ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Original Paper

Biharmonic submanifolds with mean parallel curvature on Mm(c)×R

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.03.008
More Information
  • Author Bio:

    Mi Rong, female, born in 1993, a doctoral candidate. Research field: differential geometry. E-mail: mr8231227@163.com.

  • Received Date: 20 April 2019
  • Accepted Date: 12 July 2019
  • Rev Recd Date: 12 July 2019
  • Publish Date: 31 March 2020
  • Let Mn be an n-dimensional submanifold with parallel mean curvature H of product space form Mm(c)×R, where Mm(c) is a space form with constant sectional curvature c. By using the method of Simons inequality, a series of results are obtained.
    Let Mn be an n-dimensional submanifold with parallel mean curvature H of product space form Mm(c)×R, where Mm(c) is a space form with constant sectional curvature c. By using the method of Simons inequality, a series of results are obtained.
  • loading
  • [1]
    EELLS J, SAMPSON J. H. Harmonic mappings of Riemannian manifolds[J]. Amer. J. Math., 1964, 86: 109-160.
    [2]
    JIANG G Y. 2-Harmonic maps and their first and second variational formulas[J]. Chinese Ann. Math. Ser. A(7), 1986, 4: 389-402.
    [3]
    JIANG G Y. 2-Harmonic isometric immersion between Riemannian manifolds[J]. Chinese Ann. Math. Ser. A(7), 1986, 2: 130-144.
    [4]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Properties of biharmonic submanifolds in spheres[J]. J. Geom. Symmetry Phys., 2010, 17: 87-102.
    [5]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Classification results and new examples of proper biharmonic submanifolds in spheres[J]. Note Mat., 2008, 28: 49-61.
    [6]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Biharmonic hpersurfaces in 4-dimensional space forms[J]. Math. Nachr., 2010, 283: 1696-1705.
    [7]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Classification results for biharmonic submanifolds in spheres[J]. Israel J. Math., 2008, 168, 201-220.
    [8]
    NAKAUCHI N, URAKAWA H. Biharmonic hpersurfaces in a Riemannian manifold with nonpositive Ricci curvature[J]. Ann. Glob. Anal. Geom., 2011, 40, 125-131.
    [9]
    WANG X F, WU L. Proper biharmonic submanifolds in a sphere[J]. Acta Math. Sin. (Engl. Ser)., 2012, 28: 205-218.
    [10]
    CADDEO R, MONTALDO S, ONICIUSC S. Biharmonic submanifolds in spheres[J]. Isreal J. Math., 2002, 130: 109-123.
    [11]
    ZHANG W. New examples of biharmonic submanifolds in CPn and S2n+1[J]. An. Stiint. Univ. Al. I. Cuza Iasi Mat(N.S.), 2011, 57: 207-218.
    [12]
    SASAHARA T. Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms[J]. Glasg. Math. J., 2007, 49: 497-507.
    [13]
    VRANCKEN L. Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space froms[J]. Proc. Amer. Math. Soc., 2002, 130: 1459-1466.
    [14]
    FETCU D, LOUBEAU E, MONTALDO S, et al. Biharmonic submanifolds of CPn[J]. Z. Math., 2010, 266: 505-531.
    [15]
    FETCU D. ONICIUC C. Explicit formulas for biharmonic submanifolds in Sasakian space forms[J]. Pacific J. Math., 2009, 24: 85-107.
    [16]
    OU Y L, WANG Z P. Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries[J]. J. Geom. Phys., 2006, 228: 185-199.
    [17]
    ABRESH U, ROSENBER H. The Hopf differential for constant mean curvature surfaces in S2×R and H2×R[J]. Acta Math., 2004, 193: 141-174.
    [18]
    ABRESH U, ROSENBER H. Generalized Hopf differentials[J]. Mat. Contemp., 2005, 28: 1-28.
    [19]
    ALENCAR H, DO CARMO M, TRIBUZY R. A Hopf theorem for ambient spaces of dimensions higher than three[J]. J. Differential Geom., 2010, 84: 1-17.
    [20]
    FETCU D, ONICIUC C, ROSENBEG H. Biharmonic submanifolds with parallel mean curvature in Sn× R[J]. J. Geom. Anal., 2013, 23: 2158-2176.
    [21]
    ROTH J. A note on biharmonic submanifolds of product spaces[J]. J. Geom., 2013, 104: 375C-381.
    [22]
    DANIEL B. Isometric immersions into SnR and Hn×R and applications to minimal surfaces[J]. Trans. Amer. Math. Soc., 2004, 361: 6255-6282.
    [23]
    DILLEN F, FASTENAKELS J, DER VEKEN VAN J. Surfaces inS2×R with a canonical principal direction[J]. Ann. lobal Anal. Eom., 2009, 35: 381-395.
    [24]
    Dillen F., Munteanu M.,: Constant angle surfaces inH2×R[J]. Bull. Braz. Math. Soc. (NS), 2009, 40: 85-97.
    [25]
    LI A M, LI J M. An instrinsic rigidity theorem for minimal submanifolds in a sphere[J]. Arch. Math. (Basel), 1992, 58: 582-594.
  • 加载中

