Mathematics

Conformal metrics with constant q-curvature for manifolds by Ndiaye C.B.

By Ndiaye C.B.

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M hence ¯ l e4ul (y) dVg (y) = u G(x, y)Q ¯l + C + 2 ul (x) ≤ u ¯l + C + 2 M ¯ l e4ul (y) dVg (y) G(x, y)Q Bk ¯ l e4ul (y) dVg (y). G(x, y)Q +2 M \Bk So since G is smooth out of the diagonal we have that ¯ l (y)enul (y) dVg (y). G(x, y)Q ul (x) ≤ u ¯l + C + 2 Bk Now using Jensen’s inequality we obtain , ¯ l e4ul (y) dVg (y) G(x, y)Q exp ¯ 4ul χB ||L1 (M ) |G(x, y)| exp ||Qe k ≤ Bk M ¯ l (y)e4ul (y) χB (y) Q k ¯ 4ul χB ||L1 (M ) dVg (y). ||Qe k Hence using Fubini theorem we have 4ul e ˜k B So from Bk 4¯ ul (y)dvg (x) ≤ Ce 1 dg (x, y) sup y∈M,k M 1 2π 2 ¯ 4ul χB || 1 ||Qe k L (M ) dVg (x).

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P then there exist ε > 0 and r > 0, depending only on ε, r, h, 1 h+1 ∈ + M4δ ,and points q 1 , . . , q l+1 ∈ ∂M , Bqj (2r) ⊂ ∂M × [0, 2δ ] satisfying (103) f dVg > ε, . . , Bp1 (r) Bpi (2r) ∩ Bpj (2r) = ∅ for i = j, Bpj (2¯ r) ⊂ M2δ . f dVg > ε; Bp˜ (r) h and (104) Bq+ (r) 1 H f dVg > ε, . . , Bq+ (r) f dVg > ε; Bq+i (2r) ∩ Bq+j (2r) = ∅ for i = j. 1. It characterizes some functions in for which the value of II is large negative. 1, and for k ≥ 1 given by (6), the following property holds.

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