A gap theorem for complete submanifolds with parallel mean curvature in the hyperbolic space

Abstract Let M be an n ( ≥ 7 ) -dimensional complete submanifold with parallel mean curvature in the hyperbolic space H n + p , whose mean curvature satisfies H 2 − 1 ≤ 0 . Denote by A ˚ and B R ( q ) the trace free second fundamental form of M and the geodesic ball of radius R centered at q ∈ M , respectively. We prove that if lim sup R → ∞ ∫ B R ( q ) | A ˚ | 2 d M R 2 = 0 , and if ( ∫ M | A ˚ | n d M ) 2 / n + 2 n ( n − 2 ) 3 n ( n − 1 ) H ( ∫ M | A ˚ | n / 2 d M ) 2 / n ≤ C ( n ) , then M is congruent to an n-dimensional hyperbolic space or the Euclidean space R n . Here C ( n ) is an explicit positive constant depending only on n. We also obtain a similar gap theorem in the case where n = 5 , 6 .

[1]  C. Xia,et al.  Gap theorems for minimal submanifolds of a hyperbolic space , 2016 .

[2]  H. Lawson.  Local Rigidity Theorems for Minimal Hypersurfaces , 1969 .

[3]  S. Yau,et al.  Differential equations on riemannian manifolds and their geometric applications , 1975 .

[4]  P. Berard,et al.  Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature , 1998 .

[5]  J. Gu,et al.  L n=2 Pinching Theorem for Submanifolds with Parallel Mean Curvature in H n+p (-1) , 2012 .

[6]  On the compactness of constant mean curvature hypersurfaces with finite total curvature , 1999 .

[7]  Hong-wei Xu.  $L_{n/2}$-pinching theorems for submanifolds with parallel mean curvature in a sphere , 1994 .

[8]  Richard Schoen,et al.  The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature , 1980 .

[9]  David Hoffman,et al.  Sobolev and isoperimetric inequalities for riemannian submanifolds , 2010 .

[10]  A. M. Silveira.  Stability of complete noncompact surfaces with constant mean curvature , 1987 .

[11]  An-Min Li,et al.  An intrinsic rigidity theorem for minimal submanifolds in a sphere , 1992 .

[12]  W. Fleming.  On the oriented Plateau Problem , 1962 .

[13]  M. Carmo,et al.  Stable complete minimal surfaces in $R^3$ are planes , 1979 .

[14]  L. Cheung,et al.  Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space , 2001 .

[15]  Shiing-Shen Chern,et al.  Minimal Submanifolds of a Sphere with Second Fundamental Form of Constant Length , 1970 .

[16]  Geraldo de Oliveira Filho.  Compactification of minimal submanifolds of hyperbolic space , 1993 .

[17]  Entao Zhao,et al.  A gap theorem for minimal submanifolds in Euclidean space , 2015 .

[18]  The structure of stable minimal hypersurfaces in R^n , 1997, dg-ga/9709001.

[19]  Y. Shen,et al.  On stable complete minimal hypersurfaces in Rn+1 , 1998 .

[20]  Lei Ni.  Gap theorems for minimal submanifolds in $R^{n+1}$ , 2001 .

[21]  F. Almgren.  Some Interior Regularity Theorems for Minimal Surfaces and an Extension of Bernstein's Theorem , 1966 .

[22]  Shing-Tung Yau,et al.  HEAT-EQUATIONS ON MINIMAL SUBMANIFOLDS AND THEIR APPLICATIONS , 1984 .

[23]  Walcy Santos,et al.  SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR IN SPHERES , 1994 .

[24]  James Simons,et al.  Minimal Varieties in Riemannian Manifolds , 1968 .

[25]  K. Shiohama,et al.  A general rigidity theorem for complete submanifolds , 1998, Nagoya Mathematical Journal.

[26]  Hong-wei Xu.  A rigidity theorem for submanifolds with parallel mean curvature in a sphere , 1993 .

[27]  Lin Jun-min,et al.  Global pinching theorems for even dimensional minimal submanifolds in the unit spheres , 1989 .

[28]  C. Shen.  A global pinching theorem of minimal hypersurfaces in the sphere , 1989 .

[29]  RIGIDITY AND SPHERE THEOREMS FOR SUBMANIFOLDS II , 1994 .

[30]  Eugenio Calabi,et al.  Minimal immersions of surfaces in Euclidean spheres , 1967 .

[31]  C. Xia,et al.  Rigidity of complete minimal submanifolds in a hyperbolic space , 2019 .

[32]  M. Umehara,et al.  Complete surfaces of constant mean curvature-1 in the hyperbolic 3-space , 1993 .