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Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

Ellen Krusell

TL;DR

The article extends Polyakov-Alvarez type determinant variation formulas to curvilinear polygonal domains with arbitrary positive corner angles, including slit domains. By deriving a short-time heat-trace expansion via a patchwork approach that aggregates interior, boundary, and corner contributions, it defines the ζ-regularized determinant $\det_\zeta\Delta_{(M,g)}$ for these domains and establishes a conformal variation formula. The main result is a precise Polyakov-Alvarez anomaly formula under smooth conformal changes $g\mapsto e^{2\sigma}g$, with explicit corner-angle corrections, offering a robust tool for analyzing determinants on non-smooth domains. The work connects to random conformal geometry and Loewner energy, and provides analytic machinery applicable to slit and polygonal domains in geometric analysis and mathematical physics.

Abstract

We consider the $ζ$-regularized determinant of the Friedrichs extension of the Dirichlet Laplace-Beltrami operator on curvilinear polygonal domains with corners of arbitrary positive angles. In particular, this includes slit domains. We obtain a short time asymptotic expansion of the heat trace using a classical patchwork method. This allows us to define the $ζ$-regularized determinant of the Laplacian and prove a comparison formula of Polyakov-Alvarez type for a smooth and conformal change of metric.

Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

TL;DR

The article extends Polyakov-Alvarez type determinant variation formulas to curvilinear polygonal domains with arbitrary positive corner angles, including slit domains. By deriving a short-time heat-trace expansion via a patchwork approach that aggregates interior, boundary, and corner contributions, it defines the ζ-regularized determinant for these domains and establishes a conformal variation formula. The main result is a precise Polyakov-Alvarez anomaly formula under smooth conformal changes , with explicit corner-angle corrections, offering a robust tool for analyzing determinants on non-smooth domains. The work connects to random conformal geometry and Loewner energy, and provides analytic machinery applicable to slit and polygonal domains in geometric analysis and mathematical physics.

Abstract

We consider the -regularized determinant of the Friedrichs extension of the Dirichlet Laplace-Beltrami operator on curvilinear polygonal domains with corners of arbitrary positive angles. In particular, this includes slit domains. We obtain a short time asymptotic expansion of the heat trace using a classical patchwork method. This allows us to define the -regularized determinant of the Laplacian and prove a comparison formula of Polyakov-Alvarez type for a smooth and conformal change of metric.
Paper Structure (16 sections, 6 theorems, 148 equations, 4 figures)

This paper contains 16 sections, 6 theorems, 148 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Motivation
  4. Preliminaries
  5. Curvilinear polygonal domains
  6. The Laplace-Beltrami operator and Brownian motion
  7. Basic properties of the heat kernel
  8. Domain monotnicity.
  9. Local Gaussian bounds.
  10. Kac's locality principle.
  11. Global Gaussian bounds.
  12. The McKean-Singer construction
  13. The -regularized determinant of the Laplacian
  14. Short time asymptotic expansion of the heat trace
  15. Polyakov-Alvarez type anomaly formula
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(M,g_0,(p_j),(\alpha_j))$ be a curvilinear polygonal domain, $\sigma,\psi\in C^\infty(M,g,(p_j),(\alpha_j))$, and define $g_u:=e^{2u\sigma}g_0$. Then, for each $q\in(0,1/2)$ and each $u\in \mathbb R$, where the error is locally uniform in $u$ as $t\to 0+$.

Figures (4)

  • Figure 1: Illustration of $V_{j,1},V_{j,2}\subset \hat{M}_0$.
  • Figure 2: Illustration of the set-up in Lemma \ref{['lemma:corner']}.
  • Figure 3: Partitioning of $M$.
  • Figure 4: On the top left we illustrate the set $\varphi_j(S_{1,\varepsilon})$ and on the top right we illustrate the set $S=S(\varepsilon,b,h)$ relative a ball of radius $\varepsilon$ and the $x_1$-axis. In the bottom figures we illustrate the two sets $S_{1,\varepsilon}^+\subset W_{\varepsilon}^+$ and $S_{1,\varepsilon}^-\subset W_\varepsilon^-$, which are of the type $S=S(\varepsilon,h,b)$ relative the ball $B_\varepsilon$ and the lower prong of $W_\varepsilon^+$ and $W_\varepsilon^-$ (indicated with an arrow) respectively.

Theorems & Definitions (20)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Definition 1
  • Remark 2
  • Definition 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • Remark 5
  • ...and 10 more