Search Advanced

Mathematics in Engineering

2021, Volume 3, Issue 5: 1-48. doi: 10.3934/mine.2021039
Previous Article Next Article
Research article Special Issues

Recent rigidity results for graphs with prescribed mean curvature

  • 1.
    Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Trieste 63, I-35121 Padova, Italy
  • 2.
    Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, I-20133 Milano, Italy
  • 3.
    Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici - Bloco 914, Av. Humberto Monte s/n, 60.455-760 Fortaleza, Brazil
  • 4.
    Dipartimento di Matematica, Università degli Studi di Torino, Via Carlo Alberto 10, I-10123 Torino, Italy
  • 5.
    Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, I-06123 Perugia, Italy
  • Received: 05 July 2020 Accepted: 09 September 2020 Published: 29 September 2020

  • MSC : Primary: 35B08, 35B53, 35R01; Secondary: 35R45, 53C42, 58J05

  • This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs u : M → $\mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain?? M, namely, for CMC graphs satisfying an overdetermined boundary condition.

    Citation: Bruno Bianchini, Giulio Colombo, Marco Magliaro, Luciano Mari, Patrizia Pucci, Marco Rigoli. Recent rigidity results for graphs with prescribed mean curvature[J]. Mathematics in Engineering, 2021, 3(5): 1-48. doi: 10.3934/mine.2021039

    Related Papers:

  • This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs u : M → $\mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain?? M, namely, for CMC graphs satisfying an overdetermined boundary condition.


