Minimal hulls of compact sets in ℝ³

Drinovec Drnovšek, Barbara; Forstnerič, Franc

doi:10.1090/tran/6777

Minimal hulls of compact sets in $\mathbb {R}^3$
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Trans. Amer. Math. Soc. 368 (2016), 7477-7506 Request permission

Abstract:

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb {R}^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $\mathbb {C}^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner’s tube theorem.

References

Antonio Alarcón and Franc Forstnerič, The Calabi-Yau problem, null curves, and Bryant surfaces, Math. Ann. 363 (2015), no. 3-4, 913–951. MR 3412347, DOI 10.1007/s00208-015-1189-9
H. Alexander, Polynomial approximation and hulls in sets of finite linear measure in Cn, Amer. J. Math. 93 (1971), 65–74. MR 284617, DOI 10.2307/2373448
Eric Bedford and Wilhelm Klingenberg, On the envelope of holomorphy of a $2$-sphere in $\textbf {C}^2$, J. Amer. Math. Soc. 4 (1991), no. 3, 623–646. MR 1094437, DOI 10.1090/S0894-0347-1991-1094437-0
Giovanni Bellettini, Lecture notes on mean curvature flow, barriers and singular perturbations, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 12, Edizioni della Normale, Pisa, 2013. MR 3155251, DOI 10.1007/978-88-7642-429-8
S. Bochner, A theorem on analytic continuation of functions in several variables, Ann. of Math. (2) 39 (1938), no. 1, 14–19. MR 1503384, DOI 10.2307/1968709
Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
Kenneth A. Brakke, The motion of a surface by its mean curvature, Mathematical Notes, vol. 20, Princeton University Press, Princeton, N.J., 1978. MR 0485012
Shang Quan Bu and Walter Schachermayer, Approximation of Jensen measures by image measures under holomorphic functions and applications, Trans. Amer. Math. Soc. 331 (1992), no. 2, 585–608. MR 1035999, DOI 10.1090/S0002-9947-1992-1035999-6
E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by R. A. M. Hoksbergen. MR 1111477, DOI 10.1007/978-94-009-2366-9
Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140, DOI 10.1090/gsm/121
Pascal Collin, Robert Kusner, William H. Meeks III, and Harold Rosenberg, The topology, geometry and conformal structure of properly embedded minimal surfaces, J. Differential Geom. 67 (2004), no. 2, 377–393. MR 2153082
John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
Đào Trọng Thi and A. T. Fomenko, Minimal surfaces, stratified multivarifolds, and the Plateau problem, Translations of Mathematical Monographs, vol. 84, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by E. J. F. Primrose. MR 1093903, DOI 10.1090/mmono/084
Ulrich Dierkes, Stefan Hildebrandt, and Friedrich Sauvigny, Minimal surfaces, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 339, Springer, Heidelberg, 2010. With assistance and contributions by A. Küster and R. Jakob. MR 2566897, DOI 10.1007/978-3-642-11698-8
Barbara Drinovec Drnovšek and Franc Forstnerič, The Poletsky-Rosay theorem on singular complex spaces, Indiana Univ. Math. J. 61 (2012), no. 4, 1407–1423. MR 3085613, DOI 10.1512/iumj.2012.61.4686
Julien Duval and Nessim Sibony, Polynomial convexity, rational convexity, and currents, Duke Math. J. 79 (1995), no. 2, 487–513. MR 1344768, DOI 10.1215/S0012-7094-95-07912-5
G. A. Edgar, Complex martingale convergence, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 38–59. MR 827757, DOI 10.1007/BFb0074691
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Franc Forstnerič, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), no. 4, 869–889. MR 982834, DOI 10.1512/iumj.1988.37.37042
John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
M. L. Gromov, Convex integration of differential relations. I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343 (Russian). MR 0413206
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
F. Reese Harvey and H. Blaine Lawson Jr., An introduction to potential theory in calibrated geometry, Amer. J. Math. 131 (2009), no. 4, 893–944. MR 2543918, DOI 10.1353/ajm.0.0067
F. Reese Harvey and H. Blaine Lawson Jr., Duality of positive currents and plurisubharmonic functions in calibrated geometry, Amer. J. Math. 131 (2009), no. 5, 1211–1239. MR 2555839, DOI 10.1353/ajm.0.0074
F. Reese Harvey and H. Blaine Lawson Jr., Geometric plurisubharmonicity and convexity: an introduction, Adv. Math. 230 (2012), no. 4-6, 2428–2456. MR 2927376, DOI 10.1016/j.aim.2012.03.033
F. Reese Harvey and H. Blaine Lawson Jr., $p$-convexity, $p$-plurisubharmonicity and the Levi problem, Indiana Univ. Math. J. 62 (2013), no. 1, 149–169. MR 3158505, DOI 10.1512/iumj.2013.62.4886
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Lars Hörmander, Notions of convexity, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Reprint of the 1994 edition [of MR1301332]. MR 2311920
J. Hounie, A proof of Bochner’s tube theorem, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4203–4207. MR 2538581, DOI 10.1090/S0002-9939-09-10057-6
Uroš Kuzman, Poletsky theory of discs in almost complex manifolds, Complex Var. Elliptic Equ. 59 (2014), no. 2, 262–270. MR 3170758, DOI 10.1080/17476933.2012.734300
Finnur Lárusson and Ragnar Sigurdsson, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998), 1–39. MR 1637837, DOI 10.1515/crll.1998.078
Finnur Lárusson and Ragnar Sigurdsson, Plurisubharmonicity of envelopes of disc functionals on manifolds, J. Reine Angew. Math. 555 (2003), 27–38. MR 1956593, DOI 10.1515/crll.2003.013
William H. Meeks III and Joaquín Pérez, Conformal properties in classical minimal surface theory, Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., vol. 9, Int. Press, Somerville, MA, 2004, pp. 275–335. MR 2195411, DOI 10.4310/SDG.2004.v9.n1.a8
William W. Meeks III and Shing Tung Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151–168. MR 645492, DOI 10.1007/BF01214308
Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Evgeny A. Poletsky, Plurisubharmonic functions as solutions of variational problems, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 163–171. MR 1128523
Evgeny A. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993), no. 1, 85–144. MR 1218708, DOI 10.1512/iumj.1993.42.42006
R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
Jean-Pierre Rosay, Poletsky theory of disks on holomorphic manifolds, Indiana Univ. Math. J. 52 (2003), no. 1, 157–169. MR 1970025, DOI 10.1512/iumj.2003.52.2170
Jean-Pierre Rosay, Approximation of non-holomorphic maps, and Poletsky theory of discs, J. Korean Math. Soc. 40 (2003), no. 3, 423–434. MR 1973910, DOI 10.4134/JKMS.2003.40.3.423
Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
E. Spadaro, Mean-convex sets and minimal barriers, arxiv.org/abs/1112.4288
Gabriel Stolzenberg, A hull with no analytic structure, J. Math. Mech. 12 (1963), 103–111. MR 0143061
Gabriel Stolzenberg, Uniform approximation on smooth curves, Acta Math. 115 (1966), 185–198. MR 192080, DOI 10.1007/BF02392207
Edgar Lee Stout, Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2305474, DOI 10.1007/978-0-8176-4538-0
John Wermer, The hull of a curve in $C^{n}$, Ann. of Math. (2) 68 (1958), 550–561. MR 100102, DOI 10.2307/1970155
Erlend Fornæss Wold, A note on polynomial convexity: Poletsky disks, Jensen measures and positive currents, J. Geom. Anal. 21 (2011), no. 2, 252–255. MR 2772071, DOI 10.1007/s12220-010-9130-7