On the mean curvature of spacelike submanifolds in semi-Riemannian manifolds

2006 ◽  
Vol 56 (9) ◽  
pp. 1728-1735 ◽  
Author(s):  
BingYe Wu
Keyword(s):  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Spacelike Submanifolds ◽  
The Mean

scholarly journals Mean curvature flow in Fuchsian manifolds

2019 ◽  
Vol 22 (07) ◽  
pp. 1950058
Author(s):  
Zheng Huang ◽  
Longzhi Lin ◽  
Zhou Zhang
Keyword(s):  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Closed Surface ◽  
Hyperbolic Manifolds ◽  
Warped Products ◽  
Geometric Flows ◽  
Closed Hypersurfaces ◽  
The Mean

Motivated by the goal of detecting minimal surfaces in hyperbolic manifolds, we study geometric flows in complete hyperbolic [Formula: see text]-manifolds. In general, the flows might develop singularities at some finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic [Formula: see text]-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and [Formula: see text]. We show that for a large class of closed initial surfaces, which are graphs over the totally geodesic surface [Formula: see text], the mean curvature flow exists for all time and converges to [Formula: see text]. This is among the first examples of converging mean curvature flows starting from closed hypersurfaces in Riemannian manifolds. We also provide calculations for the general warped product setting which will be useful for further works.

On Cheng's eigenvalue comparison theorem

2008 ◽  
Vol 144 (3) ◽  
pp. 673-682 ◽  
Author(s):  
G. P. BESSA ◽  
J. F. MONTENEGRO
Keyword(s):  
Comparison Theorem ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Weak Form ◽  
First Eigenvalue ◽  
Arbitrary Topology ◽  
Eigenvalue Comparison ◽  
The Mean ◽  
Normal Geodesic ◽  
Sectional And Ricci Curvatures

AbstractWe observe that Cheng's Eigenvalue Comparison Theorem for normal geodesic balls [4] is still valid if we impose bounds on the mean curvature of the distance spheres instead of bounds on the sectional and Ricci curvatures. In this version, there is a weak form of rigidity in case of equality of the eigenvalues. Namely, equality of the eigenvalues implies that the distance spheres of the same radius on each ball has the same mean curvature. On the other hand, we construct smooth metrics $g_{\kappa}$ on $[0,r]\times \mathbb{S}^{3}$, non-isometric to the standard metric canκ of constant sectional curvature κ, such that the geodesic balls $B_{g_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},g_{\kappa})$, $B_{{\rm can}_{\kappa}}(r)=([0,r]\times \mathbb{S}^{3},{\rm can}_{\kappa})$ have the same first eigenvalue, the same volume and the distance spheres $\partial B_{g_{\kappa}}(s)$ and$\partial B_{{\rm can}_{\kappa}}(s)$, $0<s\leq r$, have the same mean curvatures. In the end, we apply this version of Cheng's Eigenvalue Comparison Theorem to construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ*(M)>0 extending Veeravalli's examples,[7]

scholarly journals COMPLETE SPACELIKE SUBMANIFOLDS IN DE SITTER SPACES WITH

2013 ◽  
Vol 87 (3) ◽  
pp. 386-399 ◽  
Author(s):  
JIANCHENG LIU ◽  
JINGJING ZHANG
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Linear Relation ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
De Sitter ◽  
Spacelike Submanifolds ◽  
The Mean

AbstractIn this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation$R= aH+ b$of the normalised scalar curvature$R$and the mean curvature$H$in the de Sitter space${ S}_{p}^{n+ p} (c)$.

scholarly journals Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold

1997 ◽  
Vol 62 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Grant Cairns ◽  
Richard H. Escobales
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Final Section ◽  
General Context ◽  
Topological Properties ◽  
Normal Plane ◽  
Plane Field ◽  
The Mean ◽  
Foliated Riemannian Manifold

AbstractFor foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

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