**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.