Some equalities

In this section, we would derive some equalities which will be useful in Chapter 3. The method of proof of the propositions is mostly deduced by local coordinate method due to [1],[3]. So we provide another path. (Main proofs here followed [7])

Let (M, g) be a Riemannian manifold, g = g(t) ∈ Γ(S²T^∗M ) defined on an open interval in R and h := ^∂g_∂t.

Notations:

G(T ) := T − 1

2tr(T )g = T_ij− 1

2g_ij(g^stT_st) (Gravitation operator)

for T ∈ Γ(S²T^∗M ).

(δh)_i := (divh)_i := −g^st∇_sh_ti := −g^sth_si;t, (div^∗v)_ij := 1

2(v_i;j+ v_j;i) for h ∈ Γ(S²T^∗M ), v ∈ Γ(T^∗M ).

(∆_Lh)(X, W ) : = (∆h)(X, W ) + 2trh(R(X, ·)W, ·) − h(X, Ricc(W )) − h(W, Ricc(X))

= h_ij;s^s+ 2R_isjth^st− R_ish^s_j− R_jsh^s_i (Lichnerowicz − Laplacian) for h ∈ Γ(S²T^∗M ), Ricc(W ) := (Ricc(W, ·))^].

Firstly, we give the linearization of Ricci curvature :

Proposition 1. (Variation of Ricci formula) _∂t^∂Ricc_g = −¹₂[∆_Lh+L_(δG(h))^]g].

We need some lemmas as follows:

Lemma 1. hΠ(X, Y ), Zi = ¹₂[(∇Yh)(X, Z) + (∇Xh)(Y, Z) − (∇Zh)(X, Y )]

where Π(X, Y ) := _∂t^∂(∇_XY ), h·, ·i := g(·, ·).

proof.

hΠ(X, Y ), Zi = ∂

∂th∇_XY, Zi − h(∇_XY, Z)

= ∂

∂t[XhY, Zi − hY, ∇_XZi] − h(∇_XY, Z)

= [Xh(Y, Z) − h(Y, ∇_XZ) − g(Y, ∂

∂t∇_XZ)] − h(∇_XY, Z)

= (∇_Xh)(Y, Z) − hΠ(Z, X), Y i.

By this identity, we have

hΠ(X, Y ), Zi = (∇_Xh)(Y, Z) − [(∇_Zh)(X, Y ) − hΠ(Y, Z), Xi]

= (∇_Xh)(Y, Z) − (∇_Zh)(X, Y ) + (∇_Yh)(Z, X) − hΠ(X, Y ), Zi

=⇒ hΠ(X, Y ), Zi = 1

2[(∇_Yh)(X, Z) + (∇_Xh)(Y, Z) − (∇_Zh)(X, Y )].

Lemma 2. _∂t^∂R(X, Y )W = (∇_YΠ)(X, W ) − (∇_XΠ)(Y, W ).

proof. By Lemma 1, we have

∂

∂tR(X, Y )W = ∂

∂t(∇_Y∇_XW − ∇_X∇_YW + ∇_{[X,Y ]}W )

= [Π(Y, ∇_XW ) + ∇_Y(Π(X, W ))] − [Π(X, ∇_YW ) + ∇_X(Π(Y, W ))]

+ Π([X, Y ], W )

= (∇_YΠ)(X, W ) − (∇_XΠ)(Y, W ) + Π(T (X, Y ), W )

= (∇_YΠ)(X, W ) − (∇_XΠ)(Y, W ) where T : torsion in (M, g).

Lemma 3. _∂t^∂Rm(X, Y, W, Z) = ¹₂[h(R(X, Y )W, Z) − h(R(X, Y )Z, W ) +

∇²_Y,Wh(X, Z) − ∇²_X,Wh(Y, Z) + ∇²_X,Zh(Y, W ) − ∇²_Y,Zh(X, W )].

proof. WLOG, may assume ∇X = 0 = ∇Y = ∇Z = ∇W at a time t, at p ∈ M .

