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Height and Area Estimates for Constant Mean Curvature Graphs in \(\mathbb {E}(\kappa ,\tau )\)-Spaces

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Abstract

We obtain area growth estimates for constant mean curvature graphs in \(\mathbb {E}(\kappa ,\tau )\)-spaces with \(\kappa \le 0\), by finding sharp upper bounds for the volume of geodesic balls in \(\mathbb {E}(\kappa ,\tau )\). We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in \(\mathbb {E}(\kappa ,\tau )\) with critical mean curvature have at most cubic intrinsic area growth. We also obtain sharp upper bounds for the extrinsic area growth of graphs with zero boundary values, and study distinguished examples in detail such as invariant surfaces, k-noids, and ideal Scherk graphs. Finally, we give a relation between height and area growth of minimal graphs in the Heisenberg space (\(\kappa =0\)), and prove a Collin–Krust type estimate for such minimal graphs.

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References

  1. Albujer, A.L., Caballero, M., López, R.: Convexity of the solutions to the constant mean curvature spacelike surface equation in the Lorentz-Minkowski space. J. Differ. Equ. 258(7), 2364–2374 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bekkar, M.: Examples of minimal surfaces in Heisenberg space (French). Rend. Sem. Fac. Sci. Univ. Cagliari. 61(2), 123–130 (1991)

    MathSciNet  MATH  Google Scholar 

  3. Bekkar, M., Sari, T.: Ruled minimal surfaces in the Heisenberg space (French). Rend. Sem. Mat. Univ. Politec. Torino. 50(3), 243–254 (1993)

    MATH  Google Scholar 

  4. Capogna, L., Danielli, D., Pauls, S. D., Tyson, J.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Progress in Mathematics, vol. 259. Birkhäuser (2007). ISBN 978-3-7643-8132-5

  5. Cartier, S.: Saddle towers in Heisenberg space. Asian J. Math. (to appear)

  6. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math. (2) 104(3), 407–419 (1976)

  8. Collin, P., Krust, R.: Le problème de Dirichlet pour l’équation des surfaces minimales sur des domaines non bornés. Bull. Soc. Math. France 119(4), 443–462 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Collin, P., Rosenberg, H.: Construction of harmonic diffeomorphisms and minimal graphs. Ann. Math. (2) 172(3), 1879–1906 (2010)

  10. Daniel, B.: Isometric immersions into 3-dimensional homogeneous manifolds. Comment. Math. Helv. 82(1), 87–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Daniel, B.: The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Not. 2011(3), 674–695 (2011)

    MathSciNet  MATH  Google Scholar 

  12. Daniel, B., Hauswirth, L.: Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group. Proc. Lond. Math. Soc. 98(2), 445–470 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dillen, F., Van der Woewtyne, I., Vestraelen, L., Walrave, J.: Ruled surfaces of constant mean curvature in \(3\)-dimensional Minkowski space. Geometry and Topology of Submanifolds, vol. VIII, 145–147. World Scientific, River Edge (1996)

  14. Elbert, M.F., Rosenberg, H.: Minimal surfaces in \(M\times {\mathbb{R}}\). Ann. Glob. Anal. Geom. 34(1), 39–53 (2008)

    Article  MATH  Google Scholar 

  15. Espinar, J.M.: Finite index operators on surfaces. J. Geom. Anal. 23(1), 415–437 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fernández, I., Mira, P.: Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space. Trans. Am. Math. Soc. 361(11), 5737–5752 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Folha, A., Melo, S.: The Dirichlet problem for constant mean curvature graphs in \({\mathbb{H}}\times \mathbb{R}\) over unbounded domains. Pac. J. Math. 251(1), 37–65 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Figueroa, C., Mercuri, F., Pedrosa, R.H.L.: Invariant surfaces of the Heisenberg groups. Ann. Mat. Pura Appl. 4(177), 173–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hartman, P.: Geodesic parallel coordinates in the large. Am. J. Math. 86, 705–727 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jang, C., Park, J., Park, K.: Geodesic spheres and balls of the Heisenberg groups. Commun. Korean Math. Soc. 25(1), 83–96 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leandro, C., Rosenberg, H.: Removable singularities for sections of Riemannian submersions of prescribed mean curvature. Bull. Sci. Math. 133(4), 445–452 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee, H.: Extensions of the duality between minimal surfaces and maximal surfaces. Geom. Dedicata 151, 373–386 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lee, H., Manzano, J.M.: Generalized Calabi’s correspondence and complete spacelike surfaces. Preprint available at arXiv:1301.7241 [math.DG] (2013)

  24. Manzano, J.M.: On the classification of Killing submersions and their isometries. Pac. J. Math. 270(2), 367–692 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Manzano, J.M., Pérez, J., Rodríguez, M.: Parabolic stable surfaces with constant mean curvature. Calc. Var. Partial Differ. Eq. 42(1–2), 137–152 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Manzano, J.M., Rodríguez, M.: On complete constant mean curvature vertical multigraphs in \({\mathbb{E}}(\kappa,\tau )\). J. Geom. Anal. 25(1), 336–346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Melo, S.: Minimal graphs in \(\widetilde{{\rm PSL}}_2({\mathbb{R}})\) over unbounded domains. Bull. Braz. Math. Soc. 45(2), 91–116 (2014)

    Article  MathSciNet  Google Scholar 

  28. Morabito, F., Rodríguez, M.: Saddle towers and minimal \(k\)-noids in \({\mathbb{H}}^2\times {\mathbb{R}}\). J. Inst. Math. Jussieu 11(2), 333–349 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nelli, B., Sa Earp, R., Toubiana, E.: Minimal graphs in \({{\rm Nil}}_3\): existence and non-existence results. Calc. Var. Partial Differ. Equ. 56(2), 56:27 (2017)

  30. Plehnert, J.: Constant mean curvature \(k\)-noids in homogeneous manifolds. Illinois J. Math. 58(1), 233–249 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Pyo, J.: New complete embedded minimal surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\). Ann. Glob. Anal. Geom. 40(2), 167–176 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosenberg, H., Souam, R., Toubiana, E.: General curvature estimates for stable H-surfaces in 3-manifolds and applications. J. Differ. Geom. 84(3), 623–648 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Treibergs, A.: Entire spacelike hypersurfaces on constant mean curvature in Minkowski space. Invent. Math. 66(1), 39–56 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Massimiliano Pontecorvo for his kind hospitality during the preparation of this work, as well as the referee for reading thoroughly this manuscript pointing out some insightful remarks. This research was partially supported by PRIN-2010NNBZ78-009 and INdAM-GNSAGA. The first author was also partially supported by Spanish MCyT-Feder Research Project MTM2014-52368-P, and by the EPSRC Grant No. EP/M024512/1.

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Authors and Affiliations

  1. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    José M. Manzano

  2. Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università dell’Aquila, Via Vetoio Loc. Coppito, 67100, L’Aquila, Italy

    Barbara Nelli

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  1. José M. Manzano
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  2. Barbara Nelli
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Correspondence to José M. Manzano.

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Manzano, J.M., Nelli, B. Height and Area Estimates for Constant Mean Curvature Graphs in \(\mathbb {E}(\kappa ,\tau )\)-Spaces. J Geom Anal 27, 3441–3473 (2017). https://doi.org/10.1007/s12220-017-9810-7

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  • DOI: https://doi.org/10.1007/s12220-017-9810-7

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