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Trace of heat kernel, spectral zeta function and isospectral problem for sub-laplacians

Abstract

In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ℂn+1. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.

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References

  1. Webster S M. Pseudo-hermitian structures on a real hypersurface. J Diff Geom, 13: 25–41 (1978)

    MATH  MathSciNet  Google Scholar 

  2. Folland G B, Stein E M. Estimates for the \( \bar \partial _b \) complex and analysis on the Heisenberg group. Comm Pure Appl Math, 27: 429–522 (1974)

    MATH  Article  MathSciNet  Google Scholar 

  3. Lee J M. The Fefferman matric and pasudohermitian invariants. Trans Amer Math Soc, 296: 411–429 (1986)

    MATH  Article  MathSciNet  Google Scholar 

  4. Stanton N K. Spectral invariants of CR manifolds. Michigan Math J, 36: 267–288 (1989)

    MATH  Article  MathSciNet  Google Scholar 

  5. Jerison D, Lee J M. The Yamabe problem on CR manifolds. J Diff Geom, 25: 167–197 (1987)

    MATH  MathSciNet  Google Scholar 

  6. Beals R, Greiner P C, Stanton N K. The heat equation on a CR manifold. J Diff Geom, 20: 343–387 (1984)

    MATH  MathSciNet  Google Scholar 

  7. Folland G B. The tangential Cauchy-Riemann complex on spheres. Trans Amer Math Soc, 171: 83–133 (1972)

    MATH  Article  MathSciNet  Google Scholar 

  8. Stanton N K, Tartakoff D S. The heat equation for the \( \bar \partial _b \)-Laplacian. Comm Partial Diff Eqs, 9: 597–686 (1984)

    MATH  Article  MathSciNet  Google Scholar 

  9. Beals M, Fefferman C L, Grossman R. Strictly pesudoconvex domains in ℂn+1. Proc Symposia Pure Math, 39: 189–386 (1983)

    Google Scholar 

  10. Andrews G, Askey R, Roy R. Special Functions. In: Encyclopedia of Mathematics and Its Applications, Vol. 71. Cambridge: Cambridge University Press, 1999

    Google Scholar 

  11. Beals R. Lecture Notes on Special Functions. Book in preparation, 2008

  12. Chang D C, Li S Y. A zeta function associated to sub-Laplacian on the unit sphere in C n. J d’Analyse Math, 86: 25–48 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  13. Sunada T. Riemannian coverings and isospectral manifolds. Ann of Math, 121: 169–186 (1985)

    Article  MathSciNet  Google Scholar 

  14. Patodi V K. Curvature and the eigenforms of the Laplace operator. J Diff Geom, 5: 233–249 (1971)

    MATH  MathSciNet  Google Scholar 

  15. Gilkey P B. Invariance Theory, the heat equation, and the Atiyah-Singer index theorem. In: Mathematics Lecture Series, Vol. 11. Wilmington: Publish or Perish, 1984

    Google Scholar 

  16. Chavel I. Eigenvalues in Riemannian Geometry. In: Pure and Applied Mathematics, Vol. 115. Florida: Academic Press, 1984

    Google Scholar 

  17. Calin O, Chang D C, Greiner P C. Geometric Analysis on the Heisenberg Group and its Generalizations. AMS/IP series in Advanced Mathematics, Vol. 40. Cambridge: International Press, 2007

    Google Scholar 

  18. Beals R. Lecture Notes on Special Functions. Yale University, New Haven, 2001

    Google Scholar 

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Authors

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Correspondence to Der-Chen Chang.

Additional information

Dedicated to Professor ZHONG TongDe on the occasion of his 80th birthday

The work was supported by National Security Agency, United States Army Research Office and a Hong Kong RGC competitive earmarked research (Grant No. 600607)

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Chang, DC., Yeung, SK. Trace of heat kernel, spectral zeta function and isospectral problem for sub-laplacians. Sci. China Ser. A-Math. 52, 2570 (2009). https://doi.org/10.1007/s11425-009-0186-4

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  • DOI: https://doi.org/10.1007/s11425-009-0186-4

Keywords

  • sub-Laplacian
  • heat kernel
  • CR-isospectral problem
  • Riemannian zeta function
  • Mellin transform
  • pseudo-hermitian structure

MSC(2000)

  • Primary: 53C17
  • Secondary: 34K10, 35H20