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Transactions of the American Mathematical Society

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Local geometry of singular real analytic surfaces

Author: Daniel Grieser
Journal: Trans. Amer. Math. Soc. 355 (2003), 1559-1577
MSC (2000): Primary 14P15; Secondary 32B20, 53B20, 58J99
Published electronically: November 18, 2002
MathSciNet review: 1946405
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Abstract: Let $V\subset\mathbb{R} ^N$ be a compact real analytic surface with isolated singularities, and assume its smooth part $V_0$ is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $\mathbb{R} ^N$. We prove:

Each point of $V$ has a neighborhood which is quasi-isometric (naturally and ``almost isometrically'') to a union of metric cones and horns, glued at their tips.
A full asymptotic expansion, for any $p\in V$, of the length of $V\cap\{q:{\rm dist\,}(q,p)=r\}$ as $r\to0$.
A Gauss-Bonnet Theorem, saying that each singular point contributes $1-l/(2\pi)$, where $l$ is the coefficient of the linear term in the expansion of (2).
The $L^2$ Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on $V_0$, an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic.
As a central tool we use resolution of singularities.

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  • 1. M. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145-150. MR 41:815
  • 2. I. N. Bernstein and S. I. Gel'fand, Meromorphic property of the functions $P\sp \lambda$ (English. Russian original) Funct. Anal. Appl. 3(1969), 68-69; translation from Funkts. Anal. Prilozh. 3 (1969), 84-85. MR 40:723
  • 3. E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207-302. MR 98e:14010
  • 4. E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Inst. Hautes Etudes Sci. Publ. Math. 67 (1988), 5-42. MR 89k:32011
  • 5. L. Birbrair, Local bi-Lipschitz classification of 2-dimensional semialgebraic sets, Houston J. Math. 25 (1999), 453-472. MR 2000j:14091
  • 6. L. Bröcker, M. Kuppe, and W. Scheufler, Inner metric properties of 2-dimensional semi-algebraic sets, in: Real algebraic and analytic geometry (Segovia 1995), Rev. Mat. Univ. Complut. Madrid 10 (1997), 51-78. MR 98m:53084
  • 7. J. Brüning, Irregular spectral asymptotics, Canad. Math. Soc. Conf. Proc. 29 (2000), 85-99. MR 2002b:58032
  • 8. J. Brüning, The signature theorem for manifolds with metric horns, Journées: ``Équations aux Dérivées Partielles'' (Saint-Jean-de-Monts, 1996), Exposé II, École Polytech., Palaiseau, 1996. MR 98a:58150
  • 9. J. Brüning and M. Lesch, Hilbert complexes, J. Funct. Anal. 108 (1992), 88-132. MR 93k:58208
  • 10. J. Brüning and M. Lesch, Kähler-Hodge theory for conformal complex cones, Geom. Funct. Anal. 3 (1993), 439-473. MR 94i:58189
  • 11. J. Brüning and M. Lesch, On the spectral geometry of algebraic curves, J. reine angew. Math. 474 (1996), 25-66. MR 97d:58193
  • 12. J. Brüning, N. Peyerimhoff, and H. Schröder, The $\bar\partial$-operator on algebraic curves, Commun. Math. Phys. 129 (1990), 525-534. MR 91g:58273
  • 13. C. Callias, On the existence of small-time heat expansions for operators with irregular singularities in the coefficients, Math. Res. Letters 2 (1995), 129-146. MR 96h:58165
  • 14. J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Proc. Sympos. Pure Math. 36 (1980), 91-146. MR 83a:58081
  • 15. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geom. 18 (1983), 575-657. MR 85d:58083
  • 16. J. Cheeger, M. Goresky, and R. MacPherson,$L^2$-cohomology and intersection homology of algebraic varieties, Annals of Math. Studies 102 (1982), 303-340. MR 84f:58005
  • 17. G. Eulering, Integrale Krümmungskonstanten algebraischer Untermannigfaltigkeiten von Räumen konstanter Krümmung, Doctoral Thesis, Münster 1995.
  • 18. J. Fu, Curvature measures of subanalytic sets, Amer. J. Math. 116 (1994), 819-880. MR 95g:32016
  • 19. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. I, Academic Press, New York, 1964. MR 29:3869
  • 20. D. Grieser and M. Lesch, On the $L^2$ Stokes theorem and Hodge theory for singular algebraic varieties, to appear in Math. Nachr.
  • 21. D. Grieser,Quasiisometry of singular metrics, to appear in Houston J. Math.
  • 22. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79 (1964), 109-326. MR 33:7333
  • 23. M. W. Hirsch, Differential Topology, Springer, 1976. MR 56:6669
  • 24. P. Jeanquartier,Transformation de Mellin et développements asymptotiques, L'Enseignement Mathématique 25 (1979), 285-308. MR 81g:46049
  • 25. M. Lesch and N. Peyerimhoff, An index theorem for manifolds with metric horns, Comm. Partial Differential Equations 23 (1998), 649-684. MR 99d:58166
  • 26. J.-M. Lion and J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier 48 (1998), 755-767. MR 2000i:32011
  • 27. M. Nagase, Hodge theory of singular algebraic curves, Proc. Amer. Math. Soc. 108 (1990), 1095-1101. MR 90h:14031
  • 28. R. M. Ruiz,The basic theory of power series, Vieweg 1993. MR 94i:13012
  • 29. M. Spivak,A comprehensive introduction to differential geometry, Vol. III-IV, Publish or Perish, 1975. MR 51:8962; MR 52:15254a

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Additional Information

Daniel Grieser
Affiliation: Institut für Mathematik, Humboldt Universität zu Berlin, Sitz: Rudower Chaussee 25, 10099 Berlin, Germany
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139

Keywords: Real analytic sets, quasi-isometry, Gauss-Bonnet theorem, $L^2$ Stokes theorem, resolution of singularities
Received by editor(s): July 9, 2002
Published electronically: November 18, 2002
Additional Notes: The author gratefully acknowledges support by the Deutsche Forschungsgemeinschaft (Gerhard-Hess-Programm) and the Erwin Schrödinger Institute
Article copyright: © Copyright 2002 American Mathematical Society