|
The heat equation in several complex variables
Author(s):
Nancy K.
Stanton
Journal:
Bull. Amer. Math. Soc.
11
(1984),
65-84.
MSC (1980):
Primary 32F20, 58G11
MathSciNet review:
741724
Retrieve article in:
PDF
References |
Similar articles |
Additional information
References:
- [ABP] M. Atiyah, R. Bott and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330; errata, ibid. 28 (1975), 277-280. MR 650828
- [BG] R. Beals and P. Greiner, Pseudodifferential operators associated to hyperplane bundles, Rend. Sem. Mat. Torino 41 (1983), 3-40. MR 745972
- [BGS] R. Beals, P. C. Greiner and N. K. Stanton, The heat equation on a CR manifold (preprint). MR 788285
- [B] M. Berger, Geometry of the spectrum. I, Proc. Sympos. Pure Math., vol. 27, Part 2, Amer. Math. Soc., Providence, R.I., 1975, pp. 129-152. MR 383459
- [Ca] T. Carleman, Propriétés asymptotiques des fonctions fondamentales des membranes vibrantes, Åttonde Skand. Matematikerkongressen (Stockholm, 1934), Ohlsson, Lund, 1935, pp. 34-44.
- [Ch 1] J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259-322. MR 528965
- [Ch 2] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103-2106. MR 530173
- [CL] S. Y. Cheng and P. Li, Heat kernel estimates and lower bound of eigenvalues, Comment. Math. Helv. 56 (1981), 327-338. MR 639355
- [Dod] J. Dodziuk, Eigenvalues of the Laplacian and the heat equation, Amer. Math. Monthly 88 (1981), 686-695. MR 643271
- [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.
- [F] J. Fay, Analytic torsion and Prym differentials, Riemann Surfaces and Related Topics (Proc. 1978 Stony Brook Conf.), Ann. of Math. Stud., no. 97, Princeton Univ. Press, Princeton, N. J., 1981, pp. 107-122. MR 624809
- [FK] G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy Riemann complex, Ann. of Math. Stud., no. 75, Princeton Univ. Press, Princeton, N. I, 1972. MR 461588
- [FS] G. B. Folland and E. M. Stein, Estimates for the $\bar \partial \sbb$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522. MR 367477
- [Ga] M. P. Gaffney, Asymptotic distributions associated with the Laplacian for forms, Comm. Pure Appl. Math. 11 (1958), 535-545. MR 99541
- [Gi 1] P. Gilkey, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. in Math. 10 (1973), 344-382. MR 324731
- [Gi 2] P. Gilkey, Spectral geometry and the Kaehler condition for complex manifolds, Invent. Math. 26 (1974), 231-258. MR 346849
- [Gi Sa] P. Gilkey and J. Sacks, Spectral geometry and manifolds of constant holomorphic sectional curvature, Proc. Sympos. Pure Math., vol. 27, Part 2, Amer. Math. Soc., Providence, R. I., 1975, pp. 285-289. MR 380902
- [GW] C. S. Gordon and E. N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984). MR 739790
- [Gr St] P. C. Greiner and E. M. Stein, Estimates for the $øverline \partial $-Neumann problem, Math. Notes, no. 19, Princeton Univ. Press, Princeton, N. J., 1977. MR 499319
- [Gr] P. Greiner, An asymptotic expansion for the heat equation, Arch. Rational Mech. Anal. 41 (1971), 163-218. MR 331441
- [HP] F. R. Harvey and J. Polking, The $\bar \partial $-Neumann kernel in the ball inCn, Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, R. I., 1984, pp. 117-136. MR 740876
- [I] A. Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup. 13 (1980), 303-315. MR 597742
- [Ka] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, Part II, 1-23. MR 201237
- [Ki] K. Kimura, Kernels for the $\bar \partial $-Neumann problem on the unit ball, Thesis, Univ. of Toronto, 1983.
- [Kr] S. Krantz, Function theory of several complex variables, Wiley, New York, 1982. MR 635928
- [LY] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983), 309-318. MR 701919
- [Ma] S. Mandelstam, Dual resonance models, Phys. Rep. 13 (1974), 259-353.
- [MS] H. P. McKean, Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 43-69. MR 217739
- [Me] G. Métivier, Spectral asymptotics of the $\bar\partial$-Neumann problem, Duke Math. J. 48 (1981), 779-806. MR 782577
- [MR] A. N. Milgram and P. Rosenbloom, Harmonic forms and heat conduction, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 180-184. MR 42769
- [Mil] J. Milnor, Eigenvalues of the Laplace operator of certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 162204
- [Min] S. Minakshisundaram, Eigenfunctions on Riemannian manifolds, J. Indian Math. Soc. 17 (1953), 158-165. MR 61750
- [MP] S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242-256. MR 31145
- [Mu] W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. in Math. 28 (1978), 233-305. MR 498252
- [Pa 1]V. K. Patodi, Curvature and the fundamental solution of the heat operator, J. Indian Math. Soc. 34 (1970), 269-285. MR 488181
- [Pa 2]V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geom. 5 (1971), 233-249. MR 292114
- [Pa 3]V. K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geom. 5 (1971), 251-273. MR 290318
- [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.
- [Ran] M. Range, The $\bar\partial$-Neumann operator on the unit ball inC, Math. Ann. (to appear). MR 735527
- [Ray] D. B. Ray, Reidemeister torsion and the Laplacian on lens spaces, Adv. in Math. 4 (1970), 109-126. MR 258062
- [RS 1] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210. MR 295381
- [RS 2] D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154-177. MR 383463
- [Sch] A. S. Schwartz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1977/78), 247-252. MR 676337
- [Se 1] R. Seeley, Complex powers of an elliptic operator, Singular Integrals, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 288-307. MR 237943
- [Se 2] R. Seeley, Analytic extension of the trace associated with elliptic boundary value problems, Amer. J. Math. 91 (1969), 963-983. MR 265968
- [Si] I. M. Singer, Eigenvalues of the Laplacian and invariants of manifolds, Proc. Internat. Congress Math. I (Vancouver, 1974), pp. 187-200. MR 420746
- [Stan] M. Stanton, Intrinsic connections for Levi metrics (in preparation).
- [St 1]N. K. Stanton, Holomorphic R-torsion for fiber bundles, Amer. J. Math. 100 (1978), 547-578. MR 501224
- [St 2]N. K. Stanton, The fundamental solution of the heat equation associated with the $\bar\partial$-Neumann problem, J. Analyse Math. 34 (1978), 265-274._ MR 531278
- [St 3]N. K. Stanton, The heat equation for the $\bar\partial$-Neumann problem in a strictly pseudoconvex Siegel domain. I, II, J. Analyse Math. 38 (1980), 67-112; 39 (1981), 189-202. MR 600783
- [St 4]N. K. Stanton, The solution of the $\bar\partial$-Neumann problem in a strictly pseudoconvex Siegel domain, Invent. Math. 65 (1981), 137-174. MR 636884
- [St 5]N. K. Stanton, L’équation de la chaleur en plusieurs variables complexes, Séminaire Goulaouic-Meyer-Schwartz 1982-83, exposé VI.
- [ST] N. K. Stanton and D. S. Tartakoff, The heat equation for the $\bar\partial$, Comm. Partial Differential Equations (to appear). MR 745020
- [T] M. Taylor, Noncommutative microlocal analysis. I, Mem. Amer. Math. Soc. (to appear). MR 764508
- [U] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. École Norm. Sup. 15 (1982), 441-456. MR 690649
- [V] M. F. Vigneras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2)112 (1980), 21-32. MR 584073
- [Web] S. M. Webster, Pseudo-hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. MR 520599
- [Wey] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), 441-479. MR 1511670
Similar Articles:
Retrieve articles in Bulletin of the American Mathematical Society
with MSC
(1980):
32F20, 58G11
Retrieve articles in all Journals with MSC
(1980):
32F20, 58G11
Additional Information:
DOI:
10.1090/S0273-0979-1984-15238-8
PII:
S 0273-0979(1984)15238-8
|