Minimal varieties

Osserman, Robert

doi:10.1090/S0002-9904-1969-12357-8

Minimal varieties

Author: Robert Osserman
Journal: Bull. Amer. Math. Soc. 75 (1969), 1092-1120
MSC (1959): Primary 5304, 5374, 3547; Secondary 5301, 2880, 3061
DOI: https://doi.org/10.1090/S0002-9904-1969-12357-8
MathSciNet review: 0276875
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1. William K. Allard, On boundary regularity for Plateau’s problem, Bull. Amer. Math. Soc. 75 (1969), 522–523. MR 0241802, https://doi.org/10.1090/S0002-9904-1969-12229-9
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. MR 0200816, https://doi.org/10.2307/1970520
2. F. J. Almgren, Jr., (b) Plateau's problem, an invitation to varifold geometry, Benjamin, New York, 1966.
2. F. J. Almgren Jr., Measure theoretic geometry and elliptic variational problems, Bull. Amer. Math. Soc. 75 (1969), 285–304. MR 0240689, https://doi.org/10.1090/S0002-9904-1969-12145-2
3. E. F. Beckenbach and G. A. Hutchison, Meromorphic minimal surfaces, Pacific J. Math. 28 (1969), 17–47. MR 0238197
4. E. Beltrami, Opere, vol. 2, Hoepli, Milan, 1904, pp. 1-54.
5. 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). MR 0116267
6. Enrico Bombieri, Nuovi risultati sulle ipersuperfici minimali non parametriche, Rend. Sem. Mat. Fis. Milano 38 (1968), 203–213 (Italian). MR 0240730, https://doi.org/10.1007/BF02924490
7. E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. MR 0250205, https://doi.org/10.1007/BF01404309
8. 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). MR 0248647, https://doi.org/10.1007/BF00281503
9. Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23. MR 0057000, https://doi.org/10.2307/1969817
9. Eugenio Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111–125. MR 0233294
9. 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.
10. Shiing-shen Chern, Minimal surfaces in an Euclidean space of 𝑁 dimensions, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 187–198. MR 0180926
10. 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. MR 0248648
11. 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. MR 0273546
12. Shiing-shen Chern and Robert Osserman, Complete minimal surfaces in euclidean 𝑛-space, J. Analyse Math. 19 (1967), 15–34. MR 0226514, https://doi.org/10.1007/BF02788707
13. Ennio De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 79–85 (Italian). MR 0178385
14. 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). MR 0187158
15. Manfredo P. do Carmo and Nolan R. Wallach, Representations of compact groups and minimal immersions into spheres., J. Differential Geometry 4 (1970), 91–104. MR 0266104
15. Manfredo P. do Carmo and Nolan R. Wallach, Minimal immersions of spheres into spheres, Ann. of Math. (2) 93 (1971), 43–62. MR 0278318, https://doi.org/10.2307/1970752
16. 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). MR 0177319
16. Alexander Dinghas, Über einen allgemeinen Verzerrungssatz für beschränkte Minimalflächen, Jber. Deutsch. Math.-Verein. 69 (1967), no. Heft 3, Abt. 1, 152–160 (German). MR 0212695
17. Peter Dombrowski, Krümmungsgrössen gleichungsdefinierter Untermannigfaltigkeiten Riemannscher Mannigfaltigkeiten, Math. Nachr. 38 (1968), 133–180 (German). MR 0246232, https://doi.org/10.1002/mana.19680380302
18. L. P. Eisenhart, Riemannian geometry, Princeton Univ. Press, Princeton, N. J., 1949.
19. Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
20. Harley Flanders, Relations on minimal hypersurfaces, Pacific J. Math. 29 (1969), 83–93. MR 0244875
21. Wendell H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. MR 0157263, https://doi.org/10.1007/BF02849427
22. T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68–73. MR 0187183, https://doi.org/10.2307/1970471
23. David Gilbarg, Boundary value problems for nonlinear elliptic equations in 𝑛 variables, Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962) Univ. of Wisconsin Press, Madison, Wis., 1963, pp. 151–159. MR 0146506
24. Alfred Gray, Minimal varieties and almost Hermitian submanifolds, Michigan Math. J. 12 (1965), 273–287. MR 0184185
25. Stefan Hildebrandt, Über das Randverhalten von Minimalflächen, Math. Ann. 165 (1966), 1–18 (German). MR 0206837, https://doi.org/10.1007/BF01351660
26. Wu-yi Hsiang, On the compact homogeneous minimal submanifolds, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 5–6. MR 0205203
26. Wu-yi Hsiang, Remarks on closed minimal submanifolds in the standard Riemannian 𝑚-sphere, J. Differential Geometry 1 (1967), 257–267. MR 0225244
27. Howard Jenkins and James Serrin, The Dirichlet problem for the minimal surface equation in higher dimensions, J. Reine Angew. Math. 229 (1968), 170–187. MR 0222467, https://doi.org/10.1515/crll.1968.229.170
28. L. Jonker, A theorem of minimal surfaces, J. Differential Geometry 3 (1969), 351–360. MR 0267470
29. Shoshichi Kobayashi, Isometric imbeddings of compact symmetric spaces, Tôhoku Math. J. (2) 20 (1968), 21–25. MR 0227895, https://doi.org/10.2748/tmj/1178243214
30. H. B. Lawson, Jr., (a) Minimal varieties in constant curvature manifolds, Ph.D. thesis, Stanford University, Stanford, Calif., 1968.
30. H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 0238229, https://doi.org/10.2307/1970816
30. H. Blaine Lawson Jr., Some intrinsic characterizations of minimal surfaces, J. Analyse Math. 24 (1971), 151–161. MR 0284922, https://doi.org/10.1007/BF02790373
30. H. B. Lawson, Jr., (d) The global behavior of minimal surfaces in Ş (to appear).
30. H. B. Lawson, Jr., (e) Compact minimal surfaces in S³ Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc. Providence, R. I. (to appear).