By Lemma 2,

∂

∂thR(X, Y )W, Zi = h(R(X, Y )W, Z) + h∂

∂thR(X, Y )W, Zi

= h(R(X, Y )W, Z) + h(∇_YΠ)(X, W ) − ∇_XΠ)(Y, W ), Zi.

By Lemma 1,

h(∇_YΠ)(X, W ), Zi = h∇_Y(Π(X, W )), Zi

= 1

2Y [(∇Wh)(X, Z) + (∇Xh)(W, Z) − (∇Zh)(X, W )]

= 1

2[(∇Y∇Wh)(X, Z) + (∇Y∇Xh)(W, Z) − (∇Y∇Zh)(X, W )]

= 1

2[(∇²_Y,Wh)(X, Z) + (∇²_Y,Xh)(W, Z) − (∇²_Y,Zh)(X, W )].

Hence

∂

∂tRm(X, Y, W, Z) = h(R(X, Y )W, Z) +1

2[(∇²_Y,Wh)(X, Z) − (∇²_X,Wh)(Y, Z) + (∇²_Y,Xh)(W, Z)

− (∇²_X,Yh)(W, Z) − (∇²_Y,Zh)(X, W ) + (∇²_X,Zh)(Y, W )].

By Ricci identity

−(∇²_X,Yh)(W, Z) + (∇²_Y,Xh)(W, Z) = −[h(R(X, Y )W, Z) + h(R(X, Y )Z, W )], we obtain the result.

Lemma 4. _∂t^∂(trα) = −hh, αi + tr(^∂α_∂t) where α(t) ∈ Γ(⊗²T^∗M ).

proof. Using the local coordinate, let

α := α_ijdxⁱ⊗ dx^j. So

∂

∂t(trα) = ∂

∂t(g^ijαij) = −h^ijαij + g^ij∂α_ij

∂t = −hh, αi + tr(∂α

∂t).

Let us complete the proof of Proposition 1 as follows:

proof. By Lemma 4,

∂

∂tRicc(X, W ) = −hRm(X, ·, W, ·), hi + tr[∂

∂tRm(X, ·, W, ·)].

By Lemma 3 and Ricci identity, we have

∂

∂tRm(X, Y, W, Z) = 1

2[h(R(X, Y )W, Z) − h(R(X, Y )Z, W ) + h(R(Y, W )X, Z) + h(R(Y, W )Z, X)] + 1

2[(∇²_W,Yh)(X, Z) − (∇²_X,Wh)(Y, Z) + (∇²_X,Zh)(Y, W ) − (∇²_Y,Zh)(X, W )].

We could observe that

tr(∇²_X,·h(·, W )) = −(∇δh)(X, W ), tr(∇²_X,Wh(·, ·)) = ∇²_X,W(trh) = Hess(trh)(X, W ) and

tr(∇²_·,·h(X, W )) = (∆h)(X, W ).

Substituting these into the preceding equation, and use

tr(h(R(W, ·)·, X)) = −h(X, RicW ), tr(h(R(X, ·)W, ·)) = hRm(X, ·, W, ·), hi.

We have

∂

∂tRicc(X, W ) = −1

2tr[h(R(X, ·)W, ·) + h(R(X, ·)·, W ) + h(R(W, ·)X, ·) + h(R(W, ·)·, X)]

− 1

2[(∇δh)(X, W ) + Hess(trh)(X, W ) + (∇δh)(W, X) + (∆h)(X, W )].

Because

L_(ω^]₎g(X, W ) = ∇ω(X, W ) + ∇ω(W, X) and

L_(δG(h))^]g = L_(δh)^]g + Hess(trh), by L_{(df )}^]g = L_{(∇f )}g = 2Hess(f ), we obtain the proof!

And then it’s sufficient to obtain the linearization of Bian(g, R) which will be used in Chapter 3.

Proposition 2. If T ∈ Γ(S²T^∗M ) is independent of t, then (_∂t^∂δG(T ))Z =

−T ((δG(h))^], Z) + hh, ∇T (·, ·, Z) − ¹₂∇_ZT i.

