Related papers: Interface asymptotics of Partial Bergman kernels a…

This article is concerned with asymptotics of equivariant Bergman kernels and partial Bergman kernels for polarized projective Kahler manifolds invariant under a Hamiltonian holomorphic $S^1$ action. Asymptotics of partial Bergman kernel…

Partial Bergman kernels $\Pi_{k, E}$ are kernels of orthogonal projections onto subspaces $\mathcal{k} \subset H^0(M, L^k)$ of holomorphic sections of the $k$th power of an ample line bundle over a Kahler manifold $(M, \omega)$. The…

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function…

This is largely a survey of results obtained jointly with Boris Hanin and Peng Zhou on interfaces in spectral asymptotics, both for Schr\"odinger operators on $L^2({\mathbb R}^d)$ and for Toeplitz Hamiltonians acting on holomorphic sections…

Let X be a strictly pseudoconcave domain in a closed polarized complex manifold (Y,L) where L is a (semi-)positive line bundle over Y. Any given Hermitian metric on L, together with a volume form, induces by restriction to X a Hilbert space…

We prove nontangential asymptotic limits of the Bergman kernel on the diagonal, and the Bergman metric and its holomorphic sectional curvature at exponentially flat infinite type boundary points of smooth bounded pseudoconvex domains in…

We study the behaviors of the relative Bergman kernel metrics on holomorphic families of degenerating hyperelliptic Riemann surfaces and their Jacobian varieties. Near a node or cusp, we obtain precise asymptotic formulas with explicit…

We compute the full off-diagonal asymptotics of the equivariant and partial Bergman kernels associated with a circle action on a prequantized K\"ahler manifold with bounded geometry at infinity, then use these results to compute the…

In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an $h$-extendible boundary point.

The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic…

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb{R}$-action…

Let $( X ,d ,p ) $ be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized K\"ahler manifolds $( M_l ,\omega_l ,\mathcal{L}_l ,h_l ,p_l ) $ with $Ric ( h_l ) =2\pi \omega_l $, $Ric ( \omega_l ) \geq -\Lambda…

Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ over a complex manifold $M$ which may not be compact, we generalize the scaling method in arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and…

In this thesis, we introduce complex manifolds with local spectral gaps and study their asymptotic behavior using the scaling method. With these asymptotics, we obtain an asymptotic expansion for the Bergman kernel of a Hermitian…

We consider the critical behavior at an interface which separates two semi-infinite subsystems belonging to different universality classes, thus having different set of critical exponents, but having a common transition temperature. We…

In this paper, we develop a new scaling method to study spectral and Bergman kernels for the k-th tensor power of a line bundle over a complex manifold under local spectral gap condition. In particular, we establish a simple proof of the…

We establish local asymptotic estimates of partial Bergman kernels on closed, $S^1$-symmetric K\"{a}hler manifolds. The main result concerns the scaling asymptotics of partial Bergman kernels at generic off-diagonal points in which they are…

We consider the scaling limit of a generic ferromagnetic system with a continuous phase transition, on the half plane with boundary conditions leading to the equilibrium of two different phases below criticality. We use general properties…

In this paper, we study the asymptotic behavior of the heat kernel with respect to the Witten Laplacian. We introduce the localization and the scaling technique in semi-classical analysis, and study the semi-classical asymptotic behavior of…

In this master thesis, we give a new proof on the pointwise asymptotic expansion for Bergman kernel of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfy local spectral gap…