    加载中


    [1] Almgren FJ Jr (1966) Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem. Ann Math 85: 277-292.
    [2] Altschuler S, Wu L (1994) Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc Var 2: 101-111.
    [3] Alías LJ, Mastrolia P, Rigoli M (2016) Maximum Principles and Geometric Applications, Cham: Springer.
    [4] Bao H, Shi Y (2014) Gauss maps of translating solitons of mean curvature flow. P Am Math Soc 142: 4333-4339.
    [5] Barbosa E (2018) On CMC free-boundary stable hypersurfaces in a Euclidean ball. Math Ann 372: 179-187.
    [6] Berestycki H, Caffarelli LA, Nirenberg L (1998) Further qualitative properties for elliptic equations in unbounded domains. Ann Scuola Norm Sup Pisa Cl Sci 25: 69-94.
    [7] Berestycki H, Caffarelli LA, Nirenberg L (1997) Monotonicity for elliptic equations in unbounded Lipschitz domains. Commun Pure Appl Math 50: 1089-1111.
    [8] Bernstein S (1915) Sur un théorème de géomètrie et son application aux équations aux dérivées partielles du type elliptique. Commun Soc Math de Kharkov 15: 38-45.
    [9] Bianchini B, Mari L, Rigoli M (2015) Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds. J Funct Anal 268: 1-72.
    [10] Bianchini B, Mari L, Pucci P, et al. (2021) Geometric Analysis of Quasilinear Inequalities on Complete Manifolds, Cham: Birh?user/Springer.
    [11] Bombieri E, De Giorgi E, Miranda M (1969) Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche. Arch Rational Mech Anal 32: 255-267.
    [12] Bombieri E, De Giorgi E, Giusti E (1969) Minimal cones and the Bernstein problem. Invent Math 7: 243-268.
    [13] Bombieri E, Giusti E (1972) Harnack's inequality for elliptic differential equations on minimal surfaces. Invent Math 15: 24-46.
    [14] Bonorino L, Casteras JB, Klaser P, et al. (2020) On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations. Calc Var 59: 135.
    [15] Borbely A (2017) Stochastic Completeness and the Omori-Yau Maximum Principle. J Geom Anal 27: 3228-3239.
    [16] Brooks R (1981) A relation between growth and the spectrum of the Laplacian. Math Z 178: 501-508.
    [17] Casteras JB, Heinonen E, Holopainen I, et al. (2020) Asymptotic Dirichlet problems in warped products. Math Z 295: 211-248.
    [18] Casteras JB, Heinonen E, Holopainen I (2017) Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures. J Geom Anal 27: 1106-1130.
    [19] Casteras JB, Heinonen E, Holopainen I (2019) Dirichlet problem for f -minimal graphs. arXiv: 1605.01935v2.
    [20] Casteras JB, Heinonen E, Holopainen I (2020) Existence and non-existence of minimal graphic and p-harmonic functions. P Roy Soc Edinb A 150: 341-366.
    [21] Casteras JB, Holopainen I, Ripoll JB (2017) On the asymptotic Dirichlet problem for the minimal hypersurface equation in a Hadamard manifold. Potential Anal 47: 485-501.
    [22] Casteras JB, Holopainen I, Ripoll JB (2018) Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems. Math Z 290: 221-250.
    [23] Cheeger J (1970) A lower bound for the smallest eigenvalue of the Laplacian, In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), Princeton: Princeton Univ. Press, 195- 199.
    [24] Cheeger J, Colding TH (1996) Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann Math 144: 189-237.
    [25] Cheeger J, Colding TH (1997) On the structure of spaces with Ricci curvature bounded below. I. J Differ Geom 46: 406-480.
    [26] Cheeger J, Colding TH (2000) On the structure of spaces with Ricci curvature bounded below. II. J Differ Geom 54: 13-35.
    [27] Cheeger J, Colding TH (2000) On the structure of spaces with Ricci curvature bounded below. III. J Differ Geom 54: 37-74.
    [28] Cheeger J, Colding TH, Minicozzi WP (1995) Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature. Geom Funct Anal 5: 948-954.
    [29] Cheeger J, Gromoll D (1971) The splitting theorem for manifolds of nonnegative Ricci curvature. J Differ Geom 6: 119-128.
    [30] Cheng SY, Yau ST (1975) Differential equations on Riemannian manifolds and their geometric applications. Commun Pure Appl Math 28: 333-354.
    [31] Chern SS (1965) On the curvatures of a piece of hypersurface in Euclidean space. Abh Math Sem Univ Hamburg 29: 77-91.
    [32] Clutterbuck J, Schnürer OC, Schulze F (2007) Stability of translating solutions to mean curvature flow. Calc Var 29: 281-293.
    [33] Collin P, Krust R (1991) Le probléme de Dirichlet pour l'équation des surfaces minimales sur des domaines non bornés. B Soc Math Fr 119: 443-462.
