On a complete noncompact Kähler manifold $M^{n}$ (complex dimension) with nonnegative Ricci curvature and Euclidean volume growth, we prove that polynomial growth holomorphic functions of degree d has an dimension upper bound $cd^{n}$, where c depends only on n and the asymptotic volume ratio. Note that the power is sharp.
Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number $\epsilon \rightarrow 0^+$. Using Hilbert expansion, we rigorously justify that the solution of steady problem converges to that of Laplace’s equation, and the solution of unsteady problem converges to that of heat equation. This is the most difficult case of a long-term project on asymptotic analysis of kinetic equations in bounded domains. The proof relies on a detailed analysis on the boundary layer effect with geometric correction. The upshot of this paper is a novel boundary layer decomposition argument in 3D and $L^2-L^{2m}-L^{\infty }$ bootstrapping method for time-dependent problem.
We define and study cocycles on a Coxeter group in each degree generalizing the sign function. When the Coxeter group is a Weyl group, we explain how the degree three cocycle arises naturally from geometric representation theory.