Abstract
Let M be a compact n-dimensional hypersurface in the space form \({F^{n+1}(c)}\) where n = 3 or 4. We prove that if the scalar curvature of M is positive and the integral of the length of the second fundamental form of M satisfies certain inequality, then M must be diffeomorphic to the spherical space form. On the other hand, we prove that if M is homeomorphic to 2-dimensional sphere \({\mathbb{S}^2}\) and f 1, f 2, f 3 are first eigenfunctions of M such that \({\sum\nolimits_{i=1}^3|\nabla f_{i}|^2}\) is a constant, then M is isometric to a sphere with constant curvature.
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References
Catino G., Djadli Z.: Conformal deformations of integral pinched 3-manifolds. Adv. Math. 223, 393–404 (2010)
Chang S.-Y.A., Gursky M.J., Yang P.C.: A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Ht. Études Sci. 98, 105–143 (2003)
Cheng S.Y.: A characterization of the 2-sphere by eigenfunctions. Proc. Am. Math. Soc. 55, 379–381 (1976)
Cheng S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv. 51, 43–55 (1976)
Hersch J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270, 1645–1648 (1970)
Shiohama K., Xu H.W.: The topological sphere theorem for complete submanifolds. Compos. Math. 107, 221–232 (1997)
Xu H.W., Gu J.R.: A general gap theorem for submanifolds with parallel mean curvature in \({\mathbb{R}^{n+p}}\). Commun. Anal. Geom. 15, 175–193 (2007)
Xu H.W., Gu J.R.: Geometric, topological and differentiable rigidity of submanifolds in space forms. Geom. Funct. Anal. 23, 1684–1703 (2013)
Xu H.W., Tian L.: A differentiable sphere theorem inspired by rigidity of minimal submanifolds. Pac. J. Math. 254, 499–510 (2011)
Xu H.W., Tian L.: A new pinching theorem for closed hypersurfaces with constant mean curvature in S n+1. Asian J. Math. 15, 611–630 (2011)
Xu H.W., Ye F.: Differentiable sphere theorems for submanifolds of positive k-th Ricci curvature. Manuscr. Math. 138, 529–543 (2012)
Xu H.W., Zhao E.T.: Topological and differentiable sphere theorems for complete submanifolds. Commun. Anal. Geom. 17, 565–585 (2009)
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This work was supported by the Sogang University Research Grant of 2013.
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Ho, P.T. Differentiable rigidity of hypersurface in space forms. manuscripta math. 146, 463–472 (2015). https://doi.org/10.1007/s00229-014-0709-3
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DOI: https://doi.org/10.1007/s00229-014-0709-3