Before doing the linearization, we also need some Lemmas.

Lemma 5. _∂t^∂R = −hRicc, hi + δ²h − ∆(trh).

proof. By Lemma 4,

∂R

∂t = −hh, Ricci + tr( ∂

∂tRicc).

Then, by Proposition 1, we have tr(∂

∂tRicc) = −1

2tr[∆_Lh + L_(δh)^]g + Hess(trh)].

By

h(X, Ricc(W )) = −tr(h(R(W, ·)·, X)) = hh(X, ·), Ricc(W, ·)i which could be proved by orthonormal frame, and

tr(h(R(X, ·)W, ·)) = hRm(X, ·, W, ·), hi,

The last term on the RHS is derived from Lemma 7.

We have

The last equality of integrand is proved by local coordinate.

Let α = ω in δ(f α) = −hdf, αi + f (δα), α = ^∂ω_∂t in R hdf, αidV = R f (δα)dV

Therefore, we have

∂

∂t(δω) = δ∂ω

∂t + hh, ∇ωi − hδG(h), ωi.

Lemma 7. _∂t^∂dV = ¹₂(trh)dV . proof. This follows from

d

dt[log det(g_ij(t))] = tr[(g_ij(t))⁻¹d(g_ij(t)) dt ] and

∂

∂t q

det(g_ij) = ∂

∂t(e^{12log(det(gij))}).

Now we can complete the proof of Proposition 2 as follows:

proof. Firstly, we have

(δS)Z = δ(S(·, Z)) + 1

2hS, L_Zgi (1)

for S ∈ Γ(S²T^∗M ), proved by local coordinate. And _∂t^∂(L_Zg)(X, Y ) = h(∇_XZ, Y ) + h(X, ∇_YZ) + (∇_Zh)(X, Y ) by Lemma 1. By (1), set S = G(T ),

(∂

∂tδG(T ))Z = ∂

∂tδ(G(T )(·, Z)) + 1 2

∂

∂thG(T ), L_Zgi.

Let us deal with the first term on the RHS.

Because G(T ) := T − ¹₂(trT )g, by Lemma 4,

∂G(T )

∂t = 1

2[hh, T ig − (trT )h]. (2)

Then, by Lemma 6, δ(f α) = −hdf, αi + f (δα) for α 1-form,

Using the similar method in Lemma 4, we have 1

Combining two formulas, we conclude that (∂

∂tδG(T ))Z = −T ((δG(h))^], Z) + hh, ∇T (·, ·, Z)i − 1

2hh, ∇_ZT i.

2.2 The principal symbol on vector bundle

In this section, we give the generalization of notion of principle symbol in PDE. So we also have similar results, such as existence and uniqueness of solution of strictly parabolic equations. Then applying these to the problems:

Short-time existence of Ricci flow.

Definition 1. E: vector bundle over closed manifold M , v := v^αe_α ∈ Γ(E) for local frame {e_α} on E,

∂v

∂t = L(v)

where L is a linear second order differential operator. (i.e.

L : Γ(E) −→ Γ(E)

v 7−→ [a^ij_αβ∂_i∂_jv^β+ bⁱ_αβ∂_iv^β+ c_αβv^β]e_α. in local coordinates {xⁱ} and local frames {e_α} on E)

σ(L) : Π⁻¹(E) −→ Π⁻¹(E)

(x, ξ)v 7−→ σ(L)(x, ξ)v := (a^ij_αβξ_iξ_jv^β)e_α

where Π : T^∗M −→ M , is called the principal symbol on vector bundle E over M . The definition is equivalent to the condition: ∀(x, ξ) ∈ T^∗M, v ∈ Γ(E), φ : M −→ R with dφ(x) = ξ,

σ(L)(x, ξ)v = lim

s→∞s⁻²e^−sφ(x)L(e^sφv)(x).

Definition 2. ^∂v_∂t = L(v) is called strictly parabolic if there exists λ > 0 s.t.

hσ(L)(x, ξ)v, vi ≥ λ|ξ|²|v|² for all (x, ξ) ∈ T^∗M, v ∈ Γ(E).

