On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces
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- by Roberto Alonso Núñez PDF
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Abstract:
A theorem of Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalar curvature $R\geq \frac {2}{3}H^2$, then $R=H^2$, $R=\frac {8}{9}H^2$ or $R=\frac {2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $R=H^2$ and $R=\frac {8}{9}H^2$, and provide an example in the case $R=\frac {2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well-known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.References
- Hilário Alencar and Manfredo do Carmo, Hypersurfaces with constant mean curvature in spheres, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1223–1229. MR 1172943, DOI 10.1090/S0002-9939-1994-1172943-2
- João Lucas Marques Barbosa and Antônio Gervasio Colares, Stability of hypersurfaces with constant $r$-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277–297. MR 1456513, DOI 10.1023/A:1006514303828
- Qing Ming Cheng and Qian Rong Wan, Complete hypersurfaces of $\textbf {R}^4$ with constant mean curvature, Monatsh. Math. 118 (1994), no. 3-4, 171–204. MR 1309647, DOI 10.1007/BF01301688
- S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
- Shiu Yuen Cheng and Shing Tung Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195–204. MR 431043, DOI 10.1007/BF01425237
- Philip Hartman, On complete hypersurfaces of nonnegative sectional curvatures and constant $m$th mean curvature, Trans. Amer. Math. Soc. 245 (1978), 363–374. MR 511415, DOI 10.1090/S0002-9947-1978-0511415-8
- Xuan Guo Huang, Complete hypersurfaces with constant scalar curvature and constant scalar curvature and constant mean curvature in $\textbf {R}^4$, Chinese Ann. Math. Ser. B 6 (1985), no. 2, 177–184. A Chinese summary appears in Chinese Ann. Math. Ser. A 6 (1985), no. 2, 255. MR 841866
- 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
- T. Levi-Civita, Famiglie di superficie isoparametriche nell’ordinario spazio euclideo, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 26 (1937) 355–362.
- Haizhong Li, Hypersurfaces with constant scalar curvature in space forms, Math. Ann. 305 (1996), no. 4, 665–672. MR 1399710, DOI 10.1007/BF01444243
- Katsumi Nomizu and Brian Smyth, A formula of Simons’ type and hypersurfaces with constant mean curvature, J. Differential Geometry 3 (1969), 367–377. MR 266109
- Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI 10.2307/2373587
- Patrick J. Ryan, Hypersurfaces with parallel Ricci tensor, Osaka Math. J. 8 (1971), 251–259. MR 296859
- B. Segre, Famiglie di ipersuperficie isoparametriche negli spazi euclidei ad un qualunque numero di dimensioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (6) 27 (1938) 203–207.
- 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
Additional Information
- Roberto Alonso Núñez
- Affiliation: Rua Dr. Paulo Alves 110, Bl C, Apto. 402 24210-445 Niterói, Rio de Janeiro, Brazil
- Email: alonso_nunez@id.uff.br
- Received by editor(s): June 2, 2016
- Published electronically: February 10, 2017
- Communicated by: Lei Ni
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2677-2688
- MSC (2010): Primary 53C40; Secondary 53C42
- DOI: https://doi.org/10.1090/proc/13525
- MathSciNet review: 3626520