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Book Review
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Book Information
Author(s):
Anatoli\u \i T. Fomenko
Title:
Variational principles of topology. Multidimensional minimal surface theory
Additional book information:
Kluwer Academic Publishers, Dordrecht, Boston, and London, 1990, 374 pp., US$133.00. ISBN 0-7923-0230-3
References:
-
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- W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491. MR 0307015 (46:6136)
- [AA]
- W. K. Allard and F. Almgren, eds., Geometric measure theory and minimal surfaces, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986. MR 840266 (87b:00012)
- [A1]
- F. Almgren, The theory of varifolds. A variational calculus in the large for the k dimensional area integrand, multilithed notes (no longer available), 1965; see [AW].
- [A2]
- -, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. No. 165 (1976). MR 0420406 (54:8420)
- [A3]
- -, Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two, preprint, 1984. See Bull. Amer. Math. Soc. (N.S) 8 (1983), 327-328. MR 684900 (84b:49052)
- [A4]
- -, Deformations and multiple-valued functions, Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 29-130. MR 840268 (87h:49001)
- [A5]
- -, Questions and answers about area minimizing surfaces and geometric measure theory, Proc. 1990 AMS Summer Research Institute on Differential Geometry.
- [AB]
- F. Almgren and W. Browder, On smooth approximation of integral cycles (in preparation).
- [BK]
- K. A. Brakke, The motion of a surface by its mean curvature, Math. Notes, no. 20, Princeton Univ. Press, Princeton, NJ, 1978. MR 485012 (82c:49035)
- [CS]
- S. Chang, Two dimensional area minimizing currents are classical minimal surfaces, J. Amer. Math. Soc. 1 (1988), 699-778. MR 946554 (89i:49028)
- [FH]
- H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat-chains modulo two with arbitrary codimensions, Bull. Amer. Math. Soc. 76 (1970), 767-771. MR 0260981 (41:5601)
- [FF]
- H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520. MR 0123260 (23:A588)
- [FW]
- W. H. Fleming, Flat chains over a coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160-186. MR 0185084 (32:2554)
- [F1]
- A. T. Fomenko, The Plateau problem. Part I. Historical survey. Part II. The present state of the theory, Studies in the Development of Modern Mathematics, Gordon and Breach, New York, 1990. MR 1055826 (92e:01003)
- [F2]
- -, Mathematical impressions, Amer. Math. Soc., Providence, RI, 1990.
- [GE]
- E. Giusti, Minimal surfaces and functions of bounded variation, Monographs Math., vol. 80, Birkhäuser, Boston-Basel-Stuttgart, 1984. MR 0638362 (58:30685)
- [MF]
- F. Morgan, Geometric measure theory. A beginner's guide, Academic Press, New York, 1987.
- [PJ]
- J. T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds, Math. Notes., no. 27, Princeton Univ. Press, Princeton, NJ, 1981. MR 626027 (83e:49079)
- [R1]
- E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1-92. MR 0114145 (22:4972)
- [R2]
- -, A epiperimetric inequality related to the analyticity of minimal surfaces. On the analyticity of minimal surfaces, Ann. of Math. (2) 80 (1964), 1-21. MR 0171197 (30:1428)
- [T1]
- J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), 489-539. MR 0428181 (55:1208a)
- [T2]
- J. E. Taylor, ed., Computing optimal geometrices, Amer. Math. Soc., Providence, RI, 1991. MR 1164472 (93a:65021)
- [W1]
- B. White, Existence of least area mappings of N-dimensional domains, Ann. of Math. (2) 18 (1983), 179-185. MR 707165 (85e:49063)
- [W2]
- -, Mappings that minimize area in their homotopy classes, J. Differential Geom. 20 (1984), 433-446. MR 788287 (86f:49107)
- [ZW]
- W. P. Ziemer, Integral currents mod 2, Trans. Amer. Math. Soc. 105 (1962), 496-524. MR 0150267 (27:268)
Additional Information:
Reviewer(s):
Fred
Almgren
Review Information:
Journal:
Bull. Amer. Math. Soc.
26
(1992),
188-192.
DOI:
10.1090/S0273-0979-1992-00256-2
PII:
S 0273-0979(1992)00256-2
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