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.

scholarly journals Biharmonic submanifolds in manifolds with bounded curvature

2016 ◽  
Vol 27 (11) ◽  
pp. 1650089
Author(s):  
Shun Maeta
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Biharmonic Submanifolds ◽  
Bounded Curvature ◽  
Negative Constant ◽  
The Mean ◽  
Biharmonic Submanifold

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

scholarly journals The diameter of an immersed Riemannian manifold with bounded mean curvature

1981 ◽  
Vol 31 (2) ◽  
pp. 189-192 ◽  
Author(s):  
Th. Koufogiorgos ◽  
Ch. Baikoussis
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Mean Curvature Vector ◽  
Nonpositive Sectional Curvature ◽  
The Mean ◽  
Simply Connected ◽  
Bounded Norm

AbstractLet M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

scholarly journals EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

2010 ◽  
Vol 53 (2) ◽  
pp. 321-332 ◽  
Author(s):  
SUN HEJUN ◽  
QI XUERONG
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Complete Riemannian Manifold ◽  
Quadratic Polynomial ◽  
Polynomial Operator ◽  
Eigenvalue Estimates ◽  
The Mean

AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

HARNACK-TYPE INEQUALITIES FOR THE POROUS MEDIUM EQUATION ON A MANIFOLD WITH NON-NEGATIVE RICCI CURVATURE

2012 ◽  
Vol 23 (04) ◽  
pp. 1250009 ◽  
Author(s):  
JEONGWOOK CHANG ◽  
JINHO LEE
Keyword(s):  
Porous Medium ◽  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Porous Medium Equation ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Medium Equation

We derive Harnack-type inequalities for non-negative solutions of the porous medium equation on a complete Riemannian manifold with non-negative Ricci curvature. Along with gradient estimates, reparametrization of a geodesic and time rescaling of a solution are key tools to get the results.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

scholarly journals A volume estimate for strong subharmonicity and maximum principle on complete Riemannian manifolds

1998 ◽  
Vol 151 ◽  
pp. 25-36 ◽  
Author(s):  
Kensho Takegoshi
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Volume Growth ◽  
Complete Riemannian Manifold ◽  
Volume Estimate ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Abstract.A generalized maximum principle on a complete Riemannian manifold (M, g) is shown under a certain volume growth condition of (M, g) and its geometric applications are given.

scholarly journals Submanifolds and the length of the second fundamental tensor

1983 ◽  
Vol 34 (1) ◽  
pp. 1-6
Author(s):  
Thomas Hasanis
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Mean Curvature ◽  
Sufficient Condition ◽  
Fundamental Tensor ◽  
The Mean

AbstractA sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

scholarly journals Volume comparison of Bishop-Gromov type

1992 ◽  
Vol 45 (2) ◽  
pp. 241-248
Author(s):  
Sungyun Lee
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Complete Riemannian Manifold ◽  
Comparison Theorems ◽  
Volume Comparison

Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

scholarly journals Isometric immersions in the hyperbolic space with their image contained in a horoball

2001 ◽  
Vol 43 (1) ◽  
pp. 1-8 ◽  
Author(s):  
Ana Lluch
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Isometric Immersions ◽  
Special Groups ◽  
The Mean ◽  
Complex Hyperbolic Spaces ◽  
Real Hyperbolic Space ◽  
Sharp Lower Bound

We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

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