The heat equation in several complex variables
HTML articles powered by AMS MathViewer
- by Nancy K. Stanton PDF
- Bull. Amer. Math. Soc. 11 (1984), 65-84
References
- M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330. MR 650828, DOI 10.1007/BF01425417
- Richard Beals and Peter C. Greiner, Pseudodifferential operators associated to hyperplane bundles, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1983), 7–40 (1984). Conference on linear partial and pseudodifferential operators (Torino, 1982). MR 745972
- Richard Beals, Peter C. Greiner, and Nancy K. Stanton, The heat equation on a CR manifold, J. Differential Geom. 20 (1984), no. 2, 343–387. MR 788285
- M. Berger, Geometry of the spectrum. I, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 129–152. MR 0383459 [Ca] T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes, Åttonde Skand. Matematikerkongressen (Stockholm, 1934), Ohlsson, Lund, 1935, pp. 34-44.
- Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
- Jeff Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 530173, DOI 10.1073/pnas.76.5.2103
- Shiu Yuen Cheng and Peter Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv. 56 (1981), no. 3, 327–338. MR 639355, DOI 10.1007/BF02566216
- Jozef Dodziuk, Eigenvalues of the Laplacian and the heat equation, Amer. Math. Monthly 88 (1981), no. 9, 686–695. MR 643271, DOI 10.2307/2320674 [Don] H. Donnelly, Minakshisundaram ’s coefficients on Kaehler manifolds, Proc. Sympos. Pure Math., vol. 27, Part 2, Amer. Math. Soc., Providence, RI, 1975, pp. 195-203.
- John Fay, Analytic torsion and Prym differentials, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 107–122. MR 624809
- G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
- G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
- Matthew P. Gaffney, Asymptotic distributions associated with the Laplacian for forms, Comm. Pure Appl. Math. 11 (1958), 535–545. MR 99541, DOI 10.1002/cpa.3160110405
- Peter B. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Advances in Math. 10 (1973), 344–382. MR 324731, DOI 10.1016/0001-8708(73)90119-9
- Peter B. Gilkey, Spectral geometry and the Kaehler condition for complex manifolds, Invent. Math. 26 (1974), 231–258. MR 346849, DOI 10.1007/BF01418951
- Peter B. Gilkey and Jonathan Sacks, Spectral geometry and manifolds of constant holomorphic sectional curvature, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 281–285. MR 0380902
- Carolyn S. Gordon and Edward N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241–256. MR 739790
- P. C. Greiner and E. M. Stein, Estimates for the $\overline \partial$-Neumann problem, Mathematical Notes, No. 19, Princeton University Press, Princeton, N.J., 1977. MR 0499319
- Peter Greiner, An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163–218. MR 331441, DOI 10.1007/BF00276190
- F. R. Harvey and J. C. Polking, The $\bar \partial$-Neumann kernel in the ball in $\textbf {C}^{n}$, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 117–136. MR 740876, DOI 10.1090/pspum/041/740876
- Akira Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 3, 303–315. MR 597742, DOI 10.24033/asens.1384
- Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI 10.2307/2313748 [Ki] K. Kimura, Kernels for the $\bar \partial$-Neumann problem on the unit ball, Thesis, Univ. of Toronto, 1983.
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Peter Li and Shing Tung Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), no. 3, 309–318. MR 701919, DOI 10.1007/BF01213210 [Ma] S. Mandelstam, Dual resonance models, Phys. Rep. 13 (1974), 259-353.
- H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
- Guy Métivier, Spectral asymptotics for the $\bar \partial$-Neumann problem, Duke Math. J. 48 (1981), no. 4, 779–806. MR 782577, DOI 10.1215/S0012-7094-81-04843-2
- A. N. Milgram and P. C. Rosenbloom, Harmonic forms and heat conduction. I. Closed Riemannian manifolds, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 180–184. MR 42769, DOI 10.1073/pnas.37.3.180
- J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 162204, DOI 10.1073/pnas.51.4.542
- S. Minakshisundaram, Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc. (N.S.) 17 (1953), 159–165 (1954). MR 61750
- S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242–256. MR 31145, DOI 10.4153/cjm-1949-021-5
- Jeff Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), no. 2, 259–322. MR 528965, DOI 10.2307/1971113
- V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970), no. 3-4, 269–285 (1971). MR 0488181
- V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233–249. MR 292114, DOI 10.4310/jdg/1214429791
- V. K. Patodi, An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geometry 5 (1971), 251–283. MR 290318, DOI 10.4310/jdg/1214429991 [PI 1] A. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1952), 553-569. [PI 2] A. Pleijel, On Green’s functions and the eigenvalue distribution of the three dimensional membrance equation, Skand. Matematikerkongressen Lund, vol. 12, 1953, pp. 222-240. [Q] D. Quillen, Informal lecture, Arbeitstagung, Bonn, 1983.
- R. Michael Range, The $\bar \partial$-Neumann operator on the unit ball in $\textbf {C}^{n}$, Math. Ann. 266 (1984), no. 4, 449–456. MR 735527, DOI 10.1007/BF01458539
- D. B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Advances in Math. 4 (1970), 109–126 (1970). MR 258062, DOI 10.1016/0001-8708(70)90018-6
- D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
- D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI 10.2307/1970909
- A. S. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1977/78), no. 3, 247–252. MR 676337, DOI 10.1007/BF00406412
- R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
- R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math. 91 (1969), 963–983. MR 265968, DOI 10.2307/2373312
- I. M. Singer, Eigenvalues of the Laplacian and invariants of manifolds, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 187–200. MR 0420746 [Stan] M. Stanton, Intrinsic connections for Levi metrics (in preparation).
- Nancy K. Stanton, Holomorphic $R$-torsion for fiber bundles, Amer. J. Math. 100 (1978), no. 3, 547–578. MR 501224, DOI 10.2307/2373838
- Nancy K. Stanton, The fundamental solution of the heat equation associated with the $\bar \partial$-Neumann problem, J. Analyse Math. 34 (1978), 265–274 (1979). MR 531278, DOI 10.1007/BF02790015
- Nancy K. Stanton, The heat equation for the $\bar \partial$-Neumann problem in a strictly pseudoconvex Siegel domain, J. Analyse Math. 38 (1980), 67–112. MR 600783, DOI 10.1007/BF02803335
- Nancy K. Stanton, The solution of the $\bar \partial$-Neumann problem in a strictly pseudoconvex Siegel domain, Invent. Math. 65 (1981/82), no. 1, 137–174. MR 636884, DOI 10.1007/BF01389299 [St 5]N. K. Stanton, L’équation de la chaleur en plusieurs variables complexes, Séminaire Goulaouic-Meyer-Schwartz 1982-83, exposé VI.
- Nancy K. Stanton and David S. Tartakoff, The heat equation for the $\bar \partial _{b}$-Laplacian, Comm. Partial Differential Equations 9 (1984), no. 7, 597–686. MR 745020, DOI 10.1080/03605308408820343
- Michael E. Taylor, Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc. 52 (1984), no. 313, iv+182. MR 764508, DOI 10.1090/memo/0313
- Hajime Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 3, 441–456. MR 690649, DOI 10.24033/asens.1433
- Marie-France Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21–32 (French). MR 584073, DOI 10.2307/1971319
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599, DOI 10.4310/jdg/1214434345
- Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479 (German). MR 1511670, DOI 10.1007/BF01456804
Additional Information
- Journal: Bull. Amer. Math. Soc. 11 (1984), 65-84
- MSC (1980): Primary 32F20, 58G11
- DOI: https://doi.org/10.1090/S0273-0979-1984-15238-8
- MathSciNet review: 741724