Book Review

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Book Information:

Author: 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.

**[AW]**William K. Allard,*On the first variation of a varifold*, Ann. of Math. (2)**95**(1972), 417–491. MR**0307015**, https://doi.org/10.2307/1970868**[AA]**William K. Allard and Frederick J. Almgren Jr. (eds.),*Geometric measure theory and the calculus of variations*, Proceedings of Symposia in Pure Mathematics, vol. 44, American Mathematical Society, Providence, RI, 1986. MR**840266****[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. J. Almgren Jr.,*Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints*, Mem. Amer. Math. Soc.**4**(1976), no. 165, viii+199. MR**0420406**, https://doi.org/10.1090/memo/0165**[A3]**F. J. Almgren Jr.,*𝑄 valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two*, Bull. Amer. Math. Soc. (N.S.)**8**(1983), no. 2, 327–328. MR**684900**, https://doi.org/10.1090/S0273-0979-1983-15106-6**[A4]**F. Almgren,*Deformations and multiple-valued functions*, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 29–130. MR**840268**, https://doi.org/10.1090/pspum/044/840268**[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]**Kenneth A. Brakke,*The motion of a surface by its mean curvature*, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR**485012****[CS]**Sheldon Xu-Dong Chang,*Two-dimensional area minimizing integral currents are classical minimal surfaces*, J. Amer. Math. Soc.**1**(1988), no. 4, 699–778. MR**946554**, https://doi.org/10.1090/S0894-0347-1988-0946554-0**[FH]**Herbert Federer,*The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension*, Bull. Amer. Math. Soc.**76**(1970), 767–771. MR**0260981**, https://doi.org/10.1090/S0002-9904-1970-12542-3**[FF]**Herbert Federer and Wendell H. Fleming,*Normal and integral currents*, Ann. of Math. (2)**72**(1960), 458–520. MR**0123260**, https://doi.org/10.2307/1970227**[FW]**Wendell H. Fleming,*Flat chains over a finite coefficient group*, Trans. Amer. Math. Soc.**121**(1966), 160–186. MR**0185084**, https://doi.org/10.1090/S0002-9947-1966-0185084-5**[F1]**A. T. Fomenko,*The Plateau problem. Part I*, Studies in the Development of Modern Mathematics, vol. 1, Gordon and Breach Science Publishers, New York, 1990. Historical survey; Translated from the Russian. MR**1055826****[F2]**-,*Mathematical impressions*, Amer. Math. Soc., Providence, RI, 1990.**[GE]**Enrico Giusti,*Minimal surfaces and functions of bounded variation*, Department of Pure Mathematics, Australian National University, Canberra, 1977. With notes by Graham H. Williams; Notes on Pure Mathematics, 10. MR**0638362****[MF]**F. Morgan,*Geometric measure theory. A beginner's guide*, Academic Press, New York, 1987.**[PJ]**Jon T. Pitts,*Existence and regularity of minimal surfaces on Riemannian manifolds*, Mathematical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR**626027****[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**, https://doi.org/10.1007/BF02547186**[R2]**E. R. Reifenberg,*An epiperimetric inequality related to the analyticity of minimal surfaces*, Ann. of Math. (2)**80**(1964), 1–14. MR**0171197**, https://doi.org/10.2307/1970488**[T1]**Jean E. Taylor,*The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces*, Ann. of Math. (2)**103**(1976), no. 3, 489–539. MR**0428181**, https://doi.org/10.2307/1970949**[T2]**Jean E. Taylor (ed.),*Computing optimal geometries*, Selected Lectures in Mathematics, American Mathematical Society, Providence, RI, 1991. Lectures presented at the AMS Special Session held in San Francisco, California, January 16–19, 1991. MR**1164472****[W1]**Brian White,*Existence of least-area mappings of 𝑁-dimensional domains*, Ann. of Math. (2)**118**(1983), no. 1, 179–185. MR**707165**, https://doi.org/10.2307/2006958**[W2]**Brian White,*Mappings that minimize area in their homotopy classes*, J. Differential Geom.**20**(1984), no. 2, 433–446. MR**788287****[ZW]**William P. Ziemer,*Integral currents 𝑚𝑜𝑑 2*, Trans. Amer. Math. Soc.**105**(1962), 496–524. MR**0150267**, https://doi.org/10.1090/S0002-9947-1962-0150267-3

Review Information:

Reviewer: Fred Almgren

Journal: Bull. Amer. Math. Soc.

**26**(1992), 188-192

DOI: https://doi.org/10.1090/S0273-0979-1992-00256-2