THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

6 M. ARNAUDON, A. K. COULIBALY, AND A. THALMAIERFrom the covariant equation (2.8) for ∂X α t (v) and the definition of deformedparallel translation (2.2),DW (X) −1twe have for (t, v) ∈ [0, τ 0 ] × [0, u 0 ],W (X α (v)) −1t ∂X α t (v) = ˙ϕ(v) +or equivalently,(2.14)∂X α t (v) = W (X α (v)) t ˙ϕ(v)+ W (X α (v)) t∫ t= 1 2 Ric♯ (W (X) −1t0∫ t0) dt − ∇ W (X)−1Z dt,W (X α (v)) −1s ∇ ∂X α s (v)P X α s (v α),. d m X α s (v α ),W (X α (v)) −1s ∇ ∂X α s (v)P X α s (v α),. d m X α s (v α )with v α = nα, where the integer n is determined by nα < v ≤ (n + 1)α. Consequently,we haveρ(Xt α (u), Xtα′(u))∫ u 〈= dρ,=++0∫ u0∫ u0∫ u0(∂Xt α (v), ∂Xtα′(v))〉dv〈 ()〉dρ, W (X α (v)) t ˙ϕ(v), W (X α′ (v)) t ˙ϕ(v) dv〈∫ tdρ,(W (X α (v)) t〈dρ,0)〉W (X α (v)) −1s ∇ ∂X α s (v)P X α s (v α),. d m Xs α (v α ), 0 dv∫ t)〉(0, W (X α′ (v)) t W (X α′ (v)) −1s ∇ ∂X α ′ (v)P s Xs α′ (v ),. d mX α′α ′ s (v a ′) dv.0This yields, by means of boundedness of dρ and deformed parallel translation,together with (2.12) and the Burkholder-Davis-Gundy inequalities,[ (E sup ρ 2 Xt α (u), Xt (u)) ] ∫ u[ (α′≤ C E sup ρ 2 Xt α (v), Xt (v)) ]α′dvt≤τ 0 0 t≤τ 0∫ u[∫ τ0]∥+ C E∥ 2 ds dv+ C0∫ uFrom here we obtain[ (E sup ρ 2 Xt α (u), Xt (u)) ]α′≤ Ct≤τ 00E∫ u00[∫ τ00+ Cα 2 ∫ u+ Cα ′ 2∥∇ ∂X α s (v)P X α s (v α),.∥∥∇ ∂X α ′ (v)P s X α′t∥ ]∥∥2s (v ),. αds dv.′[ (E sup ρ 2 Xt α (v), Xt (v)) ]α′dvt≤τ 0[∫ τ0]E ‖∂Xs α (v)‖ 2 ds dv0 0∫ u[∫ τ0]∥E ∥∂X α′(v) ∥ 2 ds dv,where we used the fact that for v ∈ T x M, ∇ v P x,. = 0, together withsee estimate (2.7).ρ(X β s (v), X β s (v β )) ≤ Cβ, β = α, α ′ ,00s101