the normsk·kΓ2(Up;Et)andk·kΓ0(Up;E0)
respectively).
We denote
aij,γα (x, t) =Aij,γα (x, t)−Aij,γα (x,0) bi,γα (x, t) =Bαi,γ(x, t)−Bαi,γ(x,0)
cγα(x, t) =Cαγ(x, t)−Cαγ(x,0).
Then, for a section s=sαEα(·, t)∈Γ2(Up;Et), we have [ψt,p◦ Jt− J0◦ψt,p]s(x)
=
aij,γα (x, t)∂2sα
∂xixj(x) +bi,γα (x, t)∂sα
∂xi(x) +cγα(x, t)sα(x)
Eα(x,0).
From this expression follows that kψt,p◦ Jt− J0◦ψt,pkop≤ X
i,j,α,γ
kaij,γα kC0(Up)+X
i,α,γ
kbi,γα kC0(Up)+X
α,γ
kcγαkC0(Up).
Our claim is now immediately implied by the results of Lemma 5.3.5.
We use the notation (uk)k∈Nandσu,p as in Lemma 5.3.3. From that lemma follows thatσuk,p→σp in Γ2(Up;E0). We use this to find
J0σp= lim
k→∞J0σuk,p= lim
k→∞J0ψtuk,p(suk|U
p).
From Equation (5.6) follows that J0σp= lim
k→∞J0ψtuk,p(suk|U
p) = lim
k→∞ψtuk,p(Jtuksuk|U
p).
Here we used thatksukkΓ2(Up;Et)=kσukkΓ2(Up;E0) remains bounded uniformly ink. Finally, using the fact that the sectionssuare eigensections of the operators Jtu gives
J0σp= lim
k→∞ψtuk,p(Jtuksuk|U
p) = lim
k→∞λ(Jtuk)·ψtuk,p(suk|U
p) =λ·σp
because, by definition,λ= limu→∞λ1(Jtu).