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On conformal deformations of metrics on S n . (English) Zbl 0924.58120

The authors’ main result is: Let us suppose that Q>0 is a smooth function on S n , satisfying the nondegeneracy condition ΔQ(x)0 whenever Q(x)=0 and deg(G,B n+1 ,0)0, then the equation

P n w+(n-1)!=Qe nw onS n

has a solution where P n is an nth order conformal invariant operator on S n , g w =e 2w ·g 0 is a conformal change of the metric g 0 of S n , B n+1 is the unit ball in n+1 ; G:B n+1 n+1 is a map associated to the function Q by using the action of the conformal group of S n .

58H15Global deformations of structures on manifolds
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
58D27Moduli problems for differential geometric structures on spaces of mappings
35J60Nonlinear elliptic equations
58D30Spaces and manifolds of mappings in applications to physics