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Minimal varieties. (English) Zbl 0188.53801

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[1] William K. Allard, On boundary regularity for Plateau’s problem, Bull. Amer. Math. Soc. 75 (1969), 522 – 523. · Zbl 0183.11404
[2] F. J. Almgren Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277 – 292. · Zbl 0146.11905 · doi:10.2307/1970520 · doi.org
[3] F. J. Almgren, Jr., (b) Plateau’s problem, an invitation to varifold geometry, Benjamin, New York, 1966. · Zbl 0165.13201
[4] F. J. Almgren Jr., Measure theoretic geometry and elliptic variational problems, Bull. Amer. Math. Soc. 75 (1969), 285 – 304. · Zbl 0185.35202
[5] E. F. Beckenbach and G. A. Hutchison, Meromorphic minimal surfaces, Pacific J. Math. 28 (1969), 17 – 47. · Zbl 0179.11202
[6] E. Beltrami, Opere, vol. 2, Hoepli, Milan, 1904, pp. 1-54.
[7] Wilhelm Blaschke and Hans Reichardt, Einführung in die Differentialgeometrie, 2te Aufl. Die Grundlehren der mathematischen Wissenschaften, Bd. 58, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). · Zbl 0091.34001
[8] Enrico Bombieri, Nuovi risultati sulle ipersuperfici minimali non parametriche, Rend. Sem. Mat. Fis. Milano 38 (1968), 203 – 213 (Italian). · Zbl 0188.17601 · doi:10.1007/BF02924490 · doi.org
[9] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243 – 268. · Zbl 0183.25901 · doi:10.1007/BF01404309 · doi.org
[10] E. Bombieri, E. De Giorgi, and M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal. 32 (1969), 255 – 267 (Italian). · Zbl 0184.32803 · doi:10.1007/BF00281503 · doi.org
[11] Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1 – 23. · Zbl 0051.13103 · doi:10.2307/1969817 · doi.org
[12] Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111 – 125. · Zbl 0171.20504
[13] E. Calabi, (c) Quelques applications de l’analyse complexe aux surfaces d’aire minima (together with Topics in complex manifolds by H. Rossi), Les Presses de l’Univ. de Montréal, 1968.
[14] Shiing-shen Chern, Minimal surfaces in an Euclidean space of \? dimensions, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 187 – 198. · Zbl 0136.16701
[15] S. S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas, Department of Mathematics Technical Report 19 (New Series), Univ. of Kansas, Lawrence, Kan., 1968.
[16] S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59 – 75. · Zbl 0216.44001
[17] Shiing-shen Chern and Robert Osserman, Complete minimal surfaces in euclidean \?-space, J. Analyse Math. 19 (1967), 15 – 34. · Zbl 0172.22802 · doi:10.1007/BF02788707 · doi.org
[18] Ennio De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 79 – 85 (Italian). · Zbl 0168.09802
[19] Ennio De Giorgi and Guido Stampacchia, Sulle singolarità eliminabili delle ipersuperficie minimali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 38 (1965), 352 – 357 (Italian). · Zbl 0135.40003
[20] Manfredo P. do Carmo and Nolan R. Wallach, Representations of compact groups and minimal immersions into spheres., J. Differential Geometry 4 (1970), 91 – 104. · Zbl 0197.18301
[21] Manfredo P. do Carmo and Nolan R. Wallach, Minimal immersions of spheres into spheres, Ann. of Math. (2) 93 (1971), 43 – 62. · Zbl 0218.53069 · doi:10.2307/1970752 · doi.org
[22] Alexander Dinghas, Über Minimalabbildungen von Gebieten der komplexen Ebene in einen Hilbert-Raum, Jber. Deutsch. Math.-Verein. 67 (1964/1965), no. Abt. 1, 43 – 48 (German). · Zbl 0129.37302
[23] Alexander Dinghas, Über einen allgemeinen Verzerrungssatz für beschränkte Minimalflächen, Jber. Deutsch. Math.-Verein. 69 (1967), no. Heft 3, Abt. 1, 152 – 160 (German). · Zbl 0149.18101
[24] Peter Dombrowski, Krümmungsgrössen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten, Math. Nachr. 38 (1968), 133 – 180 (German). · Zbl 0172.23101 · doi:10.1002/mana.19680380302 · doi.org
[25] L. P. Eisenhart, Riemannian geometry, Princeton Univ. Press, Princeton, N. J., 1949. · Zbl 0041.29403
[26] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[27] Harley Flanders, Relations on minimal hypersurfaces, Pacific J. Math. 29 (1969), 83 – 93. · Zbl 0181.49404
[28] Wendell H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69 – 90. · Zbl 0107.31304 · doi:10.1007/BF02849427 · doi.org
[29] T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68 – 73. · Zbl 0189.22401 · doi:10.2307/1970471 · doi.org
[30] David Gilbarg, Boundary value problems for nonlinear elliptic equations in \? variables, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 151 – 159.
