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]).

scholarly journals ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

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 Nonlinear Equations of Infinite Order Defined by an Elliptic Symbol

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mauricio Bravo Vera
Keyword(s):  
Hilbert Space ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Measurable Function ◽  
Infinite Order ◽  
Formal Definition ◽  
Regularity Properties ◽  
Definition Of ◽  
The Laplace Operator ◽  
Class Of Functions

The aim of this work is to show existence and regularity properties of equations of the formf(Δ)u=U(x,u(x))onℝn, in whichfis a measurable function that satisfies some conditions of ellipticity andΔstands for the Laplace operator onℝn. Here, we define the class of functions to whichfbelongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition off(Δ)explaining how to understand this operator.

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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