scholarly journals On Stability of the Hyperbolic Space Form under the Normalized Ricci Flow

2010 ◽  
Vol 2010 (15) ◽  
pp. 2903-2924 ◽  
Author(s):  
Haozhao Li ◽  
Hao Yin
Keyword(s):  
Hyperbolic Space ◽  
Ricci Flow ◽  
Space Form ◽  
Normalized Ricci Flow ◽  
Hyperbolic Space Form

The normalized Ricci flow and homology in Lagrangian submanifolds of generalized complex space forms

2020 ◽  
Vol 17 (06) ◽  
pp. 2050094 ◽  
Author(s):  
Fatemah Mofarreh ◽  
Akram Ali ◽  
Wan Ainun Mior Othman
Keyword(s):  
Fundamental Form ◽  
Ricci Flow ◽  
Complex Space ◽  
Space Form ◽  
Second Fundamental Form ◽  
Complex Space Form ◽  
Standard Sphere ◽  
The Second Fundamental Form ◽  
Generalized Complex ◽  
Normalized Ricci Flow

In this paper, we prove that a simply connected Lagrangian submanifold in the generalized complex space form is diffeomorphic to standard sphere [Formula: see text] and the normalized Ricci flow converges to a constant curvature metric, provided its squared norm of the second fundamental form satisfies some upper bound depending only on the squared norm of the mean curvature vector field, the constant sectional curvature, and the dimension of the Lagrangian immersion of the ambient space. Next, we conclude that stable currents do not exist and homology groups vanish in a compact real submanifold of the general complex space form, provided that the second fundamental form satisfies some extrinsic conditions. We show that our results improve some previous results.

scholarly journals Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs

2012 ◽  
Vol 207 ◽  
pp. 95-138 ◽  
Author(s):  
Tae Hattori ◽  
Atsushi Kasue
Keyword(s):  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Space Form ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Infinite Networks ◽  
Hyperbolic Space Form

AbstractIn this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS

2002 ◽  
Vol 13 (02) ◽  
pp. 209-216 ◽  
Author(s):  
JUN-MUK HWANG
Keyword(s):  
Hyperbolic Space ◽  
Complex Projective Space ◽  
Space Form ◽  
Complex Hyperbolic Space ◽  
Space Forms ◽  
Totally Geodesic ◽  
Gauss Mapping ◽  
Complex Projective ◽  
Smooth Locus ◽  
Hyperbolic Space Form

In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.

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