Definition 3. Let P : Γ(E) −→ Γ(E) be a quasilinear second order differ-ential operator.

∂v

∂t = P (v)

is called parabolic at w ∈ Γ(E) if ^∂v_∂t = [DP (w)]v is parabolic.

Example: σ(∆)(x, ξ) = |ξ|²id.

Remark: From PDE, we know that if ^∂v_∂t = P (v) is strictly parabolic at w, then there exists  > 0, v(t) ∈ Γ(E) for t ∈ [0, ] s.t. ^∂v_∂t = P (v), v(0) = w.

2.3 Local solvability

In the last section, we have introduced some convection of strictly parabolic equations on manifold which would be applied in the first topic (Theorem 1).

And then we developed some notion of elliptic equation to solve the second topic as follows (Theorem 2).

Consider

F_j(x, D^αu) = 0 (∗)

f or j ∈ I_p := {1, 2...p}, |α| ≤ r, u = (u¹(x), ..., u^q(x)), x ∈ Rⁿ. where Fj ∈ C_x^m+σ∩ C_D^∞αu.

Definition 4. (∗) is called elliptic at x₀ for u₀ if L_jw := X

|β|=r,k≤q

∂F_j

∂D^βu^k(x₀, D^αu₀)D^βw^k := X

|β|=r,k≤q

c_jβkD^βw^k f or j ∈ I_p

is elliptic. (This means that ∀ξ ∈ Rⁿ\{0}, [σ^L_jk] : principal symbol of {L_j} has maximal rank, where σ_jk^L := i^rP

|β|=rc_jβkξ^β, j ∈ I_p, k ∈ I_q)

Definition 5. (∗) is called determined/overdetermined/underdetermined el-liptic if its principal symbol is bijective/injective/surjective. u0(x) is called an infinitesimal solution of (∗) at x₀ if

F_j(x, D^αu₀)|_x=x0 = 0 ∀j ∈ I_p.

To note the later description of preceding definition, this just means the infinitesimal solution of (∗) at x0 is ”very local” solution.

Lemma 8. (Local solvability) If u₀ is an infinitesimal solution of a deter-mined or underdeterdeter-mined elliptic system (∗) at x₀, then for ρ sufficiently small, there exists u ∈ C^m+r+σ which is a solution of (∗) for |x − x₀| < ρ.

Remark: We would modify the context of the following proof to achieve proof of Theorem 2 !

proof. Firstly, assume F_j(x, D^αu) is determined. WLOG., may assume u₀ = 0, x₀ = 0. Let v(y) be a function on B₁(0), ρ ∈ R,

Φ : R × C_B^m+r+σ

1(0) (R^q) −→ C_B^m+σ

1(0)(R^p)

(ρ, v) 7−→ F (ρy, ρ^r−|α|D^α_yv).

We just claim: Φ(ρ, v) = 0 for some ρ > 0. Because u(x) := ρ^rv(x

ρ) on Bρ(0) gives a solution of Fj(x, D^αu) = 0.

∂Φ

∂v(0, 0) := Φ₂(0, 0) := L_j = X

|β|=r,k≤q

c_jβkD^β f or j ∈ I_p.

As such it admits a continuous linear right inverse(see[5,Lemma 9.5]):

S : C_B^m+σ

1(0)(R^p) −→ C_B^m+r+σ

1(0) (R^q).

By the implicit function theorem(see[6,Theorem 6.1.1]), we know that for ρ sufficiently small,

v 7−→ v − S(Φ(ρ, v))

is a strict contraction for v near zero. The fixed point of this mapping is what we want. Secondly, for Fj(x, D^αu) : underdetermined. Notice that the LL^∗ : determined elliptic, so the above proof could be applied to F_j(x, D^αL^∗u).

