Book Review
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Book Information:
Author:
Udo Hertrich-Jeromin
Title:
Introduction to Möbius differential geometry
Additional book information:
London Mathematical Society Lecture Notes Series, vol. 300, Cambridge University Press,
Cambridge, UK,
2003,
xi+413 pp.,
ISBN 0-521-53569-7,
US$50.00$
Maks A. Akivis and Vladislav V. Goldberg, Conformal differential geometry and its generalizations, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1996. A Wiley-Interscience Publication. MR 1406793, DOI 10.1002/9781118032633
2. W. Blaschke, Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln, Grundlehren XXIX, Springer, Berlin, 1929.
Robert L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), no. 1, 23–53. MR 772125
Thomas E. Cecil, Lie sphere geometry, Universitext, Springer-Verlag, New York, 1992. With applications to submanifolds. MR 1219311, DOI 10.1007/978-1-4757-4096-7
D. Ferus, K. Leschke, F. Pedit, and U. Pinkall, Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic $2$-tori, Invent. Math. 146 (2001), no. 3, 507–593. MR 1869849, DOI 10.1007/s002220100173
6. G. Fubini, Applicabilità projettiva di due superficie, Palermo Rend. 41 (1916), 135-162.
U. Hertrich-Jeromin and U. Pinkall, Ein Beweis der Willmoreschen Vermutung für Kanaltori, J. Reine Angew. Math. 430 (1992), 21–34 (German). MR 1172905, DOI 10.1515/crll.1992.430.21
Rob Kusner, Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (1989), no. 2, 317–345. MR 996204
Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407, DOI 10.1007/BF01399507
U. Pinkall, Dupin hypersurfaces, Math. Ann. 270 (1985), no. 3, 427–440. MR 774368, DOI 10.1007/BF01473438
11. T. Takasu, Differentialgeometrien in den Kugelräumen, Bd. I, Tagaido Publ. Co., Kyoto, and Hafner Publ. Co., New York, 1938.
12. G. Thomsen, Über konforme Geometrie I: Grundlagen der konformen Flächentheorie, Hamb. Math. Abh. 3 (1923), 31-56.
James H. White, A global invariant of conformal mappings in space, Proc. Amer. Math. Soc. 38 (1973), 162–164. MR 324603, DOI 10.1090/S0002-9939-1973-0324603-1
T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 202066
T. J. Willmore, Surfaces in conformal geometry, Ann. Global Anal. Geom. 18 (2000), no. 3-4, 255–264. Special issue in memory of Alfred Gray (1939–1998). MR 1795097, DOI 10.1023/A:1006717506186
- 1.
- M. A. Akivis and V. V. Goldberg, Conformal differential geometry and its generalizations, Wiley, New York, 1996. MR 1406793
- 2.
- W. Blaschke, Vorlesungen über Differentialgeometrie III: Differentialgeometrie der Kreise und Kugeln, Grundlehren XXIX, Springer, Berlin, 1929.
- 3.
- R. L. Bryant, A duality theorem for Willmore surfaces, J. Differential Geometry 20 (1984), 23-53. MR 0772125
- 4.
- T. E. Cecil, Lie sphere geometry, Springer, New York, 1992. MR 1219311
- 5.
- D. Ferus, K. Leschke, F. Pedit and U. Pinkall, Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates, and energy estimates of harmonic 2-tori, Invent. Math. 146 (2001), 507-593. MR 1869849
- 6.
- G. Fubini, Applicabilità projettiva di due superficie, Palermo Rend. 41 (1916), 135-162.
- 7.
- U. Hertrich-Jeromin and U. Pinkall, Ein Beweis der Willmoreschen Vermutung für Kanaltori, J. Reine Angew. Math. 430 (1992), 21-34. MR 1172905
- 8.
- R. Kusner, Comparison surfaces for the Willmore problem, Pac. J. Math. 138 (1989), 317-345. MR 0996204
- 9.
- P. Li and S. T. Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math 69 (1982), 269-291. MR 0674407
- 10.
- U. Pinkall, Dupin hypersurfaces, Math. Ann. 270 (1985), 427-440. MR 0774368
- 11.
- T. Takasu, Differentialgeometrien in den Kugelräumen, Bd. I, Tagaido Publ. Co., Kyoto, and Hafner Publ. Co., New York, 1938.
- 12.
- G. Thomsen, Über konforme Geometrie I: Grundlagen der konformen Flächentheorie, Hamb. Math. Abh. 3 (1923), 31-56.
- 13.
- J. H. White, A global invariant of conformal mappings in space, Proc. Amer. Math. Soc. 38 (1973), 162-164. MR 0324603
- 14.
- T. J. Willmore, Note on embedded surfaces, An. Sti. Univ. Al. I. Cuza Iasi, N. Ser., Sect. Ia Mat. 11B (1965), 493-496. MR 0202066
- 15.
- -, Surfaces in conformal geometry, Ann. Global Anal. Geom 18 (2000), 255-264. MR 1795097
Review Information:
Reviewer:
Thomas E. Cecil
Affiliation:
College of the Holy Cross
Email:
cecil@mathcs.holycross.edu
Journal:
Bull. Amer. Math. Soc.
42 (2005), 549-554
Published electronically:
July 1, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
