Date & Time: Friday, 13th January at 4.30 pm
Venue: Ramanujan Hall
Title: : Embedding Cauchy-Riemann manifolds and the Yamabe invariant
Speaker:Prof. Sagun Chanillo, Rutgers University
Abstract: We focus on a smooth, orientable and compact three manifold M with no boundary. We consider a line sub-bundle V of the complexified tangent bundle of M. If the line bundle V is equipped with an almost complex structure J we say the manifold M is a CR( Cauchy-Riemann) manifold. Examples are the three sphere, Heisenberg group, strongly pseudo-convex hypersurfaces in C^2 etc.
It is an important problem in Complex Geometry to decide when a CR manifold globally embeds into some C^n. We show that global embeddability is possible when a certain conformally invariant fourth order operator called the Paneitz operator is non-negative and the CR Yamabe constant is positive. The sufficient conditions we have are also necessary for small deformations of the CR structure of 3-sphere. The well-known non-embedding example of Grauert-Andreotti-Siu will also be discussed in light of our result.
The Paneitz operator is a CR analog of the well-known Paneitz operator that has appeared in four dimensional Conformal geometry in problems related to the Q-curvature in the works of Alice Chang and Paul Yang.
This is a joint work with Hung-Lin Chiu and Paul Yang.