scholarly journals Scalar curvature rigidity and Ricci DeTurck flow on perturbations of Euclidean space

2018 ◽  
Vol 57 (5) ◽  
Author(s):  
Alexander Appleton
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Scalar Curvature Rigidity

scholarly journals Complete hypersurfaces in cylinders

1989 ◽  
Vol 46 (2) ◽  
pp. 313-318
Author(s):  
Thomas Hasanis
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar ◽  
Coordinate Functions ◽  
Bounded Below ◽  
Complete Hypersurfaces

AbstractWe consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

scholarly journals Isometric Immersions of Constant Mean Curvature and Triviality of the Normal Connection

1972 ◽  
Vol 45 ◽  
pp. 139-165 ◽  
Author(s):  
Joseph Erbacher
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Additional Assumption ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Normal Connection ◽  
Isometric Immersions ◽  
The Mean ◽  
Negative Sectional Curvature

In a recent paper [2] Nomizu and Smyth have determined the hypersurfaces Mn of non-negative sectional curvature iso-metrically immersed in the Euclidean space Rn+1 or the sphere Sn+1 with constant mean curvature under the additional assumption that the scalar curvature of Mn is constant. This additional assumption is automatically satisfied if Mn is compact. In this paper we extend these results to codimension p isometric immersions. We determine the n-dimensional submanifolds Mn of non-negative sectional curvature isometrically immersed in the Euclidean Space Rn+P or the sphere Sn+P with constant mean curvature under the additional assumptions that Mn has constant scalar curvature and the curvature tensor of the connection in the normal bundle is zero. By constant mean curvature we mean that the mean curvature normal is paral lel with respect to the connection in the normal bundle. The assumption that Mn has constant scalar curvature is automatically satisfied if Mn is compact. The assumption on the normal connection is automatically sa tisfied if p = 2 and the mean curvature normal is not zero.

scholarly journals On scalar curvature rigidity of vacuum static spaces

2015 ◽  
Vol 365 (3-4) ◽  
pp. 1257-1277 ◽  
Author(s):  
Jie Qing ◽  
Wei Yuan
Keyword(s):  
Scalar Curvature ◽  
Scalar Curvature Rigidity

scholarly journals Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space

2017 ◽  
Vol 2017 (727) ◽  
Author(s):  
Lan-Hsuan Huang ◽  
Dan A. Lee ◽  
Christina Sormani
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Positive Mass Theorem ◽  
Flat Manifold ◽  
Positive Mass ◽  
Asymptotically Flat ◽  
Intrinsic Flat

AbstractThe rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and

scholarly journals Cones in the Euclidean space with vanishing scalar curvature

2004 ◽  
Vol 76 (4) ◽  
pp. 631-638
Author(s):  
João Lucas M. Barbosa ◽  
Manfredo do Carmo
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Bounded Domains

Given a hypersurface M on a unit sphere of the Euclidean space, we define the cone based on M as the set of half-lines issuing from the origin and passing through M. By assuming that the scalar curvature of the cone vanishes, we obtain conditions under which bounded domains of such cone are stable or unstable.

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