The Department of Mathematical Sciences will host Dr. Mei-Chi Shaw of the University of Notre Dame, on Tuesday, April 1, from 11:00 a.m. – 12:00 p.m., as part of its math seminar series. Dr. Shaw will present “The Cauchy-Riemann Equations on Domains in the Complex Projective Space.” This free event will take place in Armitage Hall, Room 124. Contact sm1380@camden.rutgers.edu for more information.
Abstract
The Cauchy-Riemann equations play central role in one and several complex variables. The Cauchy-Riemann operator ∂ has been studied extensively on domains in the complex Euclidean space Cⁿ. Much less is known when the ambient manifold is not Cⁿ.
In this talk, we discuss the range of ∂ on domains in the complex projective space CPⁿ. We also study the ∂-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for ∂ on pseudoconcave domains in CPⁿ. In particular, we show that ∂ does not have closed range in L² for (2,1)-forms on the Hartogs triangle in CP². This is in sharp contrast to ∂ on the Hartogs triangle in C², where L² results have long been established by Hörmander.