Hostname: page-component-5cf477f64f-fcbfl Total loading time: 0 Render date: 2025-04-04T23:49:42.349Z Has data issue: false hasContentIssue false
  • English
  • Français

Collapsing Riemannian Metrics to Carnot-Caratheodory Metrics and Laplacians to Sub-Laplacians

Published online by Cambridge University Press:  20 November 2018

Zhong Ge
Zhong Ge*
Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the asymptotic behavior of the Laplacian on functions when the underlying Riemannian metric is collapsed to a Carnot-Carathéodory metric. We obtain a uniform short time asymptotics for the trace of the heat kernel in the case when the limit Carnot-Carathéodory metric is almost Heisenberg, the limit of which is the result of Beal-Greiner-Stanton, and Stanton-Tartakoff.

Keywords

58G2535H05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 537 - 553
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Beals, R., Greiner, P. C. and Stanton, N. K., The heat equation on a CR manifold, J. Differential Geometry 20(1984), 343–387.Google Scholar
2. Brockett, R., Control theory and singularRiemannian geometry. In: New Direction in Applied Mathematics, Springer, Berlin.Google Scholar
3. Folland, G. B. and Stein, E. M., Estimates for the ?b, complex and analysis on the Heisenberg group, Comm. Pure. Appl. Math. 27(1974), 429–522.Google Scholar
4. Fukaya, K., Collapsing of Riemannian manifolds and eigenvalues of Laplacian operators, Invent. Math. 87(1987),517–547.Google Scholar
5. Gaveau, B., Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certain groupes nilpotents, Acta Math. 139(1977), 95–153.Google Scholar
6. Ge, Zhong, On a constrained variational problem and the spaces of horizontal paths, Pacific J. of Math. 149(1991), 61–94. 7 , Horizontal paths spaces and Carnot-Carathéodory metrics, Pacific J. of Math., to appear.Google Scholar
8. Ge, Zhong, Betti numbers, characteristic classes and sub-Riemannian geometry, III1. J. Math., to appear.Google Scholar
9. Ge, Zhong, A sub-elliptic Hodge theory, preprint.Google Scholar
10. Getzler, E., An analogue of Demailly's inequality for strictly pseudoconvex CR manifold, J. Differential Geom. 29(1989), 231–244.Google Scholar
11. Gromov, M., Sur structure métriques pour les variétés riemannian, Cedic-Nathan.Google Scholar
12. Hamenstadt, U., Some regularity in Carnot-Carathéodory metrics, preprint.Google Scholar
13. Hormander, L., Hypoelliptic second order differential equations, Acta Math. 199( 1967), 147–171.Google Scholar
14. Kac, M., Can one hear the shape of a drum, Amer. Math. Monthly 73(1966), 1–23.Google Scholar
15. McKeanJr, H. P.. and Singer, I. M., Curvature and the eigenvalues of the Laplacian, J. Differ. Geom. 1(1967), 43–69.Google Scholar
16. Mitchell, J., On Carnot-Carathéodory metrics, J. Differ. Geom. 21(1985), 35–45.Google Scholar
17. Montgomery, R., Shortest loops with a fixed holonomy, MSRI preprint.Google Scholar
18. Rothschild, L. and Stein, E., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137( 1976), 247–320.Google Scholar
19. Pansu, P., Métriques de Carnot-Carathéodory et quasi-isometries des espaces symétriques de rang un, Ann. Math. 129(1989), 1–60.Google Scholar
20. Stanton, N. K. and Tartakoff, D. S., The heat equation for the ,-Laplacian, Comm. Partial Diff. Equations 9(1984), 597–686.Google Scholar
21. Strichartz, R. S., Sub-Riemannian geometry, J. Differential Geometry 24(1986), 221–263.Google Scholar
22. Taylor, T. J., Some aspects of differential geometry associated with hypoelliptic second order operators, Pacific J. Math. 136(1989), 355–387.Google Scholar
23. Vershik, A. M. and Gershkovich, C. Ya., The geometry of the non-holonomic sphere for the three-dimensional Lie group. In: Global Analysis-Studies and Applications III, (éd. Yu. G. Borisovich, Yu. E. Gilikikh), Lecture Notes in Math. 1334, Springer-Verlag, 1988.Google Scholar