Minimal surfaces in minimally convex domains
HTML articles powered by AMS MathViewer
- by Antonio Alarcón, Barbara Drinovec Drnovšek, Franc Forstnerič and Francisco J. López PDF
- Trans. Amer. Math. Soc. 371 (2019), 1735-1770 Request permission
Abstract:
In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb {R}^3$ can be approximated uniformly on compacts in $\mathring M=M\setminus bM$ by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb {R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb {R}^n$ for any $n\ge 3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense.References
- Antonio Alarcón, Compact complete proper minimal immersions in strictly convex bounded regular domains of $\Bbb R^3$, XVIII International Fall Workshop on Geometry and Physics, AIP Conf. Proc., vol. 1260, Amer. Inst. Phys., Melville, NY, 2010, pp. 105–111. MR 3024549
- A. Alarcón, B. Drinovec Drnovšek, F. Forstnerič, and F. J. López, Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves, Proc. Lond. Math. Soc. (3) 111 (2015), no. 4, 851–886. MR 3407187, DOI 10.1112/plms/pdv044
- 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
- A. Alarcón, F. Forstnerič, and F. J. López. New complex analytic methods in the study of nonorientable minimal surfaces in $\mathbb {R}^n$. Mem. Amer. Math. Soc., in press.
- Antonio Alarcón, Franc Forstnerič, and Francisco J. López, Embedded minimal surfaces in $\Bbb {R}^n$, Math. Z. 283 (2016), no. 1-2, 1–24. MR 3489056, DOI 10.1007/s00209-015-1586-5
- Antonio Alarcón and Francisco J. López, Minimal surfaces in $\Bbb R^3$ properly projecting into $\Bbb R^2$, J. Differential Geom. 90 (2012), no. 3, 351–381. MR 2916039
- Antonio Alarcón and Francisco J. López, Null curves in $\Bbb {C}^3$ and Calabi-Yau conjectures, Math. Ann. 355 (2013), no. 2, 429–455. MR 3010135, DOI 10.1007/s00208-012-0790-4
- Antonio Alarcón and Francisco J. López, Properness of associated minimal surfaces, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5139–5154. MR 3240920, DOI 10.1090/S0002-9947-2014-06050-9
- Antonio Alarcón and Francisco J. López, Approximation theory for nonorientable minimal surfaces and applications, Geom. Topol. 19 (2015), no. 2, 1015–1062. MR 3336277, DOI 10.2140/gt.2015.19.1015
- Antonio Alarcón and Nikolai Nadirashvili, Limit sets for complete minimal immersions, Math. Z. 258 (2008), no. 1, 107–113. MR 2350037, DOI 10.1007/s00209-007-0161-0
- Michael T. Anderson, Curvature estimates for minimal surfaces in $3$-manifolds, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 1, 89–105. MR 803196
- 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
- Yun Gang Chen, Yoshikazu Giga, and Shun’ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. MR 1100211
- 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
- Jean-Pierre Demailly, Cohomology of $q$-convex spaces in top degrees, Math. Z. 204 (1990), no. 2, 283–295. MR 1055992, DOI 10.1007/BF02570874
- Avner Dor, A domain in $\textbf {C}^m$ not containing any proper image of the unit disc, Math. Z. 222 (1996), no. 4, 615–625. MR 1406270, DOI 10.1007/PL00004548
- Barbara Drinovec Drnovšek and Franc Forstnerič, Holomorphic curves in complex spaces, Duke Math. J. 139 (2007), no. 2, 203–253. MR 2352132, DOI 10.1215/S0012-7094-07-13921-8
- Barbara Drinovec Drnovšek and Franc Forstnerič, Strongly pseudoconvex domains as subvarieties of complex manifolds, Amer. J. Math. 132 (2010), no. 2, 331–360. MR 2654777, DOI 10.1353/ajm.0.0106
- 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
- Barbara Drinovec Drnovšek and Franc Forstnerič, Minimal hulls of compact sets in $\Bbb {R}^3$, Trans. Amer. Math. Soc. 368 (2016), no. 10, 7477–7506. MR 3471098, DOI 10.1090/tran/6777
- 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
- Klaus Ecker and Gerhard Huisken, Mean curvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), no. 3, 453–471. MR 1025164, DOI 10.2307/1971452
- Klaus Ecker and Gerhard Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991), no. 3, 547–569. MR 1117150, DOI 10.1007/BF01232278
- Leonor Ferrer, Francisco Martín, and William H. Meeks III, Existence of proper minimal surfaces of arbitrary topological type, Adv. Math. 231 (2012), no. 1, 378–413. MR 2935393, DOI 10.1016/j.aim.2012.05.007
