Iterative construction and the completion of the proof of Theo-

I+C₀I− A 2πk

−1 i j

+V kⁿ N ( ˜A₁)_{i j}

where ^{V k}_NⁿA˜₁ ∈ θ∗(√

−1z) is equal to −pr_g δZ^A(H(0))
, and {s_i} is an H₁ -orthonormal basis.

2.5.4 Iterative construction and the completion of the proof of The-orem 2.1.6

Although Proposition 2.5.13 does not provide us with the ρ-balanced metric that we seek, we can use it to construct an iterative procedure which converges to one, as we discuss in the following.

We first need to estimate the Hilbert–Schmidt norm of ˜A₁ (in Proposition 2.5.13) in terms of the one of ˜E₀.

Lemma 2.5.14. There exists a constant C⁰(R, ε) which depends only on R and ε as in

Theorem 2.5.6 such that

|| ˜A₁||_HS(t)≤ C⁰(R, ε)k^1/2|| ˜E₀||_HS(t),

where|| · ||_HS(t)is defined in terms of H(t).

Proof. The equation (5.10) in [96], together with the hypothesis ¯µ_X⁰ = D_k+ E_k, D_k being a scalar matrix with D_k→ I as k → ∞ and ||E||op< ε (cf. Remark 2.5.7), implies

||X_ξ||²_L2(t)≤ C(R, ε)||ξ ||²_HS(t)

for any ξ ∈ sl in general. On the other hand, the estimate (2.36) (cf. (5.7) of [96]) implies

||ξ ||²_HS(t)≤ C_R⁰k||X_ξ||²_L2(t) (2.47) for any ξ ∈ sl in general.

Since ˜E₀ = −δZ^A(H(0)) and ^{V k}_NⁿA˜₁ = −pr_g δZ^A(H(0))
, it is sufficient to bound ||α||_HS(t)in the decomposition ξ = α + β (according to sl = g ⊕ g^⊥_t ) in terms of

||ξ ||_HS(t). Then, noting

||X_{α +β}||²_L2(t)= ||X_α+ X_β||²_L2(t)= ||X_α||²_L2(t)+ ||X_β||²_L2(t)

since g^⊥_t is defined with respect to the L²-metric induced from H(t) (cf. §2.5.2), we have

||X_α||²_L2(t)+ ||X_β||²_L2(t)≤ C(R, ε)||ξ ||²_HS(t). (2.48) Thus, by (2.47) and (2.48), there exists a constant C⁰(R, ε) > 0 such that

1 C⁰(R, ε)k

||α||²_HS(t)+ ||β ||²_HS(t)

≤ ||ξ ||²_HS(t)= ||α + β ||²_HS(t).

which implies ||α||²_HS(t)≤ C⁰(R, ε)k||α + β ||²_HS(t)≤ C⁰(R, ε)k||ξ ||²_HS(t)as required.

In what follows, we write || · ||_HS,0 for the Hilbert–Schmidt norm defined with respect to H₀ and || · ||_HS,1 for the one with respect to H₁ (which is equal to the limit H(∞) of the flow (2.41)).

In particular, Lemma 2.5.14 and (2.46) imply that we have

+ higher order terms in k

by recalling (2.32), ||A||_op≤ const, and (2.49). Now the Hilbert–Schmidt norm of

E˜₁⁰ :=

+ higher order terms in k

with respect to H1can be estimated as

|| ˜E₁⁰||_HS,1≤ 4 for all large enough k, by recalling the estimate (2.32), ||A||_op≤ const, and (2.46); we also used (cf. Proposition 2.5.13) || ˜A₁||²_op≤p

tr( ˜A₁A˜₁) = || ˜A₁||_HS,1. We then modify the constant C₀to make ˜E₁term trace free, by arguing as we did in Lemma 2.4.19. This

will change C₀by a constant of order k^{−m−12−n2}, to C₁say, satisfying the bound

|C₀−C₁| < 4N^−1/2|| ˜E₁⁰||_HS,1 ≤ 8N^−1/2|| ˜E₁⁰||_HS,0 (2.52)

as in (2.35). Hence there exists a trace free hermitian matrix ˜E₁which satisfies V kⁿ

where we used (2.34) in the first line, (2.50) in the second line, Lemma 2.5.14 and (2.49) in the third line, and (2.32) in the fourth line.

Recalling Proposition 2.5.13, the above calculations mean that we get Z

starting at H(∞) := H₁, where we observe that the error term ˜E₁(at t = 0) has now been improved to || ˜E₁||_HS,1= O(k^−m−12) by (2.56), as opposed to || ˜E₀||_HS,0 = O(k^−m) that we initially had in Corollary 2.4.20. Also note that now the projection pr_⊥,t: sl g^⊥t is onto the L²-orthogonal complement of g in sl with respect to the Fubini–Study metric induced from H⁽¹⁾(t).