2.4 Banach submanifold of solutions of F

_j

(x, D

^α

u) = 0

We adapted notations in [8] and assumed L is underdetermined. By some deduction, Φ (given in the preceding Lemma) is really a submersion, so we can write

Φ : R × kerΦ₂(0, 0) × Im(L^∗S) −→ C_B^m+σ

1(0)(R^p) (intersection with C_B^m+r+σ

1(0) (R^q) is understood on the left), and then by the implicit function theorem in [6], there is

φ : [0, ) × kerΦ2(0, 0) −→ Im(L^∗S) ∈ C^∞ that yields solutions of F_j(x, D^αu) = 0.

The submersion mapping φ satisfies F (x, D_x^αρ^r[k(^x_ρ) + φ(ρ, k)(^x_ρ)]) = 0 for x ∈ Bρ(0), k ∈ ker(Φ2(0, 0)) near 0.

Conclude it as follows:

Lemma 9. If L := Φ2(0, 0) is the highest-order constant coefficient part of the underdetermined elliptic system (∗) at x0 and the infinitesimal solution u₀, then for ρ sufficiently small, there is a Banach submanifold of solutions of F_j(x, D^αu) = 0, parametrized by functions in ker(Φ₂(0, 0)). 

Lemma 10. If R is nonsingular, then Bian(g,R) is an underdetermined el-liptic operator.

Notation:

Bian(g, R) := −div(G(R)).

proof.

Because Bian⁰(g, R)h = R^s_m(div(G(h)))_s− T_m^qsh_qs where T_m^qs := g^qkg^sl[∂R_lm

∂x^k −1 2

R_kl

∂x^m − Γⁱ_klR_im] f or h ∈ S²T^∗M, we only prove the principal symbol of div(G·) is surjective. (i.e. ∀ξ ∈ T^∗M, v ∈ T^∗M , we should solve

g^st(ξ_sp_tm− 1

2ξ_mp_st) = v_m

f or p) So we just choose

p_kl = ξ_kv_l+ ξ_lv_k g^stξ_sξ_t .

We would assume two facts as follows: (Because these two facts are not the main results in this review, we skip their proofs, but their deduction could be referred to [2])

Fact 1. If R is nonsingular, then the infinitesimal solution of Bian(g, R) = 0 exists.

So we have the following lemma by Lemma 9.

Lemma 11. If R⁻¹(0) exists, then for sufficiently small ρ > 0, the solu-tions of Bian(g,R)=0 on B_ρ(0) near a given infinitesimal solution g₀ form a submanifold of the Banach manifold of metrics on Bρ(0). 

3 Proofs

This chapter is dedicated to the two theorems as promised in Introduction.

3.1 Short-time existence of Ricci flow

First, by some calculation and Chapter 2.1, we know ^∂g_∂t = Q(g) :=

−2Ricc(g) on E := S²T^∗M isn’t strictly parabolic. For if, we have σ(L)(x, ξ)h = |ξ|²h − ξ ⊗ h(ξ^], ·) − h(ξ^], ·) ⊗ ξ + (ξ ⊗ ξ)trh

Let h := ξ ⊗ ξ, ⇒ σ(L)(x, ξ)h = 0 where ^∂h_∂t = Lh := [DQ(g)]h = ∆_Lh + L_(δG(h))^]g.

Theorem 1. (Short-time existence)If g₀ is a smooth metric on a closed Rie-mannian manifold M , then there exists a smooth solution g(t) to the Ricci flow defined on some small time interval with g(0) = g₀. (i.e. ∃ > 0, g(t) on [0, ), s.t. ^∂g_∂t = −2Ricc(g), g(0) = g0 on [0, ))

Remark: We just focus on the existence rather than uniqueness, so the proof of uniqueness can be referred to [1,P.113∼P.116].

proof. Let T ∈ Γ(S²T^∗M ) be fixed, positve definite. Denote by T the invert-ible map Γ(T^∗M ) −→ Γ(T^∗M ) which is induced by T . Let

P (g) := −2Ricc(g) + L_(T⁻¹_{δG(T ))}^]g.

By some calculations, we have

∂

∂tL_(T⁻¹_{δG(T ))}^]g = −L_{(δG(T ))}^]g + A(h, ∇h) where h := ^∂g_∂t. (That’s why we choose P (g) !)

