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Math. 41, no. 7, 891–907 (1988). [10] S. Klainerman and M. Machedon, Finite energy solutions of the Yang-Mills equations in R3+1 , Ann. of Math. (2) 142, no. 1, 39–119 (1995). [11] Sergiu Klainerman and Francesco Nicol` o, On local and global aspects of the Cauchy problem in general relativity, Classical Quantum Gravity 16, R73– R157 (1999). [12] A. Majda, Compressible ﬂuid ﬂow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984.

Let Y be a vector ﬁeld on M and let φs deﬁned by ∂s φs = Y ◦ φs , φ0 = Id, be the ﬂow of Y . A computation shows ∂s s=0 (φ∗ s g) Γk ij and similarly for = 1 kl g (Riljm Y m + Rjlim Y m + ∇i ∇j Yl + ∇j ∇i Yl ) 2 (φ∗ ˆ) k sg Γij . Recall that ∂s φ∗ V i s=0 s = [Y, V ]i = −LV Y i . 5) with φ replaced by φs . 6) ˆ of gˆ are raised and lowered with gˆ. where the indices on the Riemann tensor R ˆ in terms of Γ ˆ we have Using the deﬁnition of ∇ j ˆ m∇ ˆ n Y i = g mn (∂m ∇ ˆ nY i − Γ ˆ i ˆi ˆ ˆr ∇ g mn ∇ mn r Y + Γjm ∇n Y ) ˆ n Y i − Γr ∇ ˆ rY i + Γ ˆ nY j ) + V r ∇ ˆ rY i ˆi ∇ = g mn (∂m ∇ mn jm ˆ by Γ.

70) ≤ Let us set Λ1 (x) = xk 3 exp −C20 (x2 + x4 + xp+1 ) C22 (2 + x2 + x4 + x6 + xp ) −(x3 + x4 + x5 + x6 + x7 + xp + xp+1 + xp+2 ). 71) Then, we ﬁnd that Λ1 (0) = 0, Λ1 (0) > 0, and Λ1 (x) → −∞ as x → ∞. Thus there exists x0 ∈ (0, x1 ) such that Λ1 (x) is monotone increasing function on [0, x0 ]. For every x ∈ (0, x0 ) we deduce that if d < Λ1 (x), then F X ≤ x, and F : B(0, x) → B(0, x). 52 Dongho Chae Ann. Henri Poincar´e We now show that the mapping h → F (h) contracts in Y . 1) for h1 and h2 in X with h1 (0, r) = h2 (0, r), and take diﬀerence between them.