Polyhedral surfaces and determinant of Laplacian

Kokotov, Alexey

doi:10.1090/s0002-9939-2012-11531-x

Cited by 22 publications

(34 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This formula is stated in [9] for fixed radius R and for the heat kernel at time t as t → 0, but because of the usual scaling properties, it holds equally well for fixed t, say t = 1, and as the radius R → ∞; indeed, the quantity on the left depends only on the ratio R/t 2 . The coefficients in this expansion have been written in a nonreduced form in order to emphasize the dependence on the angle 2α.…”

Section: Neumann Boundary Conditionsmentioning

confidence: 98%

A heat trace anomaly on polygons

Mazzeo

Rowlett

2015

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Let $\Omega_0$ be a polygon in $\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $\Omega_\e$ is a family of surfaces with $\calC^\infty$ boundary which converges to $\Omega_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as $\e \searrow 0$. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.Comment: Revision includes treatment of the Neumann problem and a discussion of the higher dimensional case; some new reference

show abstract

Section: Neumann Boundary Conditionsmentioning

confidence: 98%

A heat trace anomaly on polygons

Mazzeo

Rowlett

2015

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…In [22] it was found a comparison formula (an analog of classical Polyakov formula) relating determinants of the Laplacians in two conformally equivalent flat conical metrics. This lead to the generalization of the results of [24] to the case of arbitrary flat conformal metrics with conical singularities.…”

Section: General Partmentioning

confidence: 99%

Moduli spaces of meromorphic functions and determinant of the Laplacian

Hillairet¹,

Kalvin²,

Kokotov³

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

The Hurwitz space is the moduli space of pairs (X, f ) where X is a compact Riemann surface and f is a meromorphic function on X. We study the Laplace operator ∆ |df | 2 of the flat singular Riemannian manifold (X, |df | 2 ). We define a regularized determinant for ∆ |df | 2 and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential df. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the S-matrix, S(λ), of a surface with conical singularities as a function of the spectral parameter λ and relate its behavior at λ = 0 with the Schiffer projective connection on the Riemann surface X. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move. *

show abstract

“…In recent years there has been progress towards understanding the behavior of the determinant of certain self-adjoint extensions of the Laplace operator, most notably the Friedrichs extension, on surfaces with conical singularities. This progress represents different aspects that have been studied by Kokotov [ 22 ], Hillairet and Kokotov [ 19 ], Loya et al [ 26 ], Spreafico [ 40 ], and Sher [ 39 ]. In particular, the results by Aurell and Salomonson [ 5 ] inspired our present work.…”

Section: Introductionmentioning

confidence: 95%

A Polyakov Formula for Sectors

Aldana

Rowlett

2017

J Geom Anal

View full text Add to dashboard Cite

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

show abstract

Polyhedral surfaces and determinant of Laplacian

Abstract: An explicit formula for the determinant of the Laplacian on a compact polyhedral surface of genus g > 1 is found. This formula generalizes previously known results for flat surfaces with trivial holonomy and compact polyhedral tori.

Cited by 22 publications

References 17 publications

A heat trace anomaly on polygons

A heat trace anomaly on polygons

Moduli spaces of meromorphic functions and determinant of the Laplacian

A Polyakov Formula for Sectors

Contact Info

Product

Resources

About