Correction to: A Polyakov Formula for Sectors

The Original Article was published on 05 July 2017

Correction to: J Geom Anal (2018) 28:1773–1839 https://doi.org/10.1007/s12220-017-9888-y

Let \(S_\alpha \) denote a finite circular sector of opening angle \(\alpha \in (0,\pi )\) and radius one, and let \(e^{-t \Delta _\alpha }\) denote the heat operator associated to the Dirichlet extension of the Laplacian. Based on recent joint work [2] and [3], we discovered an extra contribution to the variational Polyakov formula in [1] coming from the curved boundary component of the sector. Theorems 3 and 4 of [1] should have an added term \(+\frac{1}{4\pi }\). This calculation will appear in [2]. The corrected statements of these theorems are given below.

Theorem 1

(Theorem 3 [1]) Let \(S_{\pi /2}\subset {\mathbb {R}}^2\) be a circular sector of opening angle \(\pi /2\) and radius one. Then the variational Polyakov formula is

$$\begin{aligned} \left. \frac{\partial }{\partial \gamma } \big (-\log (\det (\Delta _{S_\gamma }))\big )\right| _{\gamma =\pi /2} = \frac{-\gamma _e}{4\pi } + \frac{2}{3 \pi }, \end{aligned}$$

where \(\gamma _e\) is the Euler-Mascheroni constant.

Theorem 2

(Theorem 4 [1]) Let \(0<\alpha < \pi \), and let

$$\begin{aligned} k_{min} = \left\lceil { \frac{-\pi }{2\alpha } }\right\rceil , \text { and } k_{max} = \left\lfloor {\frac{\pi }{2\alpha }}\right\rfloor \text { if } \frac{\pi }{2\alpha } \not \in {\mathbb {Z}}, \text { otherwise } k_{max} = \frac{\pi }{2\alpha } - 1, \end{aligned}$$

and \(W_{\alpha } =\left\{ k \in \left( {\mathbb {Z}}\bigcap \left[ k_{min}, k_{max}\right] \right) {\setminus } \left\{ \frac{\ell \pi }{\alpha } \right\} _{\ell \in {\mathbb {Z}}} \right\} .\) Then

$$\begin{aligned} {{\mathcal {S}}}(\alpha )&:=\frac{\partial }{\partial \gamma } \left. \big (-\log (\det (\Delta _{\gamma }))\big )\right| _{\gamma =\alpha } = \frac{1}{3\pi } + \frac{\pi }{12\alpha ^2} \\&\quad + \sum _{k\in W_{\alpha }} \frac{-2\gamma _e + \log (2) - \log \left( {1-\cos (2k\alpha )}\right) }{4 \pi (1-\cos (2k\alpha ))} \\&\quad - (1-\delta _{\alpha , \frac{\pi }{n}})\ \frac{2}{\alpha } \sin (\pi ^2/\alpha ) \int _{-\infty } ^\infty \frac{\gamma _e + \log (2) - \log (1+\cosh (s))}{16\pi (1+\cosh (s))(\cosh (\pi s /\alpha ) - \cos (\pi ^2/\alpha ))} ds, \end{aligned}$$

where \(n\in {\mathbb {N}}\) is arbitrary and \(\delta _{\alpha , \frac{\pi }{n}}\) denotes the Kronecker delta.

It therefore follows that the list of examples given following Theorem 4 in [1] should be revised accordingly:

  1. (1)

    \(\alpha =\frac{\pi }{4}\), \(W_{\frac{\pi }{4}}= \{-2,\pm 1,\}\), \({{\mathcal {S}}}(\frac{\pi }{4})= \frac{-5\gamma _e}{4\pi } + \frac{\log (2)}{2\pi }+ \frac{5}{3\pi } \sim 0.411167\)

  2. (2)

    \(\alpha =\frac{\pi }{3}\), \(W_{\frac{\pi }{3}}= \{-1,1\}\), \({{\mathcal {S}}}(\frac{\pi }{3})= \frac{13}{12\pi } - \frac{2\gamma _e}{3\pi } + \frac{\log (4/3)}{3\pi } \sim 0.252871\)

  3. (3)

    \(\alpha =\frac{\pi }{2}\), \(W_{\frac{\pi }{2}}=\{-1\}\), \({{\mathcal {S}}}(\frac{\pi }{2})= \frac{-\gamma _e}{4\pi } + \frac{2}{3\pi } \sim 0.166273\).

  4. (4)

    For \(\alpha \in ]\frac{\pi }{2},\pi [\), \(W_{\alpha }=\emptyset \), but \(\sin (\pi ^2/\alpha )\ne 0\). If \(\alpha =\frac{2\pi }{3}\), the integral converges rapidly, and a numerical computation gives an approximate value of 0.0075015. Hence \({{\mathcal {S}}}(\frac{2\pi }{3}) \sim \frac{1}{3\pi } + \frac{3}{16 \pi } + \frac{3}{\pi } (0.0075015) \sim 0.1729498\).

Misprint

There is a two missing in Equation (1.3) of [1]. That equation should be:

$$\begin{aligned} \partial _t \log \det (\Delta _{g_t}) = {-\frac{1}{12\pi }} \int _{M} \sigma '(t) \ \text {Scal}_t \ dA_{g_t} + \partial _t \log \text {Area}(M,g_t). \end{aligned}$$

References

  1. 1.

    Aldana, C.L., Rowlett, J.: A Polyakov formula for sectors. J. Geom. Anal. 28(2), 1773–1839 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aldana, C.L., Kirsten, K., Rowlett, J.: A Polyakov formula for surfaces with conical singularities and boundary, pre-print

  3. 3.

    Nursultanov, M., Rowlett, J., Sher, D.: The heat kernel on curvilinear polygonal domains in surfaces, arXiv:1905.00259

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Acknowledgements

We thank Klaus Kirsten and Alexander Strohmaier who separately helped us find the mistake in our formula. The first author acknowledges the hospitality of the Max Planck Institute for Mathematics in Bonn during a visit in when then missing term was found. The second author is supported by the National Science Foundation Grant DMS-1440140 during the fall 2019 semester at MSRI and by the Swedish Research Council Grant 2018-03873.

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Aldana, C.L., Rowlett, J. Correction to: A Polyakov Formula for Sectors. J Geom Anal 30, 3371–3372 (2020). https://doi.org/10.1007/s12220-019-00319-8

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