Uniqueness property for 2-dimensional minimal cones in R3

• Autores: Xiangyu Liang
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 65, Nº 1, 2021, págs. 3-59
• Idioma: inglés
• DOI: 10.5565/publicacionsmatematiques.v65i1.383612
• Enlaces
  • Texto completo
• Resumen

  • In this article we treat two closely related problems: 1) the upper semicontinuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the 2-dimensional minimal cones in R3 . Given an open set Ω ⊂ Rn, a closed set E ⊂ Ω is said to be Almgren minimal of dimension d in Ω if it minimizes the d-Hausdorff measure among all its Lipschitz deformations in Ω. We say that a d-dimensional minimal set E in an open set Ω admits upper semi-continuity if, whenever {fn(E)}n is a sequence of deformations of E in Ω that converges to a set F, then we have Hd(F) ≥ lim supn Hd(fn(E)). This guarantees in particular that E minimizes the d-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. As proved in [19], when several 2-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known 2-dimensional minimal cones in Rn to generate new families of minimal cones by taking their almost orthogonal unions. The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

• Referencias bibliográficas
  • W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95(3) (1972), 417–491. DOI: 10.2307/1970868.
  • F. J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer....
  • K. A. Brakke, Minimal cones on hypercubes, J. Geom. Anal. 1(4) (1991), 329–338. DOI: 10.1007/BF02921309.
  • H. Brezis, “Functional Analysis, Sobolev Spaces and Partial Differential Equations”, Universitext, Springer, New York, 2011. DOI: 10.1007/978-0-387-70914-7.
  • G. David, Limits of Almgren quasiminimal sets, in: “Harmonic Analysis at Mount Holyoke” (South Hadley, MA, 2001), Contemp. Math. 320, Amer....
  • G. David, H¨older regularity of two-dimensional almost-minimal sets in Rn, Ann. Fac. Sci. Toulouse Math. (6) 18(1)(2009), 65–246. DOI: 10.5802/afst.1205.
  • G. David, C1+α-regularity for two-dimensional almost-minimal sets in Rn, J. Geom. Anal. 20(4) (2010), 837–954. DOI: 10.1007/s12220-010-9138-z.
  • G. David and S. Semmes, Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Mem. Amer. Math. Soc. 144(687) (2000), 132...
  • H. Federer, “Geometric Measure Theory”, Die Grundlehren der mathematischen Wissenschaften 153, Springer Verlag New York Inc., New York, 1969....
  • V. Feuvrier, Un r´esultat d’existence pour les ensembles minimaux par optimisation sur des grilles poly´edrales, PhD thesis, Laboratoire de...
  • A. Heppes, Isogonale sph¨arische Netze, Ann. Univ. Sci. Budapest. E¨otv¨os Sect. Math. 7 (1964), 41–48.
  • E. Lamarle, Sur la stabilit´e des syst`emes liquides en lames minces, M´emoires de l’Acad´emie Royale des Sciences, des Lettres et des...
  • G. Lawlor and F. Morgan, Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms, Pacific...
  • X. Liang, Almgren-minimality of unions of two almost orthogonal planes in R4, Proc. Lond. Math. Soc. (3) 106(5)(2013), 1005–1059. DOI: 10.1112/plms/pds059.
  • X. Liang, Topological minimal sets and existence results, Calc. Var. Partial Differential Equations 47(3–4) (2013), 523–546. DOI: 10.1007/s00526-012-0526-z.
  • X. Liang, Almgren and topological minimality for the set Y ×Y , J. Funct. Anal. 266(10) (2014), 6007–6054. DOI: 10.1016/j.jfa.2014.02.033.
  • X. Liang, On the topological minimality of unions of planes of arbitrary dimension, Int. Math. Res. Not. IMRN 2015(23) (2015), 12490–12539....
  • X. Liang, Measure and sliding stability for 2-dimensional minimal cones in Euclidean spaces, Preprint (2018). arXiv:1808.09691.
  • X. Liang, Minimality for unions of 2-dimensional minimal cones with nonisolated singularities, Preprint (2018). arXiv:1808.09687.
  • X. Liang, Sliding stability and uniqueness for the set Y × Y , In preparation.
  • P. Mattila, “Geometry of Sets and Measures in Euclidean Spaces”, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44,...
  • F. Morgan, Examples of unoriented area-minimizing surfaces, Trans. Amer. Math. Soc. 283(1) (1984), 225–237. DOI: 10.2307/1999999.
  • F. Morgan, Soap films and mathematics, in: “Differential Geometry: Partial Differential Equations on Manifolds” (Los Angeles, CA, 1990), Proc....
  • J. R. Munkres, “Topology”, Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
  • E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math. 104(1–2) (1960), 1–92....
  • J. E. Taylor, The structure of singularities in soap-bubble-like and soap-filmlike minimal surfaces, Ann. of Math. (2) 103(3) (1976), 489–539....
  • H. Whitney, “Geometric Integration Theory”, Princeton University Press, Princeton, N. J., 1957.