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 0792302303.
 [AW]
William
K. Allard, On the first variation of a varifold, Ann. of Math.
(2) 95 (1972), 417–491. MR 0307015
(46 #6136)
 [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
(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]
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
(54 #8420)
 [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
(84b:49052), http://dx.doi.org/10.1090/S027309791983151066
 [A4]
F.
Almgren, Deformations and multiplevalued 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
(87h:49001), http://dx.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 (82c:49035)
 [CS]
Sheldon
XuDong Chang, Twodimensional area minimizing
integral currents are classical minimal surfaces, J. Amer. Math. Soc. 1 (1988), no. 4, 699–778. MR 946554
(89i:49028), http://dx.doi.org/10.1090/S08940347198809465540
 [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
(41 #5601), http://dx.doi.org/10.1090/S000299041970125423
 [FF]
Herbert
Federer and Wendell
H. Fleming, Normal and integral currents, Ann. of Math. (2)
72 (1960), 458–520. MR 0123260
(23 #A588)
 [FW]
Wendell
H. Fleming, Flat chains over a finite coefficient
group, Trans. Amer. Math. Soc. 121 (1966), 160–186. MR 0185084
(32 #2554), http://dx.doi.org/10.1090/S00029947196601850845
 [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 (92e:01003)
 [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
(58 #30685)
 [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
(83e:49079)
 [R1]
E.
R. Reifenberg, Solution of the Plateau Problem for
𝑚dimensional surfaces of varying topological type, Acta Math.
104 (1960), 1–92. MR 0114145
(22 #4972)
 [R2]
E.
R. Reifenberg, An epiperimetric inequality related to the
analyticity of minimal surfaces, Ann. of Math. (2) 80
(1964), 1–14. MR 0171197
(30 #1428)
 [T1]
Jean
E. Taylor, The structure of singularities in soapbubblelike and
soapfilmlike minimal surfaces, Ann. of Math. (2)
103 (1976), no. 3, 489–539. MR 0428181
(55 #1208a)
 [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
(93a:65021)
 [W1]
Brian
White, Existence of leastarea mappings of 𝑁dimensional
domains, Ann. of Math. (2) 118 (1983), no. 1,
179–185. MR
707165 (85e:49063), http://dx.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 (86f:49107)
 [ZW]
William
P. Ziemer, Integral currents
𝑚𝑜𝑑 2, Trans. Amer.
Math. Soc. 105
(1962), 496–524. MR 0150267
(27 #268), http://dx.doi.org/10.1090/S00029947196201502673
 [AW]
 W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417491. 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), 327328. MR 684900 (84b:49052)
 [A4]
 , Deformations and multiplevalued functions, Geometric Measure Theory and the Calculus of Variations, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 29130. 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), 699778. MR 946554 (89i:49028)
 [FH]
 H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flatchains modulo two with arbitrary codimensions, Bull. Amer. Math. Soc. 76 (1970), 767771. MR 0260981 (41:5601)
 [FF]
 H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458520. MR 0123260 (23:A588)
 [FW]
 W. H. Fleming, Flat chains over a coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160186. 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, BostonBaselStuttgart, 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 mdimensional surfaces of varying topological type, Acta Math. 104 (1960), 192. 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), 121. MR 0171197 (30:1428)
 [T1]
 J. E. Taylor, The structure of singularities in soapbubblelike and soapfilmlike minimal surfaces, Ann. of Math. (2) 103 (1976), 489539. 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 Ndimensional domains, Ann. of Math. (2) 18 (1983), 179185. MR 707165 (85e:49063)
 [W2]
 , Mappings that minimize area in their homotopy classes, J. Differential Geom. 20 (1984), 433446. MR 788287 (86f:49107)
 [ZW]
 W. P. Ziemer, Integral currents mod 2, Trans. Amer. Math. Soc. 105 (1962), 496524. MR 0150267 (27:268)
Review Information
Reviewer:
Fred Almgren
Journal:
Bull. Amer. Math. Soc. 26 (1992), 188192
DOI:
http://dx.doi.org/10.1090/S027309791992002562
PII:
S 02730979(1992)002562
