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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:

[AW]
W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) \textbf{95} (1972), 417--491.
[AA]
W. K. Allard and F. Almgren, eds., \emph{Geometric measure theory and minimal surfaces}, Amer. Math. Soc., Providence, RI, 1986.
[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]
F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. No. 165 (1976).
[A3]
F. Almgren, $\bold Q$ 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.) \textbf{8} (1983), 327--328.
[A4]
F. Almgren, \emph{Deformations and multiple-valued functions}, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986 pp.~29--130.
[A5]
F. Almgren, \emph{Questions and answers about area minimizing surfaces and geometric measure theory}.
[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.
[CS]
S. Chang, Two dimensional area minimizing currents are classical minimal surfaces, J. Amer. Math. Soc. \textbf{1} (1988), 699--778.
[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. \textbf{76} (1970), 767--771.
[FF]
H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) \textbf{72} (1960), 458--520.
[FW]
W. H. Fleming, Flat chains over a coefficient group, Trans. Amer. Math. Soc. \textbf{121} (1966), 160--186.
[F1]
A. T. Fomenko, The Plateau problem. {\rm Part I}. {\it Historical survey}. {\rm Part II}. {\it The present state of the theory}, Studies in the Development of Modern Mathematics, Gordon and Breach, New York, 1990.
[F2]
A. T. Fomenko, Mathematical impressions, Amer. Math. Soc., Providence, RI, 1990.
[GE]
E. Giusti, Minimal surfaces and functions of bounded variation, Monographs Math., vol. 80, Birkh\"auser, Boston-Basel-Stuttgart, 1984.
[MF]
F. Morgan, Geometric measure theory. A beginner's guide, Academic Press, New York, 1987.
[PJ]
J. T. Pitts, \emph{Existence and regularity of minimal surfaces on Riemannian manifolds}, Princeton Univ. Press, Princeton, NJ, 1981.
[R1]
E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. \textbf{104} (1960), 1--92.
[R2]
E. R. Reifenberg, A epiperimetric inequality related to the analyticity of minimal surfaces. On the analyticity of minimal surfaces, Ann. of Math. (2) \textbf{80} (1964), 1--21.
[T1]
J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) \textbf{103} (1976), 489--539.
[T2]
J. E. Taylor, ed., Computing optimal geometrices, Amer. Math. Soc., Providence, RI, 1991.
[W1]
B. White, Existence of least area mappings of N-dimensional domains, Ann. of Math. (2) \textbf{18} (1983), 179--185.
[W2]
B. White, Mappings that minimize area in their homotopy classes, J. Differential Geom. \textbf{20} (1984), 433--446.
[ZW]
W. P. Ziemer, Integral currents mod $2$, Trans. Amer. Math. Soc. \textbf{105} (1962), 496--524.

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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