    [34] Collin P, Rosenberg H (2010) Construction of harmonic diffeomorphisms and minimal graphs. Ann Math 172: 1879-1906.
    [35] Colombo G, Magliaro M, Mari L, et al. (2020) Bernstein and half-space properties for minimal graphs under Ricci lower bounds. arXiv: 1911.12054.
    [36] Colombo G, Magliaro M, Mari L, et al. (2020) A splitting theorem for capillary graphs under Ricci lower bounds. arXiv: 2007.15143.
    [37] Colombo G, Mari L, Rigoli M (2020) Remarks on mean curvature flow solitons in warped products. Discrete Cont Dyn S 13: 1957-1991.
    [38] D'Ambrosio L, Mitidieri E (2010) A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv Math 224: 967-1020.
    [39] D'Ambrosio L, Mitidieri E (2012) A priori estimates and reduction principles for quasilinear elliptic problems and applications. Adv Differential Equ 17: 935-1000.
    [40] Dajczer M, de Lira JHS (2015) Entire bounded constant mean curvature Killing graphs. J Math Pure Appl 103: 219-227.
    [41] Dajczer M, de Lira JHS (2017) Entire unbounded constant mean curvature Killing graphs. B Braz Math Soc 48: 187-198.
    [42] De Giorgi E (1965) Una estensione del teorema di Bernstein. Ann Scuola Norm Sup Pisa 19: 79-85.
    [43] De Giorgi E (1965) Errata-Corrige: "Una estensione del teorema di Bernstein". Ann Scuola Norm Sup Pisa Cl Sci 19: 463-463.
    [44] Ding Q (2020) Liouville type theorems for minimal graphs over manifolds. arXiv: 1911.10306.
    [45] Ding Q, Jost J, Xin Y (2016) Minimal graphic functions on manifolds of nonnegative Ricci curvature. Commun Pure Appl Math 69: 323-371.
    [46] Ding Q, Jost J, Xin Y (2016) Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature. Am J Math 138: 287-327.
    [47] Do Carmo MP, Lawson HB Jr (1983) On Alexandrov-Bernstein theorems in hyperbolic space. Duke Math J 50: 995-1003.
    [48] Do Carmo MP, Peng CK (1979) Stable complete minimal surfaces in $\mathbb{R}.3$ are planes. B Am Math Soc 1: 903-906.
    [49] Dupaigne L, Ghergu M, Goubet O, et al. (2012) Entire large solutions for semilinear elliptic equations. J Differ Equations 253: 2224-2251.
    [50] Espinar JM, Farina A, Mazet L (2015) f -extremal domains in hyperbolic space. arXiv: 1511.02659.
    [51] Espinar JM, Mazet L (2019) Characterization of f -extremal disks. J Differ Equations 266: 2052- 2077.
    [52] do Espírito Santo N, Fornari S, Ripoll JB (2010) The Dirichlet problem for the minimal hypersurface equation in M × $\mathbb{R}$ with prescribed asymptotic boundary. J Math Pure Appl 93: 204-221.
    [53] Farina A (2015) A Bernstein-type result for the minimal surface equation. Ann Scuola Norm Sup Pisa XIV: 1231-1237.
    [54] Farina A (2018) A sharp Bernstein-type theorem for entire minimal graphs. Calc Var 57: 123.
    [55] Farina A (2007) Liouville-type theorems for elliptic problems, In: Handbook of Differential Equations, Elsevier, 60-116.
    [56] Farina A (2002) Propriétés qualitatives de solutions d'équations et systémes d'équations nonlinéaires, Habilitation à diriger des recherches, Paris VI.
    [57] Farina A, Franz G, Mari L, Splitting and half-space properties for graphs with prescribed mean curvature. Preprint.
    [58] Farina A, Mari L, Valdinoci E (2013) Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds. Commun Part Diff Eq 38: 1818-1862,
    [59] Farina A, Sciunzi B, Valdinoci E (2008) Bernstein and De Giorgi type problems: New results via a geometric approach. Ann Scuola Norm Sup Pisa Cl Sci 7: 741-791.
    [60] Farina A, Serrin J (2011) Entire solutions of completely coercive quasilinear elliptic equations. J Differ Equations 250: 4367-4408.
    [61] Farina A, Serrin J (2011) Entire solutions of completely coercive quasilinear elliptic equations, II. J Differ Equations 250: 4409-4436.
    [62] Farina A, Valdinoci E (2010) Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems. Arch Ration Mech Anal 195: 1025-1058.
    [63] Filippucci R (2009) Nonexistence of positive weak solutions of elliptic inequalities. Nonlinear Anal 8: 2903-2916.
    [64] Finn R (1986) Equilibrium Capillary Surfaces, New York: Springer-Verlag.
    [65] Fischer-Colbrie D, Schoen R (1980) The structure of complete stable minimal surfaces in 3- manifolds of non negative scalar curvature. Commun Pure Appl Math 33: 199-211.
    [66] Flanders H (1966) Remark on mean curvature. J London Math Soc 41: 364-366.
