scholarly journals Vanishing theorems for L2 harmonic forms on complete submanifolds in Euclidean space

2015 ◽  
Vol 425 (2) ◽  
pp. 774-787 ◽  
Author(s):  
Hezi Lin
Keyword(s):  
Euclidean Space ◽  
Vanishing Theorems ◽  
Harmonic Forms ◽  
Complete Submanifolds

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

scholarly journals Total curvature of complete submanifolds of Euclidean space

2005 ◽  
Vol 57 (2) ◽  
pp. 171-200 ◽  
Author(s):  
Franki Dillen ◽  
Wolfgang Kühnel
Keyword(s):  
Euclidean Space ◽  
Total Curvature ◽  
Complete Submanifolds

scholarly journals L 2 harmonic forms and finiteness of ends

2013 ◽  
Vol 85 (2) ◽  
pp. 457-471 ◽  
Author(s):  
PENG ZHU
Keyword(s):  
Riemannian Manifolds ◽  
Poincaré Inequality ◽  
Vanishing Theorems ◽  
Poincare Inequality ◽  
Harmonic Forms ◽  
Minimal Hypersurfaces ◽  
Weighted Poincaré Inequality ◽  
Complete Riemannian Manifolds

In this paper, we obtain vanishing theorems and finitely many ends theorems of complete Riemannian manifolds with weighted Poincaré inequality, applying them to minimal hypersurfaces.

scholarly journals EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

2017 ◽  
pp. 49-53
Author(s):  
Irina Alexandrova ◽  
Irina Alexandrova ◽  
Sergey Stepanov ◽  
Sergey Stepanov ◽  
Irina Tsyganok ◽  
...  
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Differential Forms ◽  
Conformal Killing ◽  
Riemannian Symmetric Space ◽  
Vanishing Theorems ◽  
Harmonic Forms ◽  
Exterior Differential ◽  
Rank One ◽  
Riemannian Symmetric Spaces

In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

scholarly journals L2 harmonic 1-forms on complete submanifolds in Euclidean space

2009 ◽  
Vol 32 (3) ◽  
pp. 432-441 ◽  
Author(s):  
Hai-Ping Fu ◽  
Zhen-Qi Li
Keyword(s):  
Euclidean Space ◽  
Complete Submanifolds
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