scholarly journals A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds

2009 ◽  
Vol 19 (3) ◽  
pp. 719-736 ◽  
Author(s):  
Guofang Wei ◽  
Rugang Ye
Keyword(s):  
Maximum Principle ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Neumann Type ◽  
The Laplace Operator

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

2020 ◽  
pp. 84-102
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifolds ◽  
Laplacian Operator ◽  
Measure Concentration ◽  
Eigenvalues Of The Laplacian ◽  
The Laplace Operator ◽  
Concentration Phenomenon

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.

scholarly journals Some Inequalities for the Omori-Yau Maximum Principle

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Kyusik Hong
Keyword(s):  
Riemannian Manifold ◽  
Maximum Principle ◽  
Laplace Operator ◽  
Order Term ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Zeroth Order ◽  
Exhaustion Function ◽  
Semielliptic Operator ◽  
The Laplace Operator

We generalize A. Borbély’s condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semielliptic operatorLwith bounded coefficients and no zeroth order term. Also, we consider a new sufficient condition for the existence of a tamed exhaustion function. From these results, we may remark that the existence of a tamed exhaustion function is more general than the hypotheses in the version of the Omori-Yau maximum principle that was given by A. Ratto, M. Rigoli, and A. G. Setti.

Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds

1949 ◽  
Vol 1 (3) ◽  
pp. 242-256 ◽  
Author(s):  
S. Minakshisundaram ◽  
Å. Pleijel
Keyword(s):  
Differential Equation ◽  
Riemannian Manifold ◽  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Line Element ◽  
Metric Tensor ◽  
Local Coordinates ◽  
Image Position ◽  
The Laplace Operator ◽  
Beltrami Laplace Operator

Let V be a connected, compact, differentiable Riemannian manifold. If V is not closed we denote its boundary by S. In terms of local coordinates (xi), i = 1, 2, … Ν, the line-element dr is given by where gik (x1, x2, … xN) are the components of the metric tensor on V We denote by Δ the Beltrami-Laplace-Operator and we consider on V the differential equation (1) Δu + λu = 0.

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