[31] Alfred Gray, Minimal varieties and almost Hermitian submanifolds, Michigan Math. J. 12 (1965), 273 – 287. · Zbl 0132.16702
[32] Stefan Hildebrandt, Über das Randverhalten von Minimalflächen, Math. Ann. 165 (1966), 1 – 18 (German). · Zbl 0138.42302 · doi:10.1007/BF01351660 · doi.org
[33] Wu-yi Hsiang, On the compact homogeneous minimal submanifolds, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 5 – 6. · Zbl 0178.55904
[34] Wu-yi Hsiang, Remarks on closed minimal submanifolds in the standard Riemannian \?-sphere, J. Differential Geometry 1 (1967), 257 – 267. · Zbl 0168.42904
[35] Howard Jenkins and James Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math. 229 (1968), 170 – 187. · Zbl 0159.40204 · doi:10.1515/crll.1968.229.170 · doi.org
[36] L. Jonker, A theorem of minimal surfaces, J. Differential Geometry 3 (1969), 351 – 360. · Zbl 0194.52504
[37] Shoshichi Kobayashi, Isometric imbeddings of compact symmetric spaces, Tôhoku Math. J. (2) 20 (1968), 21 – 25. · Zbl 0175.48301 · doi:10.2748/tmj/1178243214 · doi.org
[38] H. B. Lawson, Jr., (a) Minimal varieties in constant curvature manifolds, Ph.D. thesis, Stanford University, Stanford, Calif., 1968.
[39] H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187 – 197. · Zbl 0174.24901 · doi:10.2307/1970816 · doi.org
[40] H. Blaine Lawson Jr., Some intrinsic characterizations of minimal surfaces, J. Analyse Math. 24 (1971), 151 – 161. · Zbl 0251.53003 · doi:10.1007/BF02790373 · doi.org
[41] H. B. Lawson, Jr., (d) The global behavior of minimal surfaces in Ş (to appear).
[42] H. B. Lawson, Jr., (e) Compact minimal surfaces in S3 Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc. Providence, R. I. (to appear).
[43] H. B. Lawson, Jr., (f) Complete minimal surfaces in S3 (to appear).
[44] R. Lipschitz, Ausdehnung der Theorie der Miniamalflächen, J. Reine Angew. Math. 78 (1874), 1-45. · JFM 06.0519.01
[45] Mario Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in \? variabili, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 233 – 249 (Italian). · Zbl 0137.08201
[46] Mario Miranda, Sul minimo dell’integrale del gradiente di una funzione, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 626 – 665 (Italian). · Zbl 0166.09604
[47] M. Miranda, Comportamento delle successioni convergenti di frontiere minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 238 – 257 (Italian). · Zbl 0154.37102
[48] Mario Miranda, Diseguaglianze di Sobolev sulle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 69 – 79 (Italian). · Zbl 0175.11802
[49] Mario Miranda, Una maggiorazione integrale per le curvature delle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 91 – 107 (Italian). · Zbl 0175.11803
[50] M. Miranda, Sulle singolarità delle frontiere minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 180 – 188 (Italian). · Zbl 0162.43301
[51] Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0142.38701
[52] Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577 – 591. · Zbl 0111.09302 · doi:10.1002/cpa.3160140329 · doi.org
[53] Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195 – 270. · Zbl 0135.21701
[54] Robert Osserman, Global properties of minimal surfaces in \?³ and \?\(^{n}\), Ann. of Math. (2) 80 (1964), 340 – 364. · Zbl 0134.38502 · doi:10.2307/1970396 · doi.org
[55] Robert Osserman, Some properties of solutions to the minimal surface system for arbitrary codimension, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 283 – 291.
[56] R. Osserman, Minimal surfaces, Uspehi Mat. Nauk 22 (1967), no. 4 (136), 55 – 136 (Russian).
