Suppose is a compact Riemannian manifold without boundary of dimension . We further assume that admits no conformal Killing vector fields.
In this entry, we discuss a beautiful result due to Dahl-Gicquaud-Humbert recently published in Duke Math. J. They proved that
where the infimum is taken over all smooth vector fields on with .
Their proof goes as follows: First by the compactness of , for some constant . We now use the the Bochner-type formula for the conformal Killing operator on manifolds without boundary, i.e.,
which now yields
thanks to . Using the standard norm for , we rewrite the preceeding inequality as follows
We now argue by contradiction. Assume that there exists a sequence of vector fields such that
Since is compact, it follows that is uniformly bounded in . Therefore, it is standard to conclude that there exists some such that
- in weakly and
- in strongly.
Consequently, . In particular, . For any smooth vector field , using the divergence theorem, we obtain
Hence solves the equation in the sense of distributions. In particular, is a nonzero conformall Killing vector fields, (see this topic for an argument). This is a contradiction.
When the manifold has boundary , the situation is now complicated since
which eventually implies
We shall come back to this later.