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 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.

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.

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.

Maximum Principles for Inhomogeneous Elliptic Inequalities on Complete Riemannian Manifolds

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Dimitri Mugnai ◽  
Patrizia Pucci
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Weak Solutions ◽  
Maximum Principles ◽  
Complete Riemannian Manifolds ◽  
Elliptic Inequalities

AbstractWe prove some maximum principle results for weak solutions of elliptic inequalities, possibly inhomogeneous, on complete Riemannian manifolds.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

Complete spacelike hypersurfaces in a Robertson–Walker spacetime

2011 ◽  
Vol 151 (2) ◽  
pp. 271-282 ◽  
Author(s):  
ALMA L. ALBUJER ◽  
FERNANDA E. C. CAMARGO ◽  
HENRIQUE F. DE LIMA
Keyword(s):  
Maximum Principle ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Rigidity Results ◽  
Complete Spacelike Hypersurfaces ◽  
Uniqueness Results ◽  
Spacelike Hypersurfaces ◽  
Generalized Maximum Principle ◽  
Non Parametric

AbstractIn this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

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