## Taiwanese Journal of Mathematics

### A Classification Theorem for Complete PMC Surfaces with Non-negative Gaussian Curvature in $M^n(c) \times \mathbb{R}$

#### Abstract

Let $M^{n}(c)$ be an $n$-dimensional space form with constant sectional curvature $c$. Alencar-do Carmo-Tribuzy [5] classified all parallel mean curvature (abbrev. PMC) surfaces with non-negative Gaussian curvature $K$ in $M^n(c) \times \mathbb{R}$ with $c \lt 0$. Later on, Fetcu-Rosenberg [28] generalized their results for $c \neq 0$. However, the classification to PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ is still open. In this paper, we give a complete classification to the PMC surfaces in $M^n(c) \times \mathbb{R}$ with $K \equiv 0$ whose tangent plane spans the constant angle with factor $\mathbb{R}$.

#### Article information

Source
Taiwanese J. Math., Volume 20, Number 1 (2016), 205-226.

Dates
First available in Project Euclid: 1 July 2017

https://projecteuclid.org/euclid.twjm/1498874429

Digital Object Identifier
doi:10.11650/tjm.20.2016.5766

Mathematical Reviews number (MathSciNet)
MR3462875

Zentralblatt MATH identifier
1357.53067

#### Citation

Hou, Zhong Hua; Qiu, Wang Hua. A Classification Theorem for Complete PMC Surfaces with Non-negative Gaussian Curvature in $M^n(c) \times \mathbb{R}$. Taiwanese J. Math. 20 (2016), no. 1, 205--226. doi:10.11650/tjm.20.2016.5766. https://projecteuclid.org/euclid.twjm/1498874429