30. H. B. Lawson, Jr., (f) Complete minimal surfaces in S³ (to appear).
31. R. Lipschitz, Ausdehnung der Theorie der Miniamalflächen, J. Reine Angew. Math. 78 (1874), 1-45.
32. Mario Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in 𝑛 variabili, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 233–249 (Italian). MR 0181918
32. Mario Miranda, Sul minimo dell’integrale del gradiente di una funzione, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 626–665 (Italian). MR 0188839
32. M. Miranda, Comportamento delle successioni convergenti di frontiere minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 238–257 (Italian). MR 0222735
32. Mario Miranda, Diseguaglianze di Sobolev sulle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 69–79 (Italian). MR 0221350
32. Mario Miranda, Una maggiorazione integrale per le curvature delle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 91–107 (Italian). MR 0222778
32. M. Miranda, Sulle singolarità delle frontiere minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 180–188 (Italian). MR 0221351
33. 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. MR 0202511
34. Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 0159138, https://doi.org/10.1002/cpa.3160140329
35. Johannes C. C. Nitsche, On new results in the theory of minimal surfaces, Bull. Amer. Math. Soc. 71 (1965), 195–270. MR 0173993, https://doi.org/10.1090/S0002-9904-1965-11276-9
36. Robert Osserman, Global properties of minimal surfaces in 𝐸³ and 𝐸ⁿ, Ann. of Math. (2) 80 (1964), 340–364. MR 0179701, https://doi.org/10.2307/1970396
36. 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. MR 0262946
36. R. Osserman, Minimal surfaces, Uspehi Mat. Nauk 22 (1967), no. 4 (136), 55–136 (Russian). MR 0238198
37. Tominosuke Ôtsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature., Amer. J. Math. 92 (1970), 145–173. MR 0264565, https://doi.org/10.2307/2373502
38. Max Pinl, Über einen Satz von G. Ricci-Curbastro und die Gausssche Krümmung der Minimalflächen, Arch. Math. (Basel) 4 (1953), 369–373 (German). MR 0059585, https://doi.org/10.1007/BF01899252
39. R. C. Reilly, The Gauss map in the study of submanifolds of spheres, Ph.D. thesis, University of California, Berkeley, Calif., 1969.
40. 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. MR 0076694
41. 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. MR 0215998
42. 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.
43. James Serrin, Addendum to: “A priori estimates for solutions of the minimal surface equation”, Arch. Rational Mech. Anal. 28 (1967/1968), 149–154. MR 0219867, https://doi.org/10.1007/BF00283862
43. 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. MR 0282058, https://doi.org/10.1098/rsta.1969.0033
44. James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 0233295, https://doi.org/10.2307/1970556
45. Guido Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Pure Appl. Math. 16 (1963), 383–421. MR 0155209, https://doi.org/10.1002/cpa.3160160403
46. Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 0198393, https://doi.org/10.2969/jmsj/01840380
47. Masaru Takeuchi and Shoshichi Kobayashi, Minimal imbeddings of 𝑅-spaces, J. Differential Geometry 2 (1968), 203–215. MR 0239007
48. Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349–371 (Italian). MR 0169102
Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387–399 (Italian). MR 0169103
Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139–163 (Italian). MR 0169104
48. Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349–371 (Italian). MR 0169102
Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387–399 (Italian). MR 0169103
Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139–163 (Italian). MR 0169104
48. Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 349–371 (Italian). MR 0169102
Dionisio Triscari, Sull’esistenza di cilindri con frontiera di misura minima, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 387–399 (Italian). MR 0169103
Dionisio Triscari, Sulle singolarità delle frontiere orientate di misura minima nello spazio euclideo a 4 dimensioni, Matematiche (Catania) 18 (1963), 139–163 (Italian). MR 0169104
49. Erhard Heinz and Friedrich Tomi, Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen, Math. Z. 111 (1969), 372–386 (German). MR 0266066, https://doi.org/10.1007/BF01110748
50. Robert Hermann, A uniqueness theorem for minimal submanifolds, J. Differential Geometry 2 (1968), 191–195. MR 0240731
51. Stefan Hildebrandt, Boundary behavior of minimal surfaces, Arch. Rational Mech. Anal. 35 (1969), 47–82. MR 0248650, https://doi.org/10.1007/BF00248494
52. W.-Y. Hsiang and H. B. Lawson, Jr., Minimal submanifolds of low cohomogeneity to appear).
53. David Kinderlehrer, Minimal surfaces whose boundaries contain spikes, J. Math. Mech. 19 (1969/1970), 829–853. MR 0256277
54. H. B. Lawson, Jr., Rigidity theorems in rank-1 symmetric spaces (to appear).
55. Johannes C. C. Nitsche, Concerning the isolated character of solutions of Plateau’s problem., Math. Z. 109 (1969), 393–411. MR 0243439, https://doi.org/10.1007/BF01110559
55. J. C. C. Nitsche, (b) The boundary behavior of minimal surfaces--Kellogg's Theorem and branch points on the boundary, Invent. Math, (to appear).
56. Robert Osserman, The nonexistence of branch points in the classical solution of Plateau’s problem, Bull. Amer. Math. Soc. 75 (1969), 1247–1248. MR 0253187, https://doi.org/10.1090/S0002-9904-1969-12377-3
56. R. Osserman, (b) A proof of the regularity everywhere of the classical solution to Plateau's problem (to appear).