[DP (g)]h = ∆h + A(h, ∇h) =⇒ σ(DP (g))(x, ξ)h = |ξ|²h

=⇒ ∂g

∂t = P (g) : strictly parabolic.

There exists a family of diffeomorphisms ψ_t: M −→ M corresponding to (−T⁻¹δG(T ))^].

Set

g(t) := (ψ_t^∗g).

We have that for all g₀: smooth, ∃ > 0, ∃ g(t): solution of ^∂g_∂t = −2Ricc(g), g(0) = g₀.

3.2 Local existence of metrics with prescribed Ricci curvature

In this section, we may omit several steps of proofs of lemmas or even not give their proofs. (It will be better to grip the main idea without tedious deduction or details)

By observing

Ricc⁰(g)h = −[1

2∆Lh + div^∗(div(G(h)))], Bian⁰(g, R)h = R_m^s(div(G(h)))_s− T_m^qsh_qs

where T_m^qs:= g^qkg^sl[∂R_lm

∂x^k − 1 2

R_kl

∂x^m − Γⁱ_klR_im] f or h ∈ S²T^∗M, we’ll consider the equation

Ricc(g) + div^∗(R⁻¹Bian(g, R)) = R. (∗∗) It’s elliptic !

Fact 2. If R is invertible s.t. R(0) is diagonal and all first partial deriva-tives of R vanish at 0, then we can choose a metric g₀ of form (g₀)_ij = δ_ij+ O(x²) s.t. Ricc(g₀)|_x=0 = R(0), Bian(g₀, R)|_x=0 = 0, and ∂_iBian(g₀, R)|_x=0 = 0 for all i ∈ I_n.

This result would reduce some tedious calculations in latter work.

Theorem 2. If R_ij is a C^m+σ(resp. C^∞, C^ω) tensor f ield(m > 2) in a neighborhood of p on Mⁿ (n ≥ 3) and R⁻¹(p) exists, then there exists a met-ric g with prescribed R as its Ricci curvature tensor locally. (More precisely, there exists g ∈ C^m+σ(resp. C^∞, C^ω) : Riemann metric s.t. Ricc(g) = R in some neighborhood of p)

proof. Because this proof is more complicated than previous ones, we divide it into several steps and lemmas.

Outlines of the proof Considering

Bian(g₀+ h, R) = 0,

we set X to be the submanifold of solutions of Bian(g₀+ h, R) = 0. By the Lemma 11, we know that X is parametrized by ρ, k(∈ kernel of the highest-order constant-coefficient part of the linearization of the Bianchi identity about g₀ at 0). Denote the constant-coefficient operator by

div₀G₀.

Let h₀ := φ(ρ, 0) be the point of X corresponding to our chosen small value of ρ where φ is defined in section 2.4 .

Step1: Given h_j for j ∈ N⁰ := NS{0}, perform the contracting iteration to form h_j.

Step2: Let h_j+1 ∈ X be the projection of h_j onto X by φ and the decom-position

C^m+2+σ(S²T^∗M ) = ker(div₀G₀) ⊕ Im(div^∗₀S).

Firstly, we pick the special continuous linear right inverses of Bianchi op-erator Bian(g, R) and (∗∗) opop-erator.

Let

F (x, D^αh) := Bian(g₀+ h, R) for h ∈ S²T^∗M (defined near 0).

Let

Φ(ρ, υ) := F (ρy, ρ^1−|α|D^α_yυ) on B₁(0), we obtain

Φ₂(0, 0)w = R(0)div₀G₀(w).

Choose S s.t.

1

2S(R(0)·) solves the Dirichlet problem for ∆₁ := −P

j∈In∂_j² as a right inverse of Φ₂(0, 0)div^∗₀.

Notice that why we not select S as a right inverse of Φ₂(0, 0)Φ₂(0, 0)^∗, it’s due to

R(0)div0G0div₀^∗(υ) = 1

2R(0)∆1(υ).

