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 On functions which satisfy some differential inequalities on Riemannian manifolds

1981 ◽  
Vol 81 ◽  
pp. 57-72 ◽  
Author(s):  
Kanji Motomiya
Keyword(s):  
Differential Geometry ◽  
Maximum Principle ◽  
Differential Equations ◽  
Riemannian Manifolds ◽  
Differential Inequalities ◽  
Important Theorem ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle

Most of the problems in differential geometry can be reduced to problems in differential equations and differential inequalities on Riemannian manifolds. Our main purpose of this paper is to study such differential inequalities on complete Riemannian manifolds. In [5], H. Omori proved a very important theorem. S. Y. Cheng and S. T. Yau gave a simplification and a generalization of it which was called the generalized maximum principle in [2] and [7], and many interesting applications in differential geometry in [2], [3], [7], and [8].

scholarly journals CHEN'S CONJECTURE AND ε-SUPERBIHARMONIC SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS

2013 ◽  
Vol 24 (04) ◽  
pp. 1350028 ◽  
Author(s):  
GLEN WHEELER
Keyword(s):  
Riemannian Manifold ◽  
Growth Condition ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
General Statement ◽  
Chen’S Conjecture ◽  
Background Space ◽  
Special Cases

Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of ε-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ε-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal.

scholarly journals Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

2001 ◽  
Vol 162 ◽  
pp. 149-167
Author(s):  
Yong Hah Lee
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Nonlinear Elliptic Equation ◽  
Complete Riemannian Manifold ◽  
Doubling Condition ◽  
Finite Covering ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Volume Doubling

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

Long time behavior of quasi-convex and pseudo-convex gradient systems on Riemannian manifolds

Filomat ◽  
2017 ◽  
Vol 31 (14) ◽  
pp. 4571-4578 ◽  
Author(s):  
P. Ahmadi ◽  
H. Khatibzadeh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Gradient Flow ◽  
Complete Riemannian Manifold ◽  
Discrete Version ◽  
Gradient System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Pseudo Convex

In this paper, we study the following gradient system on a complete Riemannian manifold M, {-x?(t) = grad'(x(t)) x(0) = x0, where ? : M ? R is a C1 function with Argmin ? ? ?. We prove that the gradient flow x(t) converges to a critical point of ? if ? is pseudo-convex, or if ? is quasi-convex and M is Hadamard. As an application to minimization, we consider a discrete version of the system to approximate a minimum point of a given pseudo-convex function ?.

New characterizations of spacelike hyperplanes in the steady state space

2020 ◽  
Vol 126 (1) ◽  
pp. 61-72
Author(s):  
Cícero P. Aquino ◽  
Halyson I. Baltazar ◽  
Henrique F. De Lima
Keyword(s):  
Steady State ◽  
State Space ◽  
Riemannian Manifolds ◽  
De Sitter ◽  
Main Hypothesis ◽  
Spacelike Hypersurfaces ◽  
Open Region ◽  
Complete Riemannian Manifolds ◽  
Generalized Maximum Principle ◽  
Hyperbolic Cylinders

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

scholarly journals EIGENVALUES OF THE LAPLACIAN ON RIEMANNIAN MANIFOLDS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250067
Author(s):  
QING-MING CHENG ◽  
XUERONG QI
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Orthonormal Basis ◽  
Smooth Boundary ◽  
Complete Riemannian Manifold ◽  
Piecewise Smooth Boundary ◽  
Dirichlet Eigenvalue ◽  
Eigenvalues Of The Laplacian ◽  
Dirichlet Eigenvalue Problem ◽  
The Laplace Operator

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

scholarly journals CHARACTERIZATIONS OF LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN EINSTEIN SPACETIMES

2013 ◽  
Vol 55 (3) ◽  
pp. 567-579 ◽  
Author(s):  
HENRIQUE F. DE LIMA ◽  
JOSEÍLSON R. DE LIMA
Keyword(s):  
Maximum Principle ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Sufficient Conditions ◽  
Analytical Tool ◽  
Isoparametric Hypersurface ◽  
De Sitter ◽  
Principal Curvatures ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle

AbstractOur purpose is to study the geometry of linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime, whose sectional curvature is supposed to obey some standard restrictions. In this setting, by using as main analytical tool a generalized maximum principle for complete non-compact Riemannian manifolds, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Applications to the de Sitter space are given.

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