Size minimizing surfaces
[Surfaces minimisantes]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 37-101.

Nous obtenons un nouveau théorème d’existence relatif au problème de Plateau dans l’espace euclidien de dimension 3. Ce faisant, nous comparons les approches d’E.R. Reifenberg d’une part, et de H. Federer et W.H. Fleming d’autre part. Un pas technique important consiste à démontrer qu’on peut approcher tout ensemble compact et rectifiable, en mesure de Hausdorff et en distance de Hausdorff, par une surface localement acyclique ayant le même bord.

We prove a new existence theorem pertaining to the Plateau problem in 3-dimensional Euclidean space. We compare the approach of E.R. Reifenberg with that of H. Federer and W.H. Fleming. A relevant technical step consists in showing that compact rectifiable surfaces are approximatable in Hausdorff measure and in Hausdorff distance by locally acyclic surfaces having the same boundary.

@article{ASENS_2009_4_42_1_37_0,
     author = {Pauw, Thierry De},
     title = {Size minimizing surfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {37--101},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {1},
     year = {2009},
     doi = {10.24033/asens.2090},
     mrnumber = {2518893},
     zbl = {1184.49041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2090/}
}
TY  - JOUR
AU  - Pauw, Thierry De
TI  - Size minimizing surfaces
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 37
EP  - 101
VL  - 42
IS  - 1
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2090/
DO  - 10.24033/asens.2090
LA  - en
ID  - ASENS_2009_4_42_1_37_0
ER  - 
%0 Journal Article
%A Pauw, Thierry De
%T Size minimizing surfaces
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 37-101
%V 42
%N 1
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2090/
%R 10.24033/asens.2090
%G en
%F ASENS_2009_4_42_1_37_0
Pauw, Thierry De. Size minimizing surfaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 1, pp. 37-101. doi : 10.24033/asens.2090. http://archive.numdam.org/articles/10.24033/asens.2090/

[1] W. K. Allard, On the first variation of a varifold: boundary behavior, Ann. of Math. 101 (1975), 418-446. | MR | Zbl

[2] K. A. Brakke, The surface evolver, Experiment. Math. 1 (1992), 141-165. | EuDML | MR | Zbl

[3] R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., 1950. | MR | Zbl

[4] T. De Pauw, Nearly flat almost monotone measures are big pieces of Lipschitz graphs, J. Geom. Anal. 12 (2002), 29-61. | MR | Zbl

[5] T. De Pauw, Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents, Rev. Mat. Iberoam. 23 (2007), 143-189. | EuDML | MR | Zbl

[6] T. De Pauw, Concentrated, nearly monotonic, epiperimetric measures in Euclidean space, J. Differential Geom. 77 (2007), 77-134. | MR | Zbl

[7] T. De Pauw & R. Hardt, Size minimization and approximating problems, Calc. Var. Partial Differential Equations 17 (2003), 405-442. | MR | Zbl

[8] S. Eilenberg & N. Steenrod, Foundations of algebraic topology, Princeton University Press, 1952. | MR | Zbl

[9] L. C. Evans & R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992. | MR | Zbl

[10] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. | MR | Zbl

[11] H. Federer, Geometric measure theory, Die Grund. Math. Wiss., Band 153, Springer, 1969. | MR | Zbl

[12] H. Federer & W. H. Fleming, Normal and integral currents, Ann. of Math. 72 (1960), 458-520. | MR | Zbl

[13] V. Feuvrier, Un résultat d'existence pour les ensembles minimaux par optimisation sur des grilles polyédrales, Thèse de doctorat, Université d'Orsay, 2008.

[14] W. H. Fleming, An example in the problem of least area, Proc. Amer. Math. Soc. 7 (1956), 1063-1074. | MR | Zbl

[15] W. H. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palermo 11 (1962), 69-90. | MR | Zbl

[16] R. L. Foote, Regularity of the distance function, Proc. Amer. Math. Soc. 92 (1984), 153-155. | MR | Zbl

[17] E. Lamarle, Sur la stabilité des systèmes liquides en lames minces, Mémoires de l'Académie Royale de Belgique 35 (1864).

[18] G. Lawlor & F. Morgan, Curvy slicing proves that triple junctions locally minimize area, J. Differential Geom. 44 (1996), 514-528. | MR | Zbl

[19] F. Morgan, Size-minimizing rectifiable currents, Invent. Math. 96 (1989), 333-348. | MR | Zbl

[20] F. Morgan, Geometric measure theory, a beginner's guide, third éd., Academic Press Inc., 2000. | MR | Zbl

[21] D. Pavlica, personal communication.

[22] J. Plateau, Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, Gauthier-Villars, 1873. | JFM

[23] E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1-92. | MR | Zbl

[24] E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Ann. of Math. 80 (1964), 1-14. | MR | Zbl

[25] G. De Rham, Variétés différentiables. Formes, courants, formes harmoniques, Hermann, 1973. | Zbl

[26] L. Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, 1983. | MR | Zbl

[27] J. E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. 103 (1976), 489-539. | MR | Zbl

[28] H. Whitney, Geometric integration theory, Princeton University Press, 1957. | MR | Zbl

Cité par Sources :