By the implicit function theorem, there exists

φ : R × kerΦ₂(0, 0) −→ Im(Φ₂(0, 0)^∗S)

s.t. Φ(ρ, k + φ(ρ, k)) = 0 for ρ > 0, k ∈ kerΦ₂(0, 0): sufficiently small.

We have the property of φ :

Lemma 12. φ(0, 0) = 0 = φ₁(0, 0) = φ₂(0, 0). 

Its proof could be deduced from Φ(ρ, k + φ(ρ, k)) = 0.

Because

C^m+σ(S²T^∗M ) = ker(div₀G₀) ⊕ Im(div₀^∗S), we have that both equations

P₁ : C^m+σ(S²T^∗M ) −→ ker(div₀G₀) P₂ : C^m+σ(S²T^∗M ) −→ Im(div₀^∗S) are canonical projections.

So the above discussion could be concluded with a sequence : C^m+1+σ(T^∗M )^div

∗

→ C0 ^m+σ(S²T^∗M )Φ(ρ,−),R(0)div0G0

−→ C^m−1+σ(T^∗M )→ C^{S m+1+σ}(T^∗M ).

From definition, we have

Φ₂(0, 0) : Im(div₀^∗S)→ C^{' m−1+σ}(T^∗M ).

So these imply

P₂h = div₀^∗S(R(0)div₀G₀(h)) f or h ∈ C^m+σ(S²T^∗M ). (3) Let

H(x, D^αh) := Ricc(g0 + h) + div^∗R⁻¹(Bian(g0+ h, R)) − R and

Ψ(ρ, υ) := H(ρy, ρ^2−|α|D^α_yυ) on B₁(0).

We obtain

Ψ₂(0, 0)h = 1 2∆₂h

where ∆₂is the standard Laplacian operating componentwise on h ∈ S²T^∗M . Let T be a right inverse of Ψ₂(0, 0) chosen as follows :

Lemma 13. If m ≥ 1, then for any continuous linear right inverse S proof. (Sketch of proof of Lemma 13)

1. Let h ∈ Im(div₀^∗S). We set T h := div₀^∗S(R(0)S(Φ₂(0, 0)h)).

2. Let h ∈ ker(div₀G₀). Let N be the fundamental solution right inverse of

1

where divergence and gravitational operators are those of metric g0(x) + ρ²h(x

ρ)

for x ∈ B_ρ(0) , and let

η_ρ(h)(r) := B_h(ρy, ρ^1−|α|D^α_yr) for r ∈ C_B^m+σ

1(0)(S²T^∗M ).

Note that

kη_ρ(h) + div₀G₀k −→ 0 as ρ → 0, khk_C^m+σ

B1(0)(S²T^∗M ) → 0.

and

if Bian(g, R) = 0 f or ρ > 0 , g(x) = g₀(x) + ρ²h(x

ρ), then η_ρ(h)(Ψ(ρ, h)) = 0.

Let

φ(ρ, k) := 1

ρφ(ρ, ρk) f or ρ > 0 and X be the submanifold of R × C_B^m+σ

1(0)(S²T^∗M ) consisting of points of the form

(ρ, k + φ(ρ, k))

for ρ > 0, k ∈ ker(Φ2(0, 0)). We use this smooth submanifold X instead of using the submanifold X in the outlines of the proof. And we set X_ρ :=

X|_{ρ}×C^m+σ

B1(0)(S²T^∗M ).

Let us complete the issues of convergence of iterates and verification that the limit is what we want !

Because we are looking for a solution of Ψ(ρ, v) = 0 for some ρ > 0, that lies on X, all of the iterates will be required to lie in X_ρ.

Set

k₀ := 0,

ki+1:= Nρ(ki) := ki− P1(T Ψ(ρ, ki+ φ(ρ, ki))).

It’s clear that {k_i} ⊆ ImTT kerdiv₀G₀.

Let bB_ρ(0) be the ball of radius ρ centered at 0 in ker(div₀G₀)T Im T . Secondly, we show that the convergence of {k_i} is hold.

