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Yoonweon Lee

Publications and source records attributed to Yoonweon Lee.

14 recordsLinked to original sources
arXiv2024

The zeta-determinants and anlaytic torsion of a metric mapping torus

We use the BFK-gluing formula for zeta-determinants to compute the zeta-determinant and analytic torsion of a metric mapping torus induced from an isometry. As applications, we compute the zeta-determinants of the Laplacians defined on a Klein bottle ${\mathbb K}$ and some compact co-Kähler manifold ${\mathbb T}_φ$. We also show that a metric mapping torus and a Riemannian product manifold with a round circle have the same heat trace asymptotic expansions. We finally compute the analytic torsion of a metric mapping torus for the Witten deformed Laplacian and recover the result of J. Marcsik in \cite{Ma}.

math.DG↗
arXiv2024

The zeta-determinant of the Dirichlet-to-Neumann operator of the Steklov Problem on forms

On a compact Riemannian manifold $M$ with boundary $Y$, we express the log of the zeta-determinant of the Dirichlet-to-Neumann operator acting on $q$-forms on $Y$ as the difference of the log of the zeta-determinant of the Laplacian on $q$-forms on $M$ with absolute boundary conditions and that of the Laplacian with Dirichlet boundary conditions with some additional terms which are expressed by curvature tensors. When the dimension of $M$ is $2$ or $3$, we compute these terms explicitly. We also discuss the value of the zeta function at zero associated to the Dirichlet-to-Neumann operator by using a conformal rescaling method. As an application, we recover the result of the conformal invariance obtained in C. Guillarmou and L. Guillopé, The determinant of the Dirichlet-to-Neumann map for surfaces with boundary, Int. Math. Res. Not. IMRN 2007, no. 22, Art. ID rnm099, when the dimension of $M$ is $2$.

math.DG↗
arXiv2023

The BFK type gluing formula of zeta-determinants for the Robin Boundary condition

In this paper we discuss the BFK type gluing formula for zeta-determinants of Laplacians with respect to the Robin boundary condition on a compact Riemannian manifold. As a special case, we discuss the gluing formula with respect to the Neumann boundary condition. We also compute the difference of two zeta-determinants with respect to the Robin and Dirichlet boundary conditions. We use this result to compute the zeta-determinant of a Laplacian on a cylinder when the Robin boundary condition is imposed, which extends a result in [25]. We also discuss the gluing formula more precisely when the product structure is given near a cutting hypersurface.

math.DG↗
arXiv2020

Virtually small spectral package of a Riemannian manifold

For a Morse function on a closed orientable Riemannian manifold one introduces the {\it virtually small spectral package} an analytic object consisting of a finite number of analytic quantities derived from the pair, {\it Riemannian metric, Morse function\} which, in principle, can be calculated. One shows that they determine the {\it Torsion } of the underlying space, a parallel to the result that the dimensions of the spaces of harmonic forms calculate the {\it Euler-Poincaré characteristic} of the underlying space and extends the {\it Poincaré Duality} between harmonic forms and between Betti numbers for a closed oriented Riemannian manifold .

math.DG↗
arXiv2019

The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann operator in terms of the scalar and principal curvatures of the cutting hypersurface.

math.DG↗
arXiv2017

The refined analytic torsion and a well-posed boundary condition for the odd signature operator

In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator $\mathcal{B}$ in the sense of Seeley. We then show that the zeta-determinants of $\mathcal{B}^2$ and eta-invariants of $\mathcal{B}$ subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

math.DG↗
arXiv2014

Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map having simple fixed points and satisfying some special condition near the boundary. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

math.DG↗
arXiv2013

The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result of [15] to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions.

math.DG↗
arXiv2012

The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman and the authors, but these two constructions are completely different. Vertman used a double of de Rham complex consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$, ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the deformation method used in [6], when the odd signature operator comes from a Hermitian flat connection and all de Rham cohomologies vanish.

math.DG↗
arXiv2011

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous work ([14]) we introduced the well-posed boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$ for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions ${\mathcal P}_{-, {\mathcal L}_{0}}$ and ${\mathcal P}_{+, {\mathcal L}_{1}}$. In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition ${\mathcal P}_{-, {\mathcal L}_{0}}$ or ${\mathcal P}_{+, {\mathcal L}_{1}}$ with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.

math.DG↗
arXiv2004

The ratio of two zeta-determinants of Dirac Laplacians associated with unitary involutions on a compact manifold with cylindrical end

Given two unitary involutions $σ_{1}$ and $σ_{2}$ satisfying $G σ_{i} = - σ_{i} G$ on $ker B$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. Müller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with $σ_{1}$ and $σ_{2}$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. Müller in [M]. This result was also obtained independently by J. Park and K. Wojciechowski ([PW2]).

math.DG↗
arXiv2003

Burghelea-Friedlander-Kappeler's gluing formula and the adiabatic decomposition of the zeta-determinant of a Dirac Laplacian

In this paper we first establish the relation between the zeta-determinant of a Dirac Laplacian with the Dirichlet boundary condition and the APS boundary condition on a cylinder. Using this result and the gluing formula of the zeta-determinant given by Burghelea, Friedlander and Kappeler with some assumptions, we prove the adiabatic decomposition theorem of the zeta-determinant of a Dirac Laplacian. This result was originally proved by J. Park and K. Wojciechowski in [11] but our method is completely different from the one they presented.

math.DG↗
arXiv2003

Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion

The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under the assumption of the product structure near boundary. As applications of this result,we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.

math.DG↗