Zeta Determinants on Manifolds with Boundary

Relative Zeta Determinants and the Quillen Metric

We compute the relative zeta-function metric on the determinant line bundle for a family of elliptic boundary value problems of Dirac-type. To do this we prove a general formula relating the zeta-determinant to a Fredholm determinant over the boundary for a class of higher-order elliptic boundary value problems.

Zeta Forms and the Local Family Index Theorem

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the classical single operator spectral zeta function and determinant. In the case where F is the curvature of a superconnection the zeta form is exact, extending to families the Atiyah-Bott-Seeley zeta function formula for the pointwise index, and equivalent to the transgression formu…

紧黎曼曲面上锥形拉普拉斯算子的Krein公式

Weyl Curves and Zeta Determinants of Conic Laplacians on Riemann Surfaces

通过临界渐近边界数据的有限维辛空间，建立紧黎曼曲面上锥形拉普拉斯算子自伴扩张的Krein公式，并导出迹恒等式用于比较不同Lagrangian边界条件下的正谱zeta行列式。

正交平坦环面上Zeta正则化行列式的全局极值

Global Extrema of the Zeta Regularized Determinant on Orthogonal Flat Tori

研究单位体积正交平坦环面中最大化Laplacian Zeta正则化行列式的问题，证明等边环面是唯一极大值点，验证了Sarnak猜想，并探讨了Neumann和Dirichlet边界条件下的类似问题。

Functional Determinant on Pseudo-Einstein 3-manifolds

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we establish an expression for the difference of determinants of the Paneitz type operators $A_θ$, related to the problem of prescribing the $Q'$-curvature, under the conformal change $θ\mapsto e^{w}θ$ with $w\in ¶$ the space of pluriharmonic functions. This generalizes the expression of the functional determinant in four dimensional Riemannian manifolds established in \cite{BO2}. We also provide an existence result of maximizers for the scaling invariant functional determinant as in \cite{CY}.

The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds

We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that…