Choose  < 1 s.t.

kΨ₂(0, 0)kkT kkP₁k < 1

6 and kT kkP₁k < 1 6. We obtain that

|Ψ₂(0, 0)(u − v) − Ψ(ρ, u) + Ψ(ρ, v)| < |u − v| if u, v ∈ C_B^m+σ

1(0)(S²T^∗M ) with |u|, |v| < δ⁰ for sufficiently small ρ, δ⁰ > 0 by MVT. ([6])

Because

limρ→0φ(ρ, k) = 0 and φ2(0, 0) = 0 by Lemma 12, we have

|k + φ(ρ, k)| < δ⁰ and kφ₂(ρ, k)k <  if k ∈ ker(div₀G₀), |k| < δ for ρ, δ sufficiently small.

These imply the following equations

|φ(ρ, k) − φ(ρ, l)| < |k − l| f or k, l ∈ bB_δ(0),

N_ρ(k) − N_ρ(l) = P₁T [Ψ₂(0, 0)(k − l) − Ψ(ρ, k) + Ψ(ρ, l)] − P₁[T Ψ₂(0, 0)(φ(ρ, k) − φ(ρ, l))]

where k, l ∈ bB_δ(0), k := k + φ(ρ, k), l := l + φ(ρ, l).

So we have |N_ρ(k) − N_ρ(l)| ≤ ¹₂|k − l|.

May decrease ρ s.t.

|T (P₁(Ψ(ρ, φ(ρ, 0))))| < δ 2.

We obtain the proof of {k_i}_i∈N: converges in bB_ρ(0) for ρ sufficiently small.

Finally, we want to show that, for ρ, δ sufficiently small, if k ∈ bB_δ(0), and N_ρ(k) = k, then Ψ(ρ, k) = 0.

Because

T is a bounded isomorphism and

kη_ρ(k) + div₀G₀k −→ 0 as ρ & 0, kkk_C^m+σ

B1(0)(S²T^∗M ) & 0,

so

|R(0)|²· |div^∗₀SS(η_ρ(k) + div₀G₀)(k)| ≤ 1 2|T k|

for all k, |k| < δ, for ρ, δ sufficiently small.

This follows from that |T k| > λ|k| for some λ, so if ρ, δ sufficiently small, then

kηρ(k) + div0G0k ≤ λ

2kdiv₀^∗SSk · |R(0)|² for |k| < δ.

It’s clear that

P₂T (Ψ(ρ, k)) = T (Ψ(ρ, k)).

By (3), Lemma 13 and the paragraph below it,

|P₂T (Ψ(ρ, k))| = |div₀^∗S(R(0)div₀G₀T Ψ(ρ, k))|

= |div₀^∗SSR(0)²[η_ρ(k) + div₀G₀]Ψ(ρ, k)|

≤ 1

2|T Ψ(ρ, k)|.

So

T Ψ(ρ, k) = 0 =⇒ Ψ(ρ, k) = 0.

Hence we complete the proof of Theorem 2.

References

[1] B. Chow, P. Lu & L. Nei, Hamilton’s Ricci Flow. American Mathematical Society. (2006)

[2] D.M. Deturck, Existence of Metrics with Prescribed Ricci Curva-ture:Local Theory. Invent. math. 65, 179∼207. (1981)

[3] D.M. Deturck, Deforming metrics in the direction of their Ricci tensors.

In ”Collected papers on Ricci flow” Edited by H.D. Cao, B. Chow, S.C.

Chu & S.T. Yau. Series in Geometry and Topology, 37. International Press. (2003)

[4] R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff.

Geo. 17, 255∼306. (1982)

[5] B. Malgrange, Equations de Lie II. J. Diff. Geo. 7, 117∼141. (1972) [6] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Lecture

Notes. (1974)

[7] P. Topping, Lecture on the Ricci flow, Warwick Lecture Notes. (2006) [8] E. Zeidler, Nonlinear Functional Analysis and its Applications

I:Fixed-Point Theorems, Springer-Verlag. (1986)

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