Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves
Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 6, p. 963-987
@article{AIHPC_2007__24_6_963_0,
     author = {Jakob, Ruben},
     title = {Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {24},
     number = {6},
     year = {2007},
     pages = {963-987},
     doi = {10.1016/j.anihpc.2006.10.003},
     zbl = {pre05247894},
     mrnumber = {2371115},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2007__24_6_963_0}
}
Jakob, Ruben. Finiteness of the set of solutions of Plateau's problem for polygonal boundary curves. Annales de l'I.H.P. Analyse non linéaire, Volume 24 (2007) no. 6, pp. 963-987. doi : 10.1016/j.anihpc.2006.10.003. http://www.numdam.org/item/AIHPC_2007__24_6_963_0/

[1] Courant R., Critical points and unstable minimal surfaces, Proc. Natl. Acad. Sci. USA 27 (1941) 51-57. | JFM 67.0368.02 | MR 3484

[2] Dierkes U., Hildebrandt S. et al. , Minimal Surfaces I, Grundlehren Math. Wiss., vol. 295, Springer, Berlin, 1991. | Zbl 0777.53012

[3] Dierkes U., Hildebrandt S. et al. , Minimal Surfaces II, Grundlehren Math. Wiss., vol. 295, Springer, Berlin, 1991. | Zbl 0777.53013

[4] Dieudonné J., Foundations of Modern Analysis, Academic Press, New York, 1960. | MR 120319 | Zbl 0100.04201

[5] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Classics in Math., third ed., Springer, Berlin, 1998. | Zbl 1042.35002

[6] Guillemin V., Pollack A., Differential Topology, Prentice-Hall, New Jersey, 1974. | MR 348781 | Zbl 0361.57001

[7] Gulliver R., Lesley F.D., On boundary branch points of minimizing surfaces, Arch. Ration. Mech. Anal. 52 (1973) 20-25. | MR 346641 | Zbl 0263.53009

[8] Heinz E., Ein Regularitätssatz für Flächen beschränkter mittlerer Krümmung, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 12 (1969) 107-118. | MR 270313 | Zbl 0192.58302

[9] Heinz E., On surfaces of constant mean curvature with polygonal boundaries, Arch. Ration. Mech. Anal. 36 (1970) 335-347. | MR 279697 | Zbl 0191.12002

[10] Heinz E., Über die analytische Abhängigkeit der Lösungen eines linearen elliptischen Randwertproblems von den Parametern, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1979) 1-20. | MR 568799 | Zbl 0436.35010

[11] Heinz E., Über eine Verallgemeinerung des Plateauschen Problems, Manuscripta Math. 28 (1979) 81-88. | MR 535696 | Zbl 0415.49027

[12] Heinz E., Ein mit der Theorie der Minimalflächen zusammenhängendes Variationsproblem, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl II. (1980) 25-35. | MR 610591 | Zbl 0456.49029

[13] Heinz E., Zum Marx-Shiffmanschen Variationsproblem, J. Reine Angew. Math. 344 (1983) 196-200. | MR 716255 | Zbl 0511.49022

[14] Heinz E., Minimalflächen mit polygonalem Rand, Math. Z. 183 (1983) 547-564. | MR 710771 | Zbl 0507.49027

[15] Heinz E., Hildebrandt S., The number of branch points of surfaces of bounded mean curvature, J. Differencial Geom. 4 (1970) 227-235. | MR 267495 | Zbl 0195.23003

[16] Hildebrandt S., Boundary behavior of minimal surfaces, Arch. Ration. Mech. Anal. 35 (1969) 47-82. | MR 248650 | Zbl 0183.39402

[17] Jakob R., H-Flächen-Index-Formel, Bonner Math. Schriften 366 (2004) 1-92. | MR 2208441 | Zbl 1087.53009

[18] Jakob R., H-surface-index-formula, Ann. Inst. H. Poincaré Analyse Non-Linéaire 22 (2005) 557-578. | Numdam | MR 2171991 | Zbl 1082.53007

[19] R. Jakob, Schwarz operators of minimal surfaces spanning polygonal boundary curves, Calc. Var., submitted for publication. | Zbl 1129.49044

[20] Jakob R., Finiteness of the set of solutions of Plateau's problem with polygonal boundary curves, Bonner Math. Schriften 379 (2006) 1-95. | MR 2338059 | Zbl 1109.53009

[21] Kato T., Perturbation Theory for Linear Operators, Springer, Berlin, 1976. | MR 407617 | Zbl 0342.47009

[22] Lojasiewics S., Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 18 (3) (1964) 449-474. | Numdam | MR 173265 | Zbl 0128.17101

[23] Nitsche J.C.C., Uniqueness and non-uniqueness for Plateau's Problem - One of the last major questions, in: Minimal Submanifolds and Geodesics, Kaigai Publications, Tokyo, 1978, pp. 143-161. | Zbl 0436.53008

[24] Sauvigny F., On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's Problem, Arch. Ration. Mech. Anal. 110 (1990) 125-140. | MR 1037345 | Zbl 0753.53006

[25] Sauvigny F., Ein Eindeutigkeitssatz für Minimalflächen im R p mit polygonalem Rand, J. Reine Angew. Math. 358 (1985) 92-96. | MR 797676 | Zbl 0553.53003

[26] Sauvigny F., Die zweite Variation von Minimalflächen im R p mit polygonalem Rand, Math. Z. 189 (1985) 167-184. | MR 779215 | Zbl 0543.53004

[27] Tomi F., On the local uniqueness of the problem of least area, Arch. Ration. Mech. Anal. 52 (1973) 312-318. | MR 346639 | Zbl 0277.49016

[28] Tomi F., On the finite solvability of Plateau's Problem, in: Lecture Notes in Math., vol. 579, Springer, 1977, pp. 679-695. | MR 454874 | Zbl 0361.53008