Geometric Analysis and Topology Seminar

I will present a recent work where we develop the concept of an elliptic pre-complex. In the scope of compact Riemannian manifolds with boundary, elliptic pre-complexes expand upon the traditional notion of an elliptic complex by allowing sequences of operators that do not form a  cochain complex, but rather satisfy an order reduction clause. Our main theorem states that every elliptic pre-complex can be rectified into an elliptic complex of pseudodifferential operators by introducing zero-order terms. This enables Hodge-like decompositions to manifest, and provides a general framework for addressing overdetermined boundary value problems.

The talk will be guided by a practical motivation for elliptic pre-complexes: the production of a compatibility operator for the range of the Killing operator, applicable in arbitrary geometry. This solves a problem initially studied by Calabi more than sixty years ago. 

Based on a joint work with Raz Kupferman.