Bulletin of the American Mathematical Society

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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.

William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. MR 307015, DOI 10.2307/1970868

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, DOI 10.1090/pspum/044

[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].

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 420406, DOI 10.1090/memo/0165

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, DOI 10.1090/S0273-0979-1983-15106-6

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, DOI 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).

Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 0485012

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, DOI 10.1090/S0894-0347-1988-0946554-0

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 260981, DOI 10.1090/S0002-9904-1970-12542-3

Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227

Wendell H. Fleming, Flat chains over a finite coefficient group, Trans. Amer. Math. Soc. 121 (1966), 160–186. MR 185084, DOI 10.1090/S0002-9947-1966-0185084-5

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.

Enrico Giusti, Minimal surfaces and functions of bounded variation, Notes on Pure Mathematics, vol. 10, Australian National University, Department of Pure Mathematics, Canberra, 1977. With notes by Graham H. Williams. MR 0638362

[MF]

F. Morgan, Geometric measure theory. A beginner's guide, Academic Press, New York, 1987.

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

E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186

E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. (2) 80 (1964), 1–14. MR 171197, DOI 10.2307/1970488

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 428181, DOI 10.2307/1970949

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

Brian White, Existence of least-area mappings of $N$-dimensional domains, Ann. of Math. (2) 118 (1983), no. 1, 179–185. MR 707165, DOI 10.2307/2006958

Brian White, Mappings that minimize area in their homotopy classes, J. Differential Geom. 20 (1984), no. 2, 433–446. MR 788287

William P. Ziemer, Integral currents $\textrm {mod}$ $2$, Trans. Amer. Math. Soc. 105 (1962), 496–524. MR 150267, DOI 10.1090/S0002-9947-1962-0150267-3

[AW]

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)