scholarly journals Mean curvature and compactification of surfaces in a negatively curved Cartan-Hadamard manifold

2014 ◽  
Vol 22 (3) ◽  
pp. 387-420
Author(s):  
Antonio Esteve ◽  
Vicente Palmer
Keyword(s):  
Mean Curvature ◽  
Hadamard Manifold ◽  
Negatively Curved

scholarly journals A NOTE ON THE UNCLOUDING THE SKY OF NEGATIVELY CURVED MANIFOLDS

2008 ◽  
Vol 77 (3) ◽  
pp. 413-424
Author(s):  
ALBERT BORBÉLY
Keyword(s):  
Convex Sets ◽  
Hadamard Manifold ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Arbitrary Convex

AbstractThe problem of finding geodesics that avoid certain obstacles in negatively curved manifolds has been studied in different situations. In this note we give a generalization of the unclouding theorem of J. Parkkonen and F. Paulin: there is a constant s0=1.534 such that for any Hadamard manifold M with curvature ≤−1 and for any family of disjoint balls or horoballs {Ca}a∈A and for any point p∈M−⋃ a∈ACa if we shrink these balls uniformly by s0 one can always find a geodesic ray emanating from p that avoids the shrunk balls. It will be shown that in the theorem above one can replace the balls by arbitrary convex sets.

On submanifolds of highly negatively curved spaces

2014 ◽  
Vol 25 (06) ◽  
pp. 1450055
Author(s):  
G. Pacelli Bessa ◽  
Stefano Pigola ◽  
Alberto G. Setti
Keyword(s):  
Mean Curvature ◽  
Curved Spaces ◽  
Negatively Curved ◽  
Curvature Estimates ◽  
Isoperimetric Ratio

We prove spectral, stochastic and mean curvature estimates for complete m-submanifolds φ : M → N of n-manifolds with a pole N in terms of the comparison isoperimetric ratio Im and the extrinsic radius rφ ≤ ∞. Our proof holds for the bounded case rφ < ∞, recovering the known results, as well as for the unbounded case rφ = ∞. In both cases, the fundamental ingredient in these estimates is the integrability over (0, rφ) of the inverse [Formula: see text] of the comparison isoperimetric radius. When rφ = ∞, this condition is guaranteed if N is highly negatively curved.

scholarly journals An estimate for the total mean curvature in negatively curved spaces

2002 ◽  
Vol 66 (2) ◽  
pp. 267-273
Author(s):  
Albert Borbély
Keyword(s):  
Compact Subset ◽  
Mean Curvature ◽  
Smooth Boundary ◽  
Convex Compact ◽  
Empty Interior ◽  
Connected Manifold ◽  
Total Mean Curvature ◽  
Negatively Curved ◽  
Nonpositively Curved ◽  
Simply Connected

Let Mn be a nonpositively curved complete simply connected manifold and D ⊂ Mn be a convex compact subset with non-empty interior and smooth boundary. It is shown that the total mean curvature ∂D can be estimated in terms of volume and curvature bound.

scholarly journals Complete submanifolds with bounded mean curvature in a Hadamard manifold

2010 ◽  
Vol 60 (1) ◽  
pp. 142-154 ◽  
Author(s):  
Manfredo P. do Carmo ◽  
Qiaoling Wang ◽  
Changyu Xia
Keyword(s):  
Mean Curvature ◽  
Hadamard Manifold ◽  
Complete Submanifolds

scholarly journals IMMERSION OF MANIFOLDS WITH UNBOUNDED IMAGE AND A MODIFIED MAXIMUM PRINCIPLE OF YAU

2008 ◽  
Vol 78 (2) ◽  
pp. 285-291 ◽  
Author(s):  
ALBERT BORBÉLY
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Ricci Curvature ◽  
Distance Function ◽  
Mean Curvature ◽  
Main Tool ◽  
Complete Riemannian Manifold ◽  
Hadamard Manifold ◽  
The Mean

AbstractLet N be a complete Riemannian manifold isometrically immersed into a Hadamard manifold M. We show that the immersion cannot be bounded if the mean curvature of the immersed manifold is small compared with the curvature of M and the Laplacian of the distance function on N grows at most linearly. The latter condition is satisfied if the Ricci curvature of N does not approach $-\infty $ too fast. The main tool in the proof is a modification of Yau’s maximum principle.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

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