We give ChernWeil definitions of the Maslov indices of bundle pairs over a Riemann surface \Sigma with boundary, which consists of symplectic vector bundle on \Sigma and a Lagrangian subbundle on \Sigma as well as its generalization for transversely intersecting Lagrangian boundary conditions. We discuss their properties and relations to the known topological definitions. As a main application, we extend Maslov index to the case with orbifold interior singularities, via curvature integral, and find also an analogous topological definition in these cases.

No abstract uploaded!

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category D(Γ_S) associated to the surface and the corresponding Seidel–Thomas braid group of D(Γ_S) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

In this paper, we extend the work in \cite{D}\cite{ChrusLiWe}\cite{ChrusCo}\cite{Co}. We weaken the asymptotic conditions on the second fundamental form, and we also give an $L^{6}-$norm bound for the difference between general data and Extreme Kerr data or Extreme Kerr-Newman data by proving convexity of the renormalized Dirichlet energy when the target has non-positive curvature. In particular, we give the first proof of the strict mass/angular momentum/charge inequality for axisymmetric Einstein/Maxwell data which is not identical with the extreme Kerr-Newman solution.

No abstract uploaded!