On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds

Details On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds

2005-09-19

arxiv.org/abs/math/0509415v1

Mathematics

Differential Geometry

16 pages

Scientific paper

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_\alpha$ were introduced via the scattering theory for Poincar\'{e} metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincar\'{e} exponent less than $\frac {n-\alpha}2$ for some $\alpha \in [2, n)$, the set of positive smooth solutions to the equation $$ P_\alpha u = u^\frac {n+\alpha}{n-\alpha} $$ is compact in the $C^\infty$ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.

Affiliated with

Also associated with

No associations

Rating

On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-298320