Heinz type estimates for graphs in Euclidean space
HTML articles powered by AMS MathViewer
- by Francisco Fontenele PDF
- Proc. Amer. Math. Soc. 138 (2010), 4469-4478 Request permission
Abstract:
Let $M^n$ be an entire graph in the Euclidean $(n+1)$-space $\mathbb R^{n+1}$. Denote by $H$, $R$ and $|A|$, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of $M^n$. We prove that if the mean curvature $H$ of $M^n$ is bounded, then $\inf _M|R|=0$, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of $M^n$ is negative, then $\inf _M|A|=0$. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate $\inf |H|,\;\inf |R|$ and $\inf |A|$ for graphs in $\mathbb R^{n+1}$ of $C^2$ real-valued functions defined on closed balls in $\mathbb R^n$.References
- J. L. M. Barbosa, G. P. Bessa, and J. F. Montenegro, On Bernstein-Heinz-Chern-Flanders inequalities, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 2, 457–464. MR 2405902, DOI 10.1017/S0305004107000643
- Shiing-shen Chern, On the curvatures of a piece of hypersurface in euclidean space, Abh. Math. Sem. Univ. Hamburg 29 (1965), 77–91. MR 188949, DOI 10.1007/BF02996311
- Marcos Dajczer, Submanifolds and isometric immersions, Mathematics Lecture Series, vol. 13, Publish or Perish, Inc., Houston, TX, 1990. Based on the notes prepared by Mauricio Antonucci, Gilvan Oliveira, Paulo Lima-Filho and Rui Tojeiro. MR 1075013
- N. V. Efimov, Hyperbolic problems in the theory of surfaces, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 177–188 (Russian). MR 0238232
- Maria Fernanda Elbert, On complete graphs with negative $r$-mean curvature, Proc. Amer. Math. Soc. 128 (2000), no. 5, 1443–1450. MR 1712913, DOI 10.1090/S0002-9939-00-05671-9
- Harley Flanders, Remark on mean curvature, J. London Math. Soc. 41 (1966), 364–366. MR 193600, DOI 10.1112/jlms/s1-41.1.364
- F. Fontenele and Sérgio L. Silva, A tangency principle and applications, Illinois J. Math. 45 (2001), no. 1, 213–228. MR 1849995, DOI 10.1215/ijm/1258138264
- F. Fontenele and Sérgio L. Silva, Sharp estimates for the size of balls in the complement of a hypersurface, Geom. Dedicata 115 (2005), 163–179. MR 2180046, DOI 10.1007/s10711-005-6909-y
- Lars Gȧrding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957–965. MR 0113978, DOI 10.1512/iumj.1959.8.58061
- Thomas Hasanis and Theodoros Vlachos, Curvature properties of hypersurfaces, Arch. Math. (Basel) 82 (2004), no. 6, 570–576. MR 2080296, DOI 10.1007/s00013-003-4648-6
- Erhard Heinz, Über Flächen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind, Math. Ann. 129 (1955), 451–454 (German). MR 71822, DOI 10.1007/BF01362385
- Tilla Klotz and Robert Osserman, Complete surfaces in $E^{3}$ with constant mean curvature, Comment. Math. Helv. 41 (1966/67), 313–318. MR 211332, DOI 10.1007/BF02566886
- Takashi Okayasu, $\textrm {O}(2)\times \textrm {O}(2)$-invariant hypersurfaces with constant negative scalar curvature in $E^4$, Proc. Amer. Math. Soc. 107 (1989), no. 4, 1045–1050. MR 990430, DOI 10.1090/S0002-9939-1989-0990430-7
- Robert C. Reilly, On the Hessian of a function and the curvatures of its graph, Michigan Math. J. 20 (1973), 373–383. MR 334045
- Isabel Maria da Costa Salavessa, Graphs with parallel mean curvature, Proc. Amer. Math. Soc. 107 (1989), no. 2, 449–458. MR 965247, DOI 10.1090/S0002-9939-1989-0965247-X
- Brian Smyth and Frederico Xavier, Efimov’s theorem in dimension greater than two, Invent. Math. 90 (1987), no. 3, 443–450. MR 914845, DOI 10.1007/BF01389174
- Shing Tung Yau, Submanifolds with constant mean curvature. I, II, Amer. J. Math. 96 (1974), 346–366; ibid. 97 (1975), 76–100. MR 370443, DOI 10.2307/2373638
- Shing Tung Yau (ed.), Seminar on Differential Geometry, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR 645728
Additional Information
- Francisco Fontenele
- Affiliation: Departamento de Geometria, Universidade Federal Fluminense, 24020-140, Niterói, Rio de Janeiro, Brazil
- Email: fontenele@mat.uff.br
- Received by editor(s): August 31, 2009
- Published electronically: August 2, 2010
- Additional Notes: This work was partially supported by CNPq (Brazil)
- Communicated by: Jon G. Wolfson
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 4469-4478
- MSC (2010): Primary 53A07, 53C42; Secondary 53A05, 35B50
- DOI: https://doi.org/10.1090/S0002-9939-2010-10590-7
- MathSciNet review: 2680071
Dedicated: To my wife, Andrea