Young-Heon Kim
The Teichmuller space Teich(S) of a surface S in genus g>1 is a real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Delta) on Teich(S) has a unique holomorphic extension to QF(S).
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The Teichmueller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Δ) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {Δ_{μ,ν}} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant o…
We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determinantal random point processes on the two-sphere are also discussed. The inequalities are obtained as corollaries of a general theorem about the maximizers of a certain non-local functional defined on the space of all positively curved Hermitian metrics on an ample line bundle ov…
Toeplitz Determinants for a Class of Holomorphic Mappings in Higher Dimensions
本文针对高维全纯映射，建立了由系数构成的Toeplitz行列式的锐界，为单叶函数子类提供了新结果，并推广了已知结论。
We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if and only if for any positive constant $C$, the number of eigenvalues of the $\overline\partial$-Neumann Laplacian less than or equal to $Ck$ grows polynomially as $k$ tends to infinity.
Heat kernel asymptotics for Kodaira Laplacians of high power of line bundle over complex manifolds
本文提出一种基于缩放技术的方法，证明了非紧Hermitian流形上线丛高次幂的Kodaira Laplacian的热核渐近，并以此直接证明覆盖流形上的全纯Morse不等式，进而通过$L^2$ Le Potier同构推广至向量丛及代数版本。
For a quasi-Fuchsian group $\Ga$ with ordinary set $Ω$, and $Δ_{n}$ the Laplacian on \n differentials on $\Ga\bkΩ$, we define a notion of a Bers dual basis $ϕ_{1},...c,ϕ_{2d}$ for $\kerΔ_{n}$. We prove that $\detΔ_{n}/\det <ϕ_{j},ϕ_{k}>$, is, up to an anomaly computed by Takhtajan and the second author in \cite{TT1}, the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta Z(n). This generalizes the D'Hoker-Phong formula $\detΔ_{n}=c_{g,n}Z(n)$, and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and t…
Fractional $\frac{1}{2}$-Laplacian of holomorphic functions via Jacobi polynomials
研究原点处有幂型奇点的光滑函数及全纯函数的分数阶1/2拉普拉斯算子，基于特定测试函数空间定义，通过对洛朗级数应用构造得含雅可比多项式的新级数，还推导了相关应用及泛函演算。
For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n=1. We introduce a holomorphic function F(n) on S_g which generalizes the classical product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly …
In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Further, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in $\mathbb{C}^n$. The obtained results provide the bounds of Toeplitz determinants for various subclasses of normalized univalent functions in higher dimensions.
The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers
研究完整梯度收缩Ricci流形上的多项式增长全纯函数和形式，通过f-拉普拉斯谱数据证明其空间维数有限，并获得线性增长全纯函数的精确维数估计。
For $n$ a nonnegative integer, we consider the $n$-Laplacian $Δ_n$ acting on the space of $n$-differentials on a confinite Riemann surface $X$ which has ramification points. The trace formula for the resolvent kernel is developed along the line à la Selberg. Using the trace formula, we compute the regularized determinant of $Δ_n+s(s+2n-1)$, from which we deduce the regularized determinant of $Δ_n$, denoted by $\det\!'Δ_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'Δ_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when $n\geq 2$. Here $Z(s)$ is…
For a Hermitian holomorphic vector bundle over a Hermitian manifold, we consider the Dolbeault Laplacian with $\overline\partial$-Neumann boundary conditions, which is a self-adjoint operator on the space of square-integrable differential forms with values in the given holomorphic bundle. We argue that some known results on the spectral properties of this operator on pseudoconvex domains in $\mathbb C^n$ continue to hold on Kähler manifolds satisfying certain bounded geometry assumptions. In particular, we will consider the Dolbeault complex for forms with values in a line bundle, where known …
Let $(E,h)$ be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of $E$. If $E$ is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.
In this paper, we derive the sharp bounds of Toeplitz determinants for a class of holomorphic mappings on the bounded starlike circular domain $Ω$ in $\mathbb{C}^n$, which extend certain known bounds for various subclasses of normalized analytic univalent functions in the unit disk to higher dimensions.
Let $(X,ω)$ be a compact Kähler manifold. Let $(L,h)$ be a hermitian holomorphic line bundle over $X$, such that $Θ_{L,h}\geq -\varepsilonω$ for a small $\varepsilon>0$, $E$ be a holomorphic line bundle over $X$. For $k\in \mathbb{N}_+$, denote by $X_k:=(X,ω^k)$ the Kähler manifold $X$ with new scaled metric $ω^k=kω$. Estimates of the number of eigenvalues smaller than $λ$ of the $\debar$-Laplacian on forms on $X_k$ with values in $L^k\otimes E$ are presented for $0\leq λ<k$. In particular, when $λ=0$, we get a numeric bound for the cohomology groups.
We prove that CR lines in an exponentially degenerate boundary are propagators of holomorphic extension. This explains, in the context of the CR geometry, why in this situation the induced Kohn-Laplacian $\Box_b$ is not hypoelliptic according to recent work by Christ.
Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq p}R^{n-i}Ω^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.
On Hankel Determinants of $β$-Spirallike Mappings in Complex Banach Spaces
研究复Banach空间单位球上归一化β-螺旋状映射的第二个Hankel行列式，利用Banach空间全纯映射与单变量β-螺旋状函数关系，先建立初始齐次展开系数估计，进而得出第二个Hankel行列式的精确界。
Large deviations of multichordal SLE$_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
本文证明了多弦 SLE$_{0+}$ 曲线的强极大偏差原理，引入了与 Laplace 算子 zeta 正则化行列式相关的 Loewner 势，并证明其最小化多弦为有理函数的实轨迹，同时作为副产品解析地证明了 Shapiro 猜想。
Spectral theory of the invariant Laplacian on the disk and the sphere -- a complex analysis approach
本文通过复分析方法，研究复化单位圆补集上不变拉普拉斯算子的全纯谱理论，给出本征空间闭Möbius不变子空间的内在分类，统一了双曲与球面拉普拉斯算子的本征值理论。
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