We summarise what we have achieved as follows. We started with an ap-proximately ρ-balanced metric H₀ ∈ B_k^K, obtained in Corollary 2.4.20 which sat-isfies δZ^A(H₀) = ˜E₀ with || ˜E₀||_HS,0 ≤ const.k^−m; ran the gradient flow (2.41) to annihilate pr_⊥,t(δZ^A), so that at the limit H₁ ∈ B^K_k of the flow we have pr_⊥,∞(δZ^A(H₁)) = 0; set ˜A₁ := −_{V k}^Nnpr_g(δZ^A(H₁)) ∈ θ∗(√

−1z) and replaced A by A₁ := A + 2πk ˜A₁, to consider the functional Z^A1 with a new constant C₁, which differs from C₀ by O(k^{−m−12−n2}); wrote ˜E₁ := −δZ^A1(H₁) with ˜E₁ satisfy-ing || ˜E₁||_HS,1 ≤ const.k^−1/2|| ˜E₀||_HS,0 = O(k^−m−12) as given in (2.55), i.e. H₁ is an approximately ρ-balanced metric of order k^−m−12. We then go back to the first step, by replacing H₀with H₁. We repeat the above process inductively, as in the following proposition.

Proposition 2.5.15. Suppose that we run the iterative procedure, starting with i = 0, to find ρ-balanced metrics as follows:

Step 1 start with an approximately ρ-balanced metric Hi∈B_k^K of order k^−m−i/2; Step 2 run the gradient flow

dH⁽ⁱ⁾(t)

dt = −pr_⊥,t

δZ^Ai(H⁽ⁱ⁾(t))

to annihilatepr_⊥,t(δZ^Ai), so that at the limit H⁽ⁱ⁾(∞) =: Hi+1∈B_k^K of the flow we havepr_⊥,∞(δZ^Ai(H_i+1)) = 0;

Step 3 set ˜A_i+1:= −_{V k}^Nnpr_g(δZ^Ai(H_i+1)) ∈ θ∗(√

−1z) and replace A_iby A_i+1:= A_i+ 2πk ˜A_i+1, to consider the functionalZ^Ai+1with a new constant C_i+1, which differs from C_iby O(k^{−m−(n+i)/2});

Step 4 observe that H_i+1satisfies||δZ^Ai+1(H_i+1)||_HS,i+1= O(k^{−m−(i+1)/2}), where || ·

||_HS,i+1is the Hilbert–Schmidt norm defined with respect to H_i+1 i.e. H_i+1 is an approximately ρ-balanced metric of order k^{−m−(i+1)/2};

Step 5 go back to the step 1, with an improved error term (i.e. the approximately ρ-balanced metric H_i+1now has order k^{−m−(i+1)/2});

so that, by repeating these steps, we get a sequence{(A_i,C_i, H_i)}_iin θ∗(√

−1z) × R × B_k^K.

Then, as i→ ∞, A_i, C_i, and H_i converges to A_∞∈ θ∗(√

−1z), C_∞∈ R, and H∞∈ B_k^K, respectively.

The proof is given in the following two lemmas, which rely on the estimates that we have established so far. We first prove the existence of A_∞and C_∞.

Lemma 2.5.16. A/k + 2π ˜A₁+ 2π ˜A₂+ · · · converges, and hence A_∞:= A + 2πk ˜A₁+ 2πk ˜A₂+ · · · exists. Also C_∞exists.

Proof. We first claim that there exist some constants γ1, γ2> 0 such that || ˜A_i||_HS,i ≤ k^−m+1(k^−1/2γ₁)ⁱ and |C_i− C_i−1| ≤ N^−1/2k^−m(k^−1/2γ₂)ⁱ. Observe that || ˜A_i||_HS,i ≤ k^−m+1(k^−1/2γ1)ⁱimplies || ˜A_i||_HS,0≤ k^−m+1(2k^−1/2γ1)ⁱby inductively using (2.46).

Note that these estimates are satisfied when i = 1; more specifically, Lemma 2.5.14, (2.46), and || ˜E₀||_HS,0 = O(k^−m) imply that there exists a constant γ > 0 such

We argue by induction; suppose that the statement holds at the (i − 1)-th step.

