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By Ndiaye C.B.

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On the other hand, since u0 satisfies ∂ G(u =0 ∀1≤ ∂ti m,2 (∂M × [0, 1]) such i ≤ m, then there exists a function w closed to G(u0 ) in the strong topology of W that (167) w(t) = 0, for some positive and small t ∈ [0, ], . Thus using the local invertibility of G around u0 , we get u = G−1 (w) is well-defined. Thus, from (167), we infer that u is a short-time solution to our initial evolution problem, thus we have the existence. The uniqueness is consequence of Local inversion Theorem. , Differential operators canonically associated to a conformal structure, Math.

We have that the Frechet derivative of G at u0 is DG(u0 )w = ∂w − Aw − 3e−3u0 w. 2 implies that the Linearization of G at u0 is bijective. Hence the Local Inversion i 0) theorem ensures that G is bijective around u0 . On the other hand, since u0 satisfies ∂ G(u =0 ∀1≤ ∂ti m,2 (∂M × [0, 1]) such i ≤ m, then there exists a function w closed to G(u0 ) in the strong topology of W that (167) w(t) = 0, for some positive and small t ∈ [0, ], . Thus using the local invertibility of G around u0 , we get u = G−1 (w) is well-defined.

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