×
Meeks, William H. III; Tinaglia, Giuseppe

Curvature estimates for constant mean curvature surfaces. (English) Zbl 1433.53014

Duke Math. J. 168, No. 16, 3057-3102 (2019).
An oriented surface \(M\) immersed in \(\mathbb{R}^3\) is said to be an \(H\)-surface if it is embedded, connected, and it has positive constant mean curvature \(H\). An \(H\)-surface that is homeomorphic to a closed disk in the Euclidean plane is called an \(H\)-disk. The main result of the paper under reviews states that there exists an upper bound for the extrinsic distance from a point in an \(H\)-disk to its boundary.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)

Cited in 3 Documents

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Keywords:

minimal surface; constant mean curvature; curvature estimates
PDF BibTeX XML Cite
\textit{W. H. Meeks III} and \textit{G. Tinaglia}, Duke Math. J. 168, No. 16, 3057--3102 (2019; Zbl 1433.53014)
Full Text: DOI arXiv

References:

[1] J. Bernstein and C. Breiner, Helicoid-like minimal disks and uniqueness, J. Reine Angew. Math. 655 (2011), 129-146. · Zbl 1225.53008
[2] T. Bourni and G. Tinaglia, Curvature estimates for surfaces with bounded mean curvature, Trans. Amer. Math. Soc. 364 (2012), no. 11, 5813-5828. · Zbl 1277.53005
[3] T. Bourni and G. Tinaglia, \(C^{1,\alpha }\)-regularity for surfaces with \(H\in L^p\), Ann. Global Anal. Geom. 46 (2014), no. 2, 159-186. · Zbl 1296.53014
[4] T. Choi, W. H. Meeks III, and B. White, A rigidity theorem for properly embedded minimal surfaces in \(\mathbb{R}^3 \), J. Differential Geom. 32 (1990), no. 1, 65-76. · Zbl 0704.53008
[5] H. I. Choi and R. Schoen, The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), no. 3, 387-394. · Zbl 0577.53044
[6] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a \(3\)-manifold, I: Estimates off the axis for disks, Ann. of Math. (2) 160 (2004), no. 1, 27-68. · Zbl 1070.53031
[7] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a \(3\)-manifold, II: Multi-valued graphs in disks, Ann. of Math. (2) 160 (2004), no. 1, 69-92. · Zbl 1070.53032
[8] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a \(3\)-manifold, III: Planar domains, Ann. of Math. (2) 160 (2004), no. 2, 523-572. · Zbl 1076.53068
[9] T. H. Colding and W. P. Minicozzi II, The space of embedded minimal surfaces of fixed genus in a \(3\)-manifold, IV: Locally simply connected, Ann. of Math. (2) 160 (2004), no. 2, 573-615. · Zbl 1076.53069
[10] T. H. Colding and W. P. Minicozzi II, The Calabi-Yau conjectures for embedded surfaces, Ann. of Math. (2) 167 (2008), no. 1, 211-243. · Zbl 1142.53012
[11] C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl. (9) 16 (1841), 309-321.
[12] M. do Carmo and C. K. Peng, Stable complete minimal surfaces in \(\mathbb{R}^3\) are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903-906. · Zbl 0442.53013
[13] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in \(3\)-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. · Zbl 0439.53060
[14] N. J. Korevaar, R. Kusner, and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465-503. · Zbl 0726.53007
[15] R. Kusner, Global geometry of extremal surfaces in three space, Ph.D. dissertation, University of California, Berkeley, Berkeley, CA, 1988.
[16] W. H. Meeks III, J. Pérez, and A. Ros, The dynamics theorem for properly embedded minimal surfaces, Math. Ann. 365 (2016), no. 3-4, 1069-1089. · Zbl 1361.53049
[17] W. H. Meeks III and H. Rosenberg, The uniqueness of the helicoid, Ann. of Math. (2) 161 (2005), no. 2, 727-758. · Zbl 1102.53005
[18] W. H. Meeks III and G. Tinaglia, The dynamics theorem for \(CMC\) surfaces in \(\mathbb{R}^3 \), J. Differential Geom. 85 (2010), no. 1, 141-173. · Zbl 1203.53009
[19] W. H. Meeks III and G. Tinaglia, Triply periodic constant mean curvature surfaces, Adv. Math. 335 (2018), no. 4, 809-837. · Zbl 1396.53087
[20] W. H. Meeks III and G. Tinaglia, Constant mean curvature surfaces in homogeneous three-manifolds, in preparation. · Zbl 1396.53087
[21] W. H. Meeks III and G. Tinaglia, The geometry of constant mean curvature surfaces in \(\mathbb{R}^3 \), preprint, arXiv:1609.08032v1 [math.DG]. · Zbl 1235.53008
[22] W. H. Meeks III and G. Tinaglia, Limit lamination theorem for \(H\)-disks, preprint, arXiv:1510.05155v2 [math.DG]. · Zbl 1420.53016
[23] W. H. Meeks III and G. Tinaglia, Limit lamination theorem for \(H\)-surfaces, to appear in J. Reine Angew. Math., preprint, arXiv:1510.07549v2 [math.DG]. · Zbl 1420.53016
[24] W. H. Meeks III and G. Tinaglia, One-sided curvature estimates for \(H\)-disks, preprint, arXiv.1408.5233v2 [math.DG]. · Zbl 1453.53009
[25] W. H. Meeks III and S. T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), no. 2, 151-168. · Zbl 0479.49026
[26] A. V. Pogorelov, On the stability of minimal surfaces (in Russian), Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 293-295; English translation in Dokl. Math. 24 (1981), 274-276. · Zbl 0495.53005
[27] H. Rosenberg, R. Souam, and E. Toubiana, General curvature estimates for stable \(H\)-surfaces in \(3\)-manifolds and applications, J. Differential Geom. 84 (2010), no. 3, 623-648. · Zbl 1198.53062
[28] R. Schoen, “Estimates for stable minimal surfaces in three-dimensional manifolds” in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, 1983, 111-126. · Zbl 0532.53042
[29] R. Schoen and L. Simon, “Regularity of simply connected surfaces with quasiconformal Gauss map” in Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press, Princeton, 1983, 127-146. · Zbl 0544.53001
[30] R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275-288. · Zbl 0323.53039
[31] B. Smyth and G. Tinaglia, The number of constant mean curvature isometric immersions of a surface, Comment. Math. Helv. 88 (2013), no. 1, 163-183. · Zbl 1260.53023
[32] G. Tinaglia, Multi-valued graphs in embedded constant mean curvature disks, Trans. Amer. Math. Soc. 359 (2007), no. 1, 143-164. · Zbl 1115.53010
[33] G. Tinaglia, Structure theorems for embedded disks with mean curvature bounded in \({L}^p\), Comm. Anal. Geom. 16 (2008), no. 4, 819-836. · Zbl 1162.53302
[34] G. Tinaglia, Curvature estimates for minimal surfaces with total boundary curvature less than \(4 \pi \), Proc. Amer. Math. Soc. 137 (2009), no. 7, 2445-2450. · Zbl 1169.53008
[35] G. Tinaglia, “On curvature estimates for constant mean curvature surfaces” in Geometric Analysis: Partial Differential Equations and Surfaces, Contemp. Math. 570, Amer. Math. Soc., Providence, 2012, 165-185. · Zbl 1268.53009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.