    [67] Fleming WH (1962) On the oriented Plateau problem. Rend Circolo Mat Palermo 9: 69-89.
    [68] Fraser A, Schoen R (2011) The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv Math 226: 4011-4030.
    [69] Fraser A, Schoen R (2016) Sharp eigenvalue bounds and minimal surfaces in the ball. Invent Math 203: 823-890.
    [70] Gálvez JA, Rosenberg H (2010) Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces. Am J Math 132: 1249-1273.
    [71] Grigor'yan A (1999) Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. B Am Math Soc 36: 135-249.
    [72] Greene RE, Wu H (1979) C∞ approximations of convex, subharmonic, and plurisubharmonic functions. Ann Sci école Norm Sup 12: 47-84.
    [73] Guan B, Spruck J (2000) Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity. Am J Math 122: 1039-1060.
    [74] Heinonen E (2019) Survey on the asymptotic Dirichlet problem for the minimal surface equation. arXiv: 1909.08437.
    [75] Heinz E (1955) Uber Flächen mit eindeutiger projektion auf eine ebene, deren krümmungen durch ungleichungen eingschr?nkt sind. Math Ann 129: 451-454.
    [76] Heinz E (1952) über die L?sungen der Minimalfl?chengleichung. Nachr Akad Wiss G?ttingen. Math-Phys Kl Math-Phys-Chem Abt 1952: 51-56.
    [77] Holopainen I, Ripoll JB (2015) Nonsolvability of the asymptotic Dirichlet problem for some quasilinear elliptic PDEs on Hadamard manifolds. Rev Mat Iberoam 31: 1107-1129.
    [78] Hopf E (1950) On S. Bernstein's theorem on surfaces z(x, y) of nonpositive curvature. P Am Math Soc 1: 80-85.
    [79] Impera D, Pigola S, Setti AG (2017) Potential theory for manifolds with boundary and applications to controlled mean curvature graphs. J Reine Angew Math 733: 121-159.
    [80] Keller JB (1957) On solutions of Δu = f (u). Commun Pure Appl Math 10: 503-510.
    [81] Korevaar NJ (1986) An easy proof of the interior gradient bound for solutions of the prescribed mean curvature equation, In: Proceedings of Symposia in Pure Mathematics, 45: 81-89.
    [82] Li H, Xiong C (2018) Stability of capillary hypersurfaces in a Euclidean ball. Pac J Math 297: 131-146.
    [83] Li H, Xiong C (2017) Stability of capillary hypersurfaces with planar boundaries. J Geom Anal 27: 79-94.
    [84] Li P, Tam LF (1992) Harmonic functions and the structure of complete manifolds. J Differ Geom 35: 359-383.
    [85] Li P, Wang J (2001) Complete manifolds with positive spectrum. J Differ Geom 58: 501-534.
    [86] Li P, Wang J (2002) Complete manifolds with positive spectrum. II. J Differ Geom 62: 143-162.
    [87] Li P, Wang J (2001) Finiteness of disjoint minimal graphs. Math Res Lett 8: 771-777.
    [88] Li P, Wang J (2004) Stable minimal hypersurfaces in a nonnegatively curved manifold. J Reine Angew Math 566: 215-230.
    [89] López R (2001) Constant mean curvature graphs on unbounded convex domains. J Differ Equations 171: 54-62.
    [90] López R (2014) Capillary surfaces with free boundary in a wedge. Adv Math 262: 476-483.
    [91] Mari L, Rigoli M, Setti AG (2010) Keller-Osserman conditions for diffusion-type operators on Riemannian Manifolds. J Funct Anal 258: 665-712.
    [92] Mari L, Rigoli M, Setti AG (2019) On the 1/H-flow by p-Laplace approximation: new estimates via fake distances under Ricci lower bounds. arXiv: 1905.00216.
    [93] Mari L, Pessoa LF (2020) Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations. Commun Anal Geom 28: 395-497.
    [94] Mari L, Pessoa LF (2019) Maximum principles at infinity and the Ahlfors-Khas'minskii duality: An overview, In: Contemporary Research in Elliptic PDEs and Related Topics, Cham: Springer, 419-455.
    [95] Mari L, Valtorta D (2013) On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting. T Am Math Soc 365: 4699-4727.
    [96] Mickle EJ (1950) A remark on a theorem of Serge Bernstein. P Am Math Soc 1: 86-89.
    [97] Miklyukov V, Tkachev V (1996) Denjoy-Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces. Commun Anal Geom 4: 547-587.
    [98] Mitidieri E, Pokhozhaev SI (2001) A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr Mat Inst Steklova 234: 1-384.
    [99] Moser J (1961) On Harnack's theorem for elliptic differential equations. Commun Pure Appl Math 14: 577-591.
    [100] Naito Y, Usami H (1997) Entire solutions of the inequality div (A(|?u|)?u) ≥ f (u). Math Z 225: 167-175.
    [101] Nelli B, Rosenberg H (2002) Minimal surfaces in $\mathbb{H}.2$ × $\mathbb{R}$. B Braz Math Soc 33: 263-292.