Catalog

    [1]
    EELLS J, SAMPSON J. H. Harmonic mappings of Riemannian manifolds[J]. Amer. J. Math., 1964, 86: 109-160.
    [2]
    JIANG G Y. 2-Harmonic maps and their first and second variational formulas[J]. Chinese Ann. Math. Ser. A(7), 1986, 4: 389-402.
    [3]
    JIANG G Y. 2-Harmonic isometric immersion between Riemannian manifolds[J]. Chinese Ann. Math. Ser. A(7), 1986, 2: 130-144.
    [4]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Properties of biharmonic submanifolds in spheres[J]. J. Geom. Symmetry Phys., 2010, 17: 87-102.
    [5]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Classification results and new examples of proper biharmonic submanifolds in spheres[J]. Note Mat., 2008, 28: 49-61.
    [6]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Biharmonic hpersurfaces in 4-dimensional space forms[J]. Math. Nachr., 2010, 283: 1696-1705.
    [7]
    BALMU??塁 A, MONTALDO S, ONICIUC C. Classification results for biharmonic submanifolds in spheres[J]. Israel J. Math., 2008, 168, 201-220.
    [8]
    NAKAUCHI N, URAKAWA H. Biharmonic hpersurfaces in a Riemannian manifold with nonpositive Ricci curvature[J]. Ann. Glob. Anal. Geom., 2011, 40, 125-131.
    [9]
    WANG X F, WU L. Proper biharmonic submanifolds in a sphere[J]. Acta Math. Sin. (Engl. Ser)., 2012, 28: 205-218.
    [10]
    CADDEO R, MONTALDO S, ONICIUSC S. Biharmonic submanifolds in spheres[J]. Isreal J. Math., 2002, 130: 109-123.
    [11]
    ZHANG W. New examples of biharmonic submanifolds in CPn and S2n+1[J]. An. Stiint. Univ. Al. I. Cuza Iasi Mat(N.S.), 2011, 57: 207-218.
    [12]
    SASAHARA T. Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms[J]. Glasg. Math. J., 2007, 49: 497-507.
    [13]
    VRANCKEN L. Minimal Lagrangian submanifolds with constant sectional curvature in indefinite complex space froms[J]. Proc. Amer. Math. Soc., 2002, 130: 1459-1466.
    [14]
    FETCU D, LOUBEAU E, MONTALDO S, et al. Biharmonic submanifolds of CPn[J]. Z. Math., 2010, 266: 505-531.
    [15]
    FETCU D. ONICIUC C. Explicit formulas for biharmonic submanifolds in Sasakian space forms[J]. Pacific J. Math., 2009, 24: 85-107.
    [16]
    OU Y L, WANG Z P. Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries[J]. J. Geom. Phys., 2006, 228: 185-199.
    [17]
    ABRESH U, ROSENBER H. The Hopf differential for constant mean curvature surfaces in S2×R and H2×R[J]. Acta Math., 2004, 193: 141-174.
    [18]
    ABRESH U, ROSENBER H. Generalized Hopf differentials[J]. Mat. Contemp., 2005, 28: 1-28.
    [19]
    ALENCAR H, DO CARMO M, TRIBUZY R. A Hopf theorem for ambient spaces of dimensions higher than three[J]. J. Differential Geom., 2010, 84: 1-17.
    [20]
    FETCU D, ONICIUC C, ROSENBEG H. Biharmonic submanifolds with parallel mean curvature in Sn× R[J]. J. Geom. Anal., 2013, 23: 2158-2176.
    [21]
    ROTH J. A note on biharmonic submanifolds of product spaces[J]. J. Geom., 2013, 104: 375C-381.
    [22]
    DANIEL B. Isometric immersions into SnR and Hn×R and applications to minimal surfaces[J]. Trans. Amer. Math. Soc., 2004, 361: 6255-6282.
    [23]
    DILLEN F, FASTENAKELS J, DER VEKEN VAN J. Surfaces inS2×R with a canonical principal direction[J]. Ann. lobal Anal. Eom., 2009, 35: 381-395.
    [24]
    Dillen F., Munteanu M.,: Constant angle surfaces inH2×R[J]. Bull. Braz. Math. Soc. (NS), 2009, 40: 85-97.
    [25]
    LI A M, LI J M. An instrinsic rigidity theorem for minimal submanifolds in a sphere[J]. Arch. Math. (Basel), 1992, 58: 582-594.

    Article Metrics

    Article views (72) PDF downloads(107)
    Proportional views

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return