Book Review
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Book Information:
Authors:
Sorin Dragomir and
Giuseppe Tomassini
Title:
Differential geometry and analysis on CR manifolds
Additional book information:
Progress in Mathematics, vol. 246,
Birkhäuser, Basel,
2006,
xiv+487 pp.,
ISBN 978-0-8176-4388-1,
US$109.00$
Takao Akahori, A new approach to the local embedding theorem of CR-structures for $n\geq 4$ (the local solvability for the operator $\overline \partial _b$ in the abstract sense), Mem. Amer. Math. Soc. 67 (1987), no. 366, xvi+257. MR 888499, DOI 10.1090/memo/0366
Aldo Andreotti and C. Denson Hill, Complex characteristic coordinates and tangential Cauchy-Riemann equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 299–324. MR 460724
M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103, DOI 10.1515/9781400883967
Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
David Catlin, Sufficient conditions for the extension of CR structures, J. Geom. Anal. 4 (1994), no. 4, 467–538. MR 1305993, DOI 10.1007/BF02922141
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI 10.1007/BF02392146
John P. D’Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993. MR 1224231
Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR 350069, DOI 10.1007/BF01406845
Charles L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 2, 395–416. MR 407320, DOI 10.2307/1970945
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
Howard Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, vol. 32, American Mathematical Society, Providence, RI, 1990. MR 1067341, DOI 10.1090/surv/032
David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
John M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), no. 1, 411–429. MR 837820, DOI 10.1090/S0002-9947-1986-0837820-2
John M. Lee and Thomas H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880, DOI 10.1090/S0273-0979-1987-15514-5
Hans Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2) 64 (1956), 514–522. MR 81952, DOI 10.2307/1969599
Hans Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math. (2) 66 (1957), 155–158. MR 88629, DOI 10.2307/1970121
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 88770, DOI 10.2307/1970051
Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9, Kinokuniya Book Store Co., Ltd., Tokyo, 1975. MR 0399517
S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599
Sidney M. Webster, On the local solution of the tangential Cauchy-Riemann equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989), no. 3, 167–182 (English, with French summary). MR 995503
- 1.
- T. Akahori, A new approach to the local embedding theorem of CR-structures for
, Mem. Amer. Math. Soc. 67(1987). MR 888499 (88i:32027)
- 2.
- A. Andreotti and C. D. Hill, Complex characteristic coordinates and the tangential Cauchy-Riemann equations, Ann. Scuola Norm. Pisa 26(1972), 299-324. MR 0460724 (57:717)
- 3.
- M. S. Baouendi, P. F. Ebenfelt, and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton Univ. Press, Princeton, NJ, 1999. MR 1668103 (2000b:32066)
- 4.
- A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Math., CRC Press, Boca Raton, Fla., 1991. MR 1211412 (94e:32035)
- 5.
- D. Catlin, Sufficient conditions for the extension of CR structures, J. Geom. Anal. 4 (1994), 467-538. MR 1305993 (95j:32028)
- 6.
- S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133(1974), 219-271. MR 0425155 (54:13112)
- 7.
- J. P. D'Angelo, Several complex variables and the geometry of real hypersurfaces, Studies in Advanced Math., CRC Press, Boca Raton, Fla., 1993. MR 1224231 (94i:32022)
- 8.
- C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26(1974), 1-65. MR 0350069 (50:2562)
- 9.
- C. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Annals of Math. (2) 103(1976), 395-416; 104(1976), 393-394. MR 0407321 (53:11097b)
- 10.
- G. B. Folland and E. M. Stein, Estimates for the
-complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27(1974), 429-522. MR 0367477 (51:3719)
- 11.
- H. Jacobowitz, An introduction to CR structures, Mathematical Surveys and Monographs, No. 32, Amer. Math. Soc., Providence, RI, 1990. MR 1067341 (93h:32023)
- 12.
- D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Diff. Geom. 25(1987), 167-197. MR 880182 (88i:58162)
- 13.
- J. M. Lee, The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296(1986), 411-429. MR 837820 (87j:32063)
- 14.
- J. M. Lee and T. Parker, The Yamabe problem, Bull. Amer. Soc. 17(1987), 37-91. MR 888880 (88f:53001)
- 15.
- H. Lewy, On the local character of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Annals of Math. 64(1956), 514-522. MR 0081952 (18:473b)
- 16.
- H. Lewy, An example of a smooth linear partial differential equation without solution, Annals of Math. 66(1957), 155-158. MR 0088629 (19:551d)
- 17.
- A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Annals of Math. (2) 65(1957), 391-404. MR 0088770 (19:577a)
- 18.
- N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya Book Store Co., Ltd., Kyoto, 1975. MR 0399517 (53:3361)
- 19.
- S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Diff. Geom. 13(1978), 25-41. MR 520599 (80e:32015)
- 20.
- S. M. Webster, On the proof of Kuranishi's embedding theorem, Ann. Inst. Henri Poincaré 6(1989), no. 3, 183-207 (French summary). MR 0995504 (90h:32042b)
Review Information:
Reviewer:
John P. D’Angelo
Affiliation:
University of Illinois, Urbana
Email:
jpda@math.uiuc.edu
Journal:
Bull. Amer. Math. Soc.
45 (2008), 177-183
Published electronically:
May 1, 2007
Additional Notes:
The reviewer was partially supported by NSF Grant DMS 05-00765.
Review copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