    [102] Nitsche JCC (1957) Elementary proof of Bernstein's theorem on minimal surfaces. Ann Math 66: 543-544.
    [103] Nunes I (2017) On stable constant mean curvature surfaces with free boundary. Math Z 287: 473-479.
    [104] Omori H (1967) Isometric immersions of Riemannian manifolds. J Math Soc JPN 19: 205-214.
    [105] Osserman R (1957) On the inequality Δu ≥ f (u). Pac J Math 7: 1641-1647.
    [106] Pigola S, Rigoli M, Setti AG (2002) Some remarks on the prescribed mean curvature equation on complete manifolds. Pac J Math 206: 195-217.
    [107] Pigola S, Rigoli M, Setti AG (2005) Maximum principles on Riemannian manifolds and applications. Mem Am Math Soc 174: 822.
    [108] Pigola S, Rigoli M, Setti AG (2008) Vanishing and Finiteness Results in Geometric Analysis. A Generalization of the B?chner Technique, Birk?user.
    [109] Pogorelov A (1981) On the stability of minimal surfaces. Soviet Math Dokl 24: 274-276.
    [110] Rigoli M, Setti AG (2001) Liouville-type theorems for?-subharmonic functions. Rev Mat Iberoam 17: 471-520.
    [111] Ripoll J, Telichevesky M (2019) On the asymptotic Plateau problem for CMC hypersurfaces in hyperbolic space. B Braz Math Soc 50: 575-585.
    [112] Ripoll J, Telichevesky M (2015) Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems. T Am Math Soc 367: 1523-1541.
    [113] Ros A, Ruiz D, Sicbaldi P (2017) A rigidity result for overdetermined elliptic problems in the plane. Commun Pure Appl Math 70: 1223-1252.
    [114] Ros A, Ruiz D, Sicbaldi P (2020) Solutions to overdetermined elliptic problems in nontrivial exterior domains. J Eur Math Soc 22: 253-281.
    [115] Ros A, Souam R (1997) On stability of capillary surfaces in a ball. Pac J Math 178: 345-361.
    [116] Ros A, Vergasta E (1995) Stability for hypersurfaces of constant mean curvature with free boundary. Geom Dedicata 56: 19-33.
    [117] Rosenberg H, Schulze F, Spruck J (2013) The half-space property and entire positive minimal graphs in M × $\mathbb{R}$. J Differ Geom 95: 321-336.
    [118] Salavessa I (1989) Graphs with parallel mean curvature. P Am Math Soc 107: 449-458.
    [119] Schoen R, Yau ST (1976) Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature. Comment Math Helv 51: 333-341.
    [120] Serrin J (1971) A symmetry theorem in potential theory. Arch Ration Mech Anal 43: 304-318.
    [121] Serrin J (2009) Entire solutions of quasilinear elliptic equations. J Math Anal Appl 352: 3-14.
    [122] Sicbaldi P (2010) New extremal domains for the first eigenvalue of the Laplacian in flat tori. Calc Var 37: 329-344.
    [123] Simon L (1997) The minimal surface equation. In: Encyclopaedia of Mathematical Sciences, Geometry, V, Berlin: Springer, 239-272.
    [124] Simon L (1989) Entire solutions of the minimal surface equation. J Differ Geom 30: 643-688.
    [125] Simons J (1968) Minimal varieties in Riemannian manifolds. Ann Math 88: 62-105.
    [126] Sternberg P, Zumbrun K (1998) A Poincaré inequality with applications to volume constrained area-minimizing surfaces. J Reine Angew Math 503: 63-85.
    [127] Sung CJA, Wang J (2014) Sharp gradient estimate and spectral rigidity for p-Laplacian. Math Res Lett 21: 885-904.
    [128] Tkachev VG (1992) Some estimates for the mean curvature of nonparametric surfaces defined over domains in Rn. Ukr Geom Sb 35: 135-150.
    [129] Tkachev VG (1991) Some estimates for the mean curvature of graphs over domains in $\mathbb{R}.n$. Dokl Akad Nauk SSSR 314: 140-143.
    [130] Usami H (1994) Nonexistence of positive entire solutions for elliptic inequalities of the mean curvature type. J Differ Equations 111: 472-480.
    [131] Yau ST (1975) Harmonic functions on complete Riemannian manifolds. Commun Pure Appl Math 28: 201-228.
    [132] Wang G, Xia C (2019) Uniqueness of stable capillary hypersurfaces in a ball. Math Ann 374: 1845-1882.
    [133] Wang JF (2003) How many theorems can be derived from a vector function - On uniqueness theorems for the minimal surface equation. Taiwanese J Math 7: 513-539.
    [134] Wang XJ (2011) Convex solutions to the mean curvature flow. Ann Math 173: 1185-1239.
    [135] Weinberger HF (1971) Remark on the preceding paper of Serrin. Arch Ration Mech Anal 43: 319-320.
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematics in Engineering
1.4 2.2

Metrics

Article views(4281) PDF downloads(404) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog