The global geometry of surfaces with prescribed mean curvature in ℝ³

Bueno, Antonio; Gálvez, José; Mira, Pablo

doi:10.1090/tran/8041

The global geometry of surfaces with prescribed mean curvature in $\mathbb {R}^3$
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by Antonio Bueno, José A. Gálvez and Pablo Mira PDF

Trans. Amer. Math. Soc. 373 (2020), 4437-4467 Request permission

Abstract:

We develop a global theory for complete hypersurfaces in $\mathbb {R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. This theory extends the usual one of constant mean curvature hypersurfaces in $\mathbb {R}^{n+1}$, and also that of self-translating solitons of the mean curvature flow. For the particular case $n=2$, we will obtain results regarding a priori height and curvature estimates, non-existence of complete stable surfaces, and classification of properly embedded surfaces with at most one end.

References

Juan A. Aledo, José M. Espinar, and José A. Gálvez, The Codazzi equation for surfaces, Adv. Math. 224 (2010), no. 6, 2511–2530. MR 2652214, DOI 10.1016/j.aim.2010.02.007
A. D. Aleksandrov, Uniqueness theorems for surfaces in the large. I, Vestnik Leningrad. Univ. 11 (1956), no. 19, 5–17 (Russian). MR 0086338
Lars Andersson, Michael Eichmair, and Jan Metzger, Jang’s equation and its applications to marginally trapped surfaces, Complex analysis and dynamical systems IV. Part 2, Contemp. Math., vol. 554, Amer. Math. Soc., Providence, RI, 2011, pp. 13–45. MR 2884392, DOI 10.1090/conm/554/10958
Lars Andersson, Marc Mars, and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv. Theor. Math. Phys. 12 (2008), no. 4, 853–888. MR 2420905, DOI 10.4310/ATMP.2008.v12.n4.a5
Lars Andersson and Jan Metzger, Curvature estimates for stable marginally trapped surfaces, J. Differential Geom. 84 (2010), no. 2, 231–265. MR 2652461
César Rosales, Antonio Cañete, Vincent Bayle, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differential Equations 31 (2008), no. 1, 27–46. MR 2342613, DOI 10.1007/s00526-007-0104-y
Pierre Bérard and Laurent Hauswirth, General curvature estimates for stable $H$-surfaces immersed into a space form, J. Math. Pures Appl. (9) 78 (1999), no. 7, 667–700 (English, with English and French summaries). MR 1711051, DOI 10.1016/S0021-7824(99)00025-2
Antonio Bueno, José A. Gálvez, and Pablo Mira, Rotational hypersurfaces of prescribed mean curvature, J. Differential Equations 268 (2020), no. 5, 2394–2413. MR 4046194, DOI 10.1016/j.jde.2019.09.009
E. B. Christoffel, Ueber die Bestimmung der Gestalt einer krummen Oberfläche durch lokale Messungen auf derselben, J. Reine Angew. Math. 64 (1865), 193–209 (German). MR 1579295, DOI 10.1515/crll.1865.64.193
Juan Dávila, Manuel del Pino, and Xuan Hien Nguyen, Finite topology self-translating surfaces for the mean curvature flow in $\Bbb {R}^3$, Adv. Math. 320 (2017), 674–729. MR 3709119, DOI 10.1016/j.aim.2017.09.014
José M. Espinar, Gradient Schrödinger operators, manifolds with density and applications, J. Math. Anal. Appl. 455 (2017), no. 2, 1505–1528. MR 3671237, DOI 10.1016/j.jmaa.2017.06.055
José M. Espinar, José A. Gálvez, and Harold Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009), no. 2, 351–386. MR 2495798, DOI 10.4171/CMH/165
D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three-manifolds, Invent. Math. 82 (1985), no. 1, 121–132. MR 808112, DOI 10.1007/BF01394782
Doris Fischer-Colbrie and Richard 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. MR 562550, DOI 10.1002/cpa.3160330206
Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black hole topology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–576. MR 2238889, DOI 10.1007/s00220-006-0019-z
José A. Gálvez and Pablo Mira, A Hopf theorem for non-constant mean curvature and a conjecture of A. D. Alexandrov, Math. Ann. 366 (2016), no. 3-4, 909–928. MR 3563227, DOI 10.1007/s00208-015-1351-4
J. A. Gálvez and P. Mira, Uniqueness of immersed spheres in three-manifolds, J. Diff. Geom., to appear. arXiv:1603.07153
J. A. Gálvez and P. Mira, Rotational symmetry of Weingarten spheres in homogeneous three-manifolds, preprint, arXiv:1807.09654
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, DOI 10.1007/978-3-642-61798-0
M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215. MR 1978494, DOI 10.1007/s000390300004
Bo Guan and Pengfei Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2) 156 (2002), no. 2, 655–673. MR 1933079, DOI 10.2307/3597202
Pengfei Guan, Zhizhang Wang, and Xiangwen Zhang, A proof of Alexandrov’s uniqueness theorem for convex surfaces in $\Bbb {R}^3$, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 2, 329–336. MR 3465378, DOI 10.1016/j.anihpc.2014.09.011
Qiang Guang, Volume growth, entropy and stability for translating solitons, Comm. Anal. Geom. 27 (2019), no. 1, 47–72. MR 3951020, DOI 10.4310/CAG.2019.v27.n1.a2
Qing Han, Nikolai Nadirashvili, and Yu Yuan, Linearity of homogeneous order-one solutions to elliptic equations in dimension three, Comm. Pure Appl. Math. 56 (2003), no. 4, 425–432. MR 1949137, DOI 10.1002/cpa.10064
Gerhard Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266. MR 772132
Tom Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108 (1994), no. 520, x+90. MR 1196160, DOI 10.1090/memo/0520
Debora Impera and Michele Rimoldi, Rigidity results and topology at infinity of translating solitons of the mean curvature flow, Commun. Contemp. Math. 19 (2017), no. 6, 1750002, 21. MR 3691501, DOI 10.1142/S021919971750002X
Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
Rafael López, Invariant surfaces in Euclidean space with a log-linear density, Adv. Math. 339 (2018), 285–309. MR 3866898, DOI 10.1016/j.aim.2018.09.029
Francisco J. López and Antonio Ros, Complete minimal surfaces with index one and stable constant mean curvature surfaces, Comment. Math. Helv. 64 (1989), no. 1, 34–43. MR 982560, DOI 10.1007/BF02564662
Francisco Martín, Jesús Pérez-García, Andreas Savas-Halilaj, and Knut Smoczyk, A characterization of the grim reaper cylinder, J. Reine Angew. Math. 746 (2019), 209–234. MR 3895630, DOI 10.1515/crelle-2016-0011
Francisco Martín, Andreas Savas-Halilaj, and Knut Smoczyk, On the topology of translating solitons of the mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2853–2882. MR 3412395, DOI 10.1007/s00526-015-0886-2
Laurent Mazet, Optimal length estimates for stable CMC surfaces in 3-space forms, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2761–2765. MR 2497490, DOI 10.1090/S0002-9939-09-09885-2
William H. Meeks III, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), no. 3, 539–552. MR 940118
William H. Meeks III, Joaquín Pérez, and Antonio Ros, Stable constant mean curvature surfaces, Handbook of geometric analysis. No. 1, Adv. Lect. Math. (ALM), vol. 7, Int. Press, Somerville, MA, 2008, pp. 301–380. MR 2483369
Pablo Padilla, The principal eigenvalue and maximum principle for second order elliptic operators on Riemannian manifolds, J. Math. Anal. Appl. 205 (1997), no. 2, 285–312. MR 1428349, DOI 10.1006/jmaa.1997.5139
A. V. Pogorelov, Extension of a general uniqueness theorem of A. D. Aleksandrov to the case of nonanalytic surfaces, Doklady Akad. Nauk SSSR (N.S.) 62 (1948), 297–299 (Russian). MR 0027569
Antonio Ros and Harold Rosenberg, Properly embedded surfaces with constant mean curvature, Amer. J. Math. 132 (2010), no. 6, 1429–1443. MR 2766524
Harold Rosenberg, Rabah Souam, and Eric Toubiana, General curvature estimates for stable $H$-surfaces in 3-manifolds and applications, J. Differential Geom. 84 (2010), no. 3, 623–648. MR 2669367
Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR 795231
J. Spruck and L. Xiao, Complete translating solitons to the mean curvature flow in $\mathbb {R}^3$ with nonnegative mean curvature, arXiv:1703.01003. American Journal of Mathematics, to appear.