A Study on Some Geometric Invariants on Manifolds(2/2)

In joint works with Chin-Yu Hsiao, we plan to generalize the asymptic formula for the Ray-Singerholomorphic torsion of Bismut-Vasserot to the case on CR manifolds with circle action. The key steps willinvolve a corresponding asymptotic formula of certain heat kernels for Kohn Laplacian associated to m-thS^1 Fourier coefficient as m goes to infinity. The asymptotic formula will also give a new proof of Morseinequalities on CR manifolds with circle action. We also plan to study the corresponding results on CRmanifolds with transversal circle action for fibration case.In joint works with Professor Yoonweon Lee, we extended the construction of the refined analytic torsionto the case on compact Riemannian manifolds with boundary. For this purpose we introduced a pair of newwell-posed boundary conditions for the odd signature operator and showed that the refined analytic torsion iswell defined under these boundary conditions. Currently we obtain these results under some technicalassumptions. We are trying to remove these assumptions. We obtain a Lefschetz fixed point formula on thesecomplexes with respect to a smooth map. The results hold under the assumption that the Riemannian metricis a product one near the boundary of the manifold. In joint work with Yoonweon Lee, we plan to remove theassumption that the Riemannian metric is a product one near the boundary of the manifold. We are studying apair of new singular cochain complexes on manifolds with boundary.