- Franc Forstnerič, Stein manifolds and holomorphic mappings, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Cham, 2017. The homotopy principle in complex analysis. MR 3700709, DOI 10.1007/978-3-319-61058-0
- Franc Forstnerič and Josip Globevnik, Discs in pseudoconvex domains, Comment. Math. Helv. 67 (1992), no. 1, 129–145. MR 1144617, DOI 10.1007/BF02566492
- M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- R. E. Greene and H. Wu, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 1, vii, 215–235 (English, with French summary). MR 382701
- 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., Plurisubharmonicity in a general geometric context, Geometry and analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, Int. Press, Somerville, MA, 2011, pp. 363–402. MR 2882430
- 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
- Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
- Gerhard Huisken and Tom Ilmanen, Higher regularity of the inverse mean curvature flow, J. Differential Geom. 80 (2008), no. 3, 433–451. MR 2472479
- Luquésio P. Jorge and William H. Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), no. 2, 203–221. MR 683761, DOI 10.1016/0040-9383(83)90032-0
- 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, Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves, Indiana Univ. Math. J. 48 (1999), no. 4, 1513–1534. MR 1757081, DOI 10.1512/iumj.1999.48.1767
- 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
- Francisco Martín, William H. Meeks III, and Nikolai Nadirashvili, Bounded domains which are universal for minimal surfaces, Amer. J. Math. 129 (2007), no. 2, 455–461. MR 2306042, DOI 10.1353/ajm.2007.0013
- 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 H. Meeks III and Harold Rosenberg, Maximum principles at infinity, J. Differential Geom. 79 (2008), no. 1, 141–165. MR 2401421
- 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
- Gwenaël Mercier and Matteo Novaga, Mean curvature flow with obstacles: existence, uniqueness and regularity of solutions, Interfaces Free Bound. 17 (2015), no. 3, 399–426. MR 3421913, DOI 10.4171/IFB/348
- Johannes C. C. Nitsche, Vorlesungen über Minimalflächen, Die Grundlehren der mathematischen Wissenschaften, Band 199, Springer-Verlag, Berlin-New York, 1975 (German). MR 0448224
- 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, 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
- 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
- Nessim Sibony, Pfaff systems, currents and hulls, Math. Z. 285 (2017), no. 3-4, 1107–1123. MR 3623742, DOI 10.1007/s00209-016-1740-8
- E. Spadaro, Mean-convex sets and minimal barriers, preprint, 2011. arXiv:1112.4288.
- Graham H. Williams, The Dirichlet problem for the minimal surface equation with Lipschitz continuous boundary data, J. Reine Angew. Math. 354 (1984), 123–140. MR 767575, DOI 10.1515/crll.1984.354.123
- 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
Additional Information
- Antonio Alarcón
- Affiliation: Departmento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, E–18071 Granada, Spain
- MR Author ID: 783655
- Email: alarcon@ugr.es
- Barbara Drinovec Drnovšek
- Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
- Email: barbara.drinovec@fmf.uni-lj.si
- Franc Forstnerič
- Affiliation: Faculty of Mathematics and Physics, University of Ljubljana, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia
- MR Author ID: 228404
- Email: franc.forstneric@fmf.uni-lj.si
- Francisco J. López
- Affiliation: Departmento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, E–18071 Granada, Spain
- Email: fjlopez@ugr.es
- Received by editor(s): August 5, 2016
- Received by editor(s) in revised form: June 27, 2017
- Published electronically: June 7, 2018
- Additional Notes: The first author was supported by the Ramón y Cajal program of the Spanish Ministry of Economy and Competitiveness.\endgraf The first and fourth authors were partially supported by the MINECO/FEDER grants MTM2014-52368 and MTM2017-89677-P, Spain.
The second and third authors were partially supported by the research program P1-0291 and grants J1-5432 and J1-7256 from ARRS, Republic of Slovenia. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1735-1770
- MSC (2010): Primary 53A10; Secondary 32B15, 32E30, 32H02
- DOI: https://doi.org/10.1090/tran/7331
- MathSciNet review: 3894033
Dedicated: Dedicated to Josip Globevnik for his seventieth birthday