Journal of the Mathematical Society of Japan

The analytic torsion of the finite metric cone over a compact manifold

Luiz HARTMANN and Mauro SPREAFICO

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 311-371.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730028

Digital Object Identifier
doi:10.2969/jmsj/06910311

Mathematical Reviews number (MathSciNet)
MR3597557

Zentralblatt MATH identifier
1369.58025

Subjects
Primary: 58J52: Determinants and determinant bundles, analytic torsion

Keywords
analytic torsion pseudo manifolds finite metric cone

Citation

HARTMANN, Luiz; SPREAFICO, Mauro. The analytic torsion of the finite metric cone over a compact manifold. J. Math. Soc. Japan 69 (2017), no. 1, 311--371. doi:10.2969/jmsj/06910311. https://projecteuclid.org/euclid.jmsj/1484730028


Export citation

References

  • J. Brüning and X. Ma, An anomaly formula for Ray–Singer metrics on manifolds with boundary, GAFA, 16 (2006), 767–837.
  • J. Brüning and X. Ma, On the gluing formula for the analytic torsion, Math. Z., 273 (2013), 1085–1117.
  • J. Brüning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal., 73 (1988), 369–415.
  • J. Cheeger, Analytic torsion and the heat equation, Ann. Math., 109 (1979), 259–322.
  • J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geom., 18 (1983), 575–657.
  • J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Proc. Sympos. Pure Math., 36 (1980), 91–146.
  • L. Hartmann, The boundary term from the analytic torsion of a cone over a $m$-dimensional sphere, Mat. Contemp., 43 (2014), 133–170.
  • L. Hartmann, T. de Melo and M. Spreafico, The analytic torsion of a disc, Ann. Glob. Anal. Geom., 42 (2012), 29–59.
  • L. Hartmann and M. Spreafico, The analytic torsion of a cone over a sphere, J. Math. Pures Appl., 93 (2010), 408–435.
  • L. Hartmann and M. Spreafico, The analytic torsion of the cone over an odd dimensional manifold, J. Geom. Phys., 61 (2011), 624–657.
  • L. Hartmann and M. Spreafico, R torsion and analytic torsion of a conical frustum, J. Gökova Geom. Topology, 6 (2012), 28–57.
  • L. Hartmann and M. Spreafico, On the Cheeger–Müller theorem for an even-dimensional cone, St. Petersburg Math. J., 27 (2016), 137–154.
  • W. Lück, Analytic and topological torsion for manifolds with boundary and symmetry, J. Diff. Geom., 37 (1993), 263–322.
  • J. Milnor, Whitehead torsion, Bull. AMS, 72 (1966), 358–426.
  • W. Müller, Analytic torsion and $R$-torsion of Riemannian manifolds, Adv. Math., 28 (1978), 233–305.
  • W. Müller and B. Vertman, The metric anomaly of analytic torsion on manifolds with conical singularities, Comm. PDE., 39 (2014), 146–191.
  • F. W. J. Olver, Asymptotics and special functions, AKP, 1997.
  • D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Adv. Math., 7 (1971), 145–210.
  • M. Spreafico, On the non homogeneous quadratic Bessel zeta function, Mathematika, 51 (2004), 123–130.
  • M. Spreafico, Zeta function and regularized determinant on a disc and on a cone, J. Geom. Phys., 54 (2005), 355–371.
  • M. Spreafico, Zeta invariants for sequences of spectral type, special functions and the Lerch formula, Proc. Roy. Soc. Edinburgh, 136A (2006), 863–887.
  • M. Spreafico, Zeta determinant for double sequences of spectral type, Proc. Amer. Math. Soc., 140 (2012), 1881–1896.
  • G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, 1922.