The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs.
Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We...
The author propose what is the principal part of linear systems of partial differential equations in the Cauchy problem through the normal form of systems in the meromorphic formal symbol class and the theory of weighted determinant. As applications, he choose the necessary and sufficient conditions for the analytic well-posedness ( Cauchy-Kowalevskaya theorem ) and well-posedness (Levi condition).
Let be a compact CR manifold of dimension with a contact form , and its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form on conformal to which has a constant Webster curvature. This problem is equivalent to the existence of a function such that , on . D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where and is not locally CR equivalent to the sphere of . In a join work with R. Yacoub, the CR Yamabe problem...