Combined with Lemma 2.5.14 and (2.46), the argument in (2.54) at the i-th step implies

|| ˜A_i||_HS,i≤ C⁰(R, ε)k^1/2|| ˜E_i−1||_HS,i

for all i ≥ 2. Then the induction hypothesis and (2.32) imply 1+8 

2 for all large enough k. We can thus take

γ₁:= max

by arguing as in (2.50) and (2.52). The induction hypothesis and (2.58) imply that

|C_i−C_i−1| ≤ 32N^−1/2

Having established the claim as above, we thus have

||A/k +2π ˜A₁+2π ˜A₂+· · · ||_HS,0≤γ₁

k + k^−m+1(2γ1k^−1/2) + k^−m+1(2γ1k^−1/2)²+ · · ·

< ∞

for all large enough k, and

|C_∞| ≤γ₂

k + N^−1/2k^−m(γ2k^−1/2) + N^−1/2k^−m(γ2k^−1/2)²+ · · ·

< ∞.

We now prove the existence of H_∞.

Lemma 2.5.17. Repeating the procedure as given in Proposition 2.5.15 infinitely many times moves H₀ by a finite distance in B_k^K with respect to the bi-invariant metric, i.e.dist(H_∞, H₀) < ∞.

Proof. Consider first the case i = 1. Recall that we use the limit H₁= H(∞) of the first gradient flow (2.41) as the initial condition for the second gradient flow (2.57). By proceeding as we did in (2.45), we get

dist(H₂, H₀) ≤ 1

Inductively continuing as described in Proposition 2.5.15, we have dist(Hi+1, H_j) ≤

1

λ1 || ˜E_j||_{HS, j}+ · · · + || ˜E_i||_HS,i
for i > j, and also dist(Hi+1, H₀) ≤ 1

λ₁ || ˜E₀||_HS,0+ || ˜E₁||_HS,1+ · · · + || ˜E_i||_HS,i
. (2.59) Now the estimates as in (2.53)-(2.55) at the i-th step (and also Lemma 2.5.16) implies that we have Thus we find that there exists a constant c > 0, independent of k, such that

|| ˜E_i||_HS,i≤ (ck^−1/2)ⁱ⁺¹|| ˜E₀||_HS,0, and hence we get

for all large enough i, where we recall || ˜E₀||_HS,0 = O(k^−m) (cf. Corollary 2.4.20) and (2.42). Thus the sequence {H_i}_iis Cauchy inB^K_k with respect to the bi-invariant metric, and hence the limit H_∞∈B_k^K exists. 1 + O(k^−m+n+2), and hence it suffices to evaluate its derivatives.

We fix a local trivialisation of the line bundle L to write h_FS(H0(k,m))= e^−φm,k, and

for a constant C₁(φm,k) which depends only on (first and second derivatives of) φm,k. Higher order derivatives can be similarly bounded in terms of C^l-norms of φm,k; namely we get ∑ie^−kφm,k

Observe that (2.4) implies e^−kφm,k/2|s_i| ≤ 1. Thus we get, again using (2.4),

Thus, inductively continuing, we get 

Writing h = e^−φ for the hermitian metric corresponding to the extremal metric ω (i.e. ω = −√

−1∂ ¯∂ log h), we have φm,k→ φ in C^∞as k → ∞ (cf. the proof of Corol-lary 2.3.15). Thus we get ||ω_H∞− ω_(m)||_Cl,ω_H0(k,m)≤ C_lk−m+n+(l+2)−1 for a constant C_l which depends only on l, as claimed.

We thus get

||ω_H∞(k,m)− ω||_Cl,ω≤ 2||ω_H∞(k,m)− ω_H0(k,m)||_Cl,ω_H0(k,m)+ ||ω_H0(k,m)− ω||_Cl,ω

≤ ˜C_l(k^−m+n+l+1+ k⁻¹).

Thus, given l ∈ N, we can choose m to be large enough so that the sequence {ωH_∞(k,m)}_k converges to ω in C^l, establishing all the statements claimed in Theorem 2.1.6.

Remark 2.5.18. It is tempting to say that, given such ω_H∞(k,m)’s, there exists a sequence {ω_k}_k which converges to ω in C^∞by diagonal argument. However, k must be chosen to be large enough for ω_H

∞(k,m) to be well-defined, and how large k must be depends on m (cf. §2.3), and hence on l. Thus, by diagonal argument, we can only claim the existence of ω_k’s (with ω_k → ω in C^∞) satisfying ¯∂ grad^1,0_ωkρ_k(ωk) = 0 for infinitely many k’s rather than for all sufficiently large k’s.

We finally note that, if we have the uniqueness theorem as mentioned in Remark 2.1.9, it follows that ω_H

∞(k,m) = ω_H∞(k,m⁰₎ for all m and m⁰, and hence we can say that

the sequence converges in C^∞(cf. §4.2 of [40]).