Abstract
Let G be a generalized gradient on a compact, Riemannian n-manifold M, carrying an 0(n)-irreducible tensor bundle F to another such bundle E, and suppose that G*G is elliptic but not conformally covariant. Let Tr0 denote the trace of the compression to the Hodge sector R(G) in the decomposition L2(E) = R(G) ⊕N(G*). Then if ω∈C∞(M), Tt0 ω exp(-t G G*) has a small-time asymptotic expansion in powers of t, plus a log t term in the case of even n. All coefficients except that of t0 are integrals of local expressions. For G = d: C∞(M) → C∞(Λ1(M)) and n = 4, the coefficient of log t is nonzero for some ω whenever the scalar curvature is not constant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
AHLFORS, L.: ‘Conditions for quasiconformal deformation in several variables’, in: Contributions to Analysis. A Collection of Erpers Dedicated to L. Bers, Academic Press, New York 1974, 19–25.
BEALS, M., C. FEFFERMAN, and R. GROSSMAN: ‘Strictly pseudoconvex domains in Cn’, Bull. Amer. Math. Soc. 8 (1983), 125–322.
BEALS, M., P. GREINER, and N. STANTON: ‘The heat equation on a CR manifold’, J. Diff. Geom. 20 (1984), 343–387.
BERGER, M., P. GAUDUCHON, and E. MAZET: Le Spectre d’une variete Riemannienne, Springer-Verlag, Berlin 1971.
BRANSON, T.: ‘Conformally covariant equations on differential forms’, Comm. Partial Differential Equations 7 (1982), 393–431.
BRANSON, T.: ‘Differential operators canonically associated to a conformal structure’, Math. Scand. 57 (1985), 293–345.
BRANSON, T.: ‘Geometry of the Ahlfors operator’, preprint, University of Iowa, 1987.
BRANSON, T.: ‘Group representations arising from Lorentz conformal geometry’, J. Funct. Anal., to appear.
BRANSON, T. and B. ØRSTED: ‘Conformal indices of Riemannian manifolds’, Compositio Math. 60 (1986), 261–293.
BRANSON, T. and B. ØRSTED: ‘Conformal deformation and the heat operator’, Indiana U. Math. J., to appear.
FEFFERMAN, C. and A. SANCHEZ-CALLE: ‘Fundamental solutions for second order subelliptic operators’, Ann. Math. 124 (1986), 247–272.
FEGAN, H.: ‘Conformally invariant first order differential operators’, Quart. J. Math. Oxford 27 (1976), 371–378.
GILKEY, P.: Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish, Wilmington, Delaware, 1984.
HADAMARD, J.: Le probleme de Cauchy et les equations aux derivees partielles lineaires hyperboliques, Hermann et Cie, Paris 1932.
HÖRMANDER, L.: ‘Hypoelliptic second order differential equations’, Acta Math. 119 (1967), 147–171.
JERISON, D. and A. SANCHEZ-CALLE: ‘Estimates for the heat kernel for a sum of squares of vector fields’, Indiana U. Math. J. 35 (1986), 835–854.
MELROSE, R.: ‘The trace of the wave group’, Cont. Math. 27 (1984), 127–169.
MINAKSHISUNDARAM, S. and Å. PLEUEL: ‘Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds’, Canad. J. Math. 1 (1949), 242–256.
ØRSTED, B. and A. PIERZCHALSKI: ‘The Ahlfors Laplacian on a Riemannian manifold’, preprint, Sonderforschungsbereich Göttingen 1987.
PIERZCHALSKI, A.: ‘On quasiconformal deformations of manifolds and hypersurfaces’, Ber. Univ. Jyväskylä Math. Inst. 28 (1984), 79–94.
RAY, D. and I. SINGER: ‘R-torsion and the Laplacian on Riemannian manifolds’, Advances in Math. 7 (1971), 145–210.
SANCHEZ-CALLE, A.: ‘Fundamental solutions and geometry of sums of squares of vector fields’, Invent. Math. 78 (1984), 143–160.
STANTON, N.: ‘The heat equation for the Mathb-Laplacian’, Comm. Partial Differential Equations 90 (1984), 597–686.
STEIN, E. and G. WEISS: ‘Generalization of the Cauchy-Riemann equations and representations of the rotation group’, Amer. J. Math. 90 (1968), 163–196.
STRICHARTZ, R.: ‘Sub-Riemannian geometry’, J. Biff. Geom. 24 (1986), 221–263.
XU, C.: ‘On the asymptotic expansion of the trace of the heat kernel for a subelliptic operator’, Ph. D. Thesis, M.I.T., 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Kluwer Academic Publishers
About this chapter
Cite this chapter
Branson, T.P., Ørsted, B. (1989). Generalized Gradients and Asymptotics of the Functional Trace. In: Ławrynowicz, J. (eds) Deformations of Mathematical Structures. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2643-1_23
Download citation
DOI: https://doi.org/10.1007/978-94-009-2643-1_23
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7693-7
Online ISBN: 978-94-009-2643-1
eBook Packages: Springer Book Archive