[57] Tominosuke Ôtsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature., Amer. J. Math. 92 (1970), 145 – 173. · Zbl 0196.25102 · doi:10.2307/2373502 · doi.org
[58] Max Pinl, Über einen Satz von G. Ricci-Curbastro und die Gausssche Krümmung der Minimalflächen, Arch. Math. (Basel) 4 (1953), 369 – 373 (German). · Zbl 0053.29402 · doi:10.1007/BF01899252 · doi.org
[59] R. C. Reilly, The Gauss map in the study of submanifolds of spheres, Ph.D. thesis, University of California, Berkeley, Calif., 1969.
[60] Gregorio Ricci-Curbastro, Opere. Vol. I. Note e memorie, Edizioni Cremonese, Roma, 1956 (Italian). A cura dell’Unione Matematica Italiana e col contributo del Consiglio Nazionale delle Recerche. · Zbl 0070.16902
[61] Leo Sario, Kiyoshi Noshiro, Kikuji Matsumoto, and Mitsuru Nakai, Value distribution theory, In collaboration with Tadashi Kuroda, Kikuji Matsumoto and Mitsuru Nakai, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.
[62] A. H. Schoen, Infinite regular warped polyhedra (IRWP) and infinite periodic minimal surfaces (IPMS), Notices Amer. Math. Soc. 15 (1968), 727; A fifth intersection-free infinite periodic minimal surface (IPMS) of cubic symmetry, Notices Amer. Math. Soc. 16 (1969), 519.
[63] James Serrin, Addendum to: ”A priori estimates for solutions of the minimal surface equation”, Arch. Rational Mech. Anal. 28 (1967/1968), 149 – 154. · Zbl 0157.18201 · doi:10.1007/BF00283862 · doi.org
[64] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413 – 496. · Zbl 0181.38003 · doi:10.1098/rsta.1969.0033 · doi.org
[65] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62 – 105. · Zbl 0181.49702 · doi:10.2307/1970556 · doi.org
[66] Guido Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963), 383 – 421. · Zbl 0138.36903 · doi:10.1002/cpa.3160160403 · doi.org
[67] Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380 – 385. · Zbl 0145.18601 · doi:10.2969/jmsj/01840380 · doi.org
[68] Masaru Takeuchi and Shoshichi Kobayashi, Minimal imbeddings of \?-spaces, J. Differential Geometry 2 (1968), 203 – 215. · Zbl 0165.24901
[69] Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349 – 371 (Italian). Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387 – 399 (Italian). Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139 – 163 (Italian).
[70] Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349 – 371 (Italian). Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387 – 399 (Italian). Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139 – 163 (Italian).
[71] Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349 – 371 (Italian). Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387 – 399 (Italian). Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139 – 163 (Italian).
[72] Erhard Heinz and Friedrich Tomi, Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen, Math. Z. 111 (1969), 372 – 386 (German). · Zbl 0172.38601 · doi:10.1007/BF01110748 · doi.org
[73] Robert Hermann, A uniqueness theorem for minimal submanifolds, J. Differential Geometry 2 (1968), 191 – 195. · Zbl 0175.19002
[74] Stefan Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969), 47 – 82. · Zbl 0183.39402 · doi:10.1007/BF00248494 · doi.org
[75] W.-Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity to appear). · Zbl 0219.53045
[76] David Kinderlehrer, Minimal surfaces whose boundaries contain spikes, J. Math. Mech. 19 (1969/1970), 829 – 853. · Zbl 0193.39301
[77] H. B. Lawson, Jr., Rigidity theorems in rank-1 symmetric spaces (to appear).
[78] Johannes C. C. Nitsche, Concerning the isolated character of solutions of Plateau’s problem., Math. Z. 109 (1969), 393 – 411. · Zbl 0189.10804 · doi:10.1007/BF01110559 · doi.org
[79] J. C. C. Nitsche, (b) The boundary behavior of minimal surfaces–Kellogg’s Theorem and branch points on the boundary, Invent. Math, (to appear). · Zbl 0195.23101
[80] Robert Osserman, The nonexistence of branch points in the classical solution of Plateau’s problem, Bull. Amer. Math. Soc. 75 (1969), 1247 – 1248. · Zbl 0183.39401
[81] R. Osserman, (b) A proof of the regularity everywhere of the classical solution to Plateau’s problem (to appear). · Zbl 0194.22302
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