#### References

• U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$, Acta Math. 193 (2004), no. 2, 141–174.
• ––––, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28.
• H. Alencar, M. do Carmo, I. Fernández and R. Tribuzy, A theorem of H. Hopf and the Cauchy-Riemann inequality II, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 4, 525–532.
• H. Alencar, M. do Carmo and R. Tribuzy, A theorem of Hopf and the Cauchy-Riemann inequality, Comm. Anal. Geom. 15 (2007), no. 2, 283–298.
• ––––, A Hopf theorem for ambient spaces of dimensions higher than three, J. Differential Geom. 84 (2010), no. 1, 1–17.
• ––––, Surfaces of $M^2_k \times \mb{R}$ invariant under a one-parameter group of isometries, Ann. Mat. Pura Appl. (4) 193 (2014), no. 2, 517–527.
• C. P. Aquino, H. F. de Lima and E. A. Lima, Jr., On the angle of complete CMC hypersurfaces in Riemannian product spaces, Differential Geom. Appl. 33 (2014), 139–148.
• J. O. Baek, Q.-M. Cheng and Y. J. Suh, Complete space-like hypersurfaces in locally symmetric Lorentz spaces, J. Geom. Phys. 49 (2004), no. 2, 231–247.
• M. Batista, Simons type equation in $\mb{S}^{2} \times \mb{R}$ and $\mb{H}^{2} \times \mb{R}$ and applications, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1299–1322.
• M. Batista, M. P. Cavalcante and D. Fetcu, Constant Mean Curvature Surfaces in $\mb{M}^2(c) \times \mb{R}$ and Finite Total Curvature, arXiv:1402.1231.
• M. do Carmo, Some recent developments on Hopf's holomorphic form, Results Math. 60 (2011), no. 1-4, 175–183.
• M. do Carmo and I. Fernández, A Hopf theorem for open surfaces in product spaces, Forum Math. 21 (2009), no. 6, 951–963.
• B.-Y. Chen, Riemannian submanifolds: A Survey, in: Handbook of Differential Geometry, Vol. I, 187–418, North-Holland, Amsterdam, 2000.
• H. Chen, G. Y. Chen and H. Z. Li, Some pinching theorems for minimal submanifolds in $\mb{S}^{m}(1) \times \mb{R}$, Sci. China Math. 56 (2013), no. 8, 1679–1688.
• S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), no. 3, 195–204.
• S. S. Chern, On surfaces of constant mean curvature in a three-dimensional space of constant curvature, in: Geometric Dynamics (Rio de Janeiro, 1981), 104–108, Lecture Notes in Math. 1007, Springer, Berlin, 1983.
• S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in: Functional Analysis and Related Fields, 59–75, Springer, New York, 1970.
• B. Daniel, Isometric immersions into $\mb{S}^{n} \times \mb{R}$ and $\mb{H}^{n} \times \mb{R}$ and applications to minimal surfaces, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6255–6282.
• F. Dillen, J. Fastenakels and J. Van der Veken, Surfaces in $\mb{S}^2 \times \mb{R}$ with a canonical principal direction, Ann. Global Anal. Geom. 35 (2009), no. 4, 381–396.
• F. Dillen, J. Fastenakels, J. Van der Veken and L. Vrancken, Constant angle surfaces in $\mb{S}^2 \times \mb{R}$, Monatsh. Math. 152 (2007), no. 2, 89–96.
• F. Dillen and M. I. Munteanu, Constant angle surfaces in $\mb{H}^2 \times \mb{R}$, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 1, 85–97.
• F. Dillen, M. I. Munteanu and A. I. Nistor, Canonical coordinates and principal directions for surfaces in $\mb{H}^2 \times \mb{R}$, Taiwanese J. Math. 15 (2011), no. 5, 2265–2289.
• J. M. Espinar, J. A. Gálvez and H. Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009), no. 2, 351–386.
• J. M. Espinar and H. Rosenberg, Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 \times \mb{R}$, J. Differential Geom. 82 (2009), no. 3, 611–628.
• ––––, Complete constant mean curvature surfaces in homogeneous spaces, Comment. Math. Helv. 86 (2011), no. 3, 659–674.
• M. J. Ferreira and R. Tribuzy, Parallel mean curvature surfaces in symmetric spaces, Ark. Mat. 52 (2014), no. 1, 93–98.
• D. Fetcu, C. Oniciuc and H. Rosenberg, Biharmonic submanifolds with parallel mean curvature in $\mb{S}^{n} \times \mb{R}$, J. Geom. Anal. 23 (2013), no. 4, 2158–2176.
• D. Fetcu and H. Rosenberg, A note on surfaces with parallel mean curvature, C. R. Math. Acad. Sci. Paris 349 (2011), no. 21-22, 1195–1197.
• ––––, Surfaces with parallel mean curvature in $\mb{S}^{3} \times \mb{R}$ and $\mb{H}^{3} \times \mb{R}$, Michigan Math. J. 61 (2012), no. 4, 715–729.
• ––––, On complete submanifolds with parallel mean curvature in product spaces, Rev. Mat. Iberoam. 29 (2013), no. 4, 1283–1306.
• Y. Fu and A. I. Nistor, Constant angle property and canonical principal directions for surfaces in $\mb{M}^2(c) \times \mb{R}_1$, Mediterr. J. Math. 10 (2013), no. 2, 1035–1049.
• H. Hopf, Differential Geometry in the Large, Lectures Notes in Mathematics 1000, Springer-Verlag, Berlin, 1983.
• Z. H. Hou and W.-H. Qiu, Submanifolds with parallel mean curvature vector field in product spaces, Vietnam J. Math. 43 (2015), no. 4, 705–723.
• A. Huber, On subharmonic functions and differential geometry in the large, Comment. Math. Helv. 32 (1957), no. 1, 13–72.
• H. F. de Lima and E. A. Lima, Jr., Generalized maximum principles and the unicity of complete spacelike hypersurfaces immersed in a Lorentzian product space, Beitr. Algebra Geom. 55 (2014), no. 1, 59–75.
• K. Nomizu and B. Smyth, A formula of Simons' type and hypersurfaces with constant mean curvature, J. Differential Geom. 3 (1969), 367–377.
• J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), no. 1, 62–105.
• B. Smyth, Submanifolds of constant mean curvature, Math. Ann. 205 (1973), no. 4, 265–280.
• S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), no. 2, 201–228.