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We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $frac{1}{sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kahler potential $kvarphi$ in a $frac{1}{sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
In this paper, we study the asymptotics of Ohsawa-Takegoshi extension operator and orthogonal Bergman projector associated with high tensor powers of a positive line bundle. More precisely, for a fixed submanifold in a complex manifold, we consider
This paper deals with some special integral transforms of Bargmann-Fock type in the setting of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. The construction is based on the well-known Fueter mapping theorem. In part
The Petrowsky type equation $y_{tt}^eps+eps y_{xxxx}^eps - y_{xx}^eps=0$, $eps>0$ encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order $sqrt{eps}$ occurring at the extre
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
We study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate.