scholarly journals Structure theorems of the scalar curvature equation on subdomains of a compact Riemannian manifold

1997 ◽  
Vol 21 (1) ◽  
pp. 153-168
Author(s):  
Shin Kato
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Compact Riemannian Manifold ◽  
Curvature Equation ◽  
Structure Theorems

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

scholarly journals TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

2012 ◽  
Vol 86 (3) ◽  
pp. 405-415 ◽  
Author(s):  
SEOUNG DAL JUNG ◽  
HUILI LIU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Compact Riemannian Manifold ◽  
Killing Field ◽  
Nonzero Constant ◽  
Conformal Field ◽  
Infinitesimal Automorphisms

AbstractLet ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

scholarly journals First Eigenvalues of Geometric Operator under The Ricci-Bourguignon Flow

2017 ◽  
Vol 24 (1) ◽  
pp. 51-60
Author(s):  
Shahroud Azami
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Compact Riemannian Manifold ◽  
Nonnegative Curvature ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Geometric Operator

Let $(M,g(t))$ be a compact Riemannian manifold  and  the metric $g(t)$ evolve by the Ricci-Bourguignon flow. We find the formula variation of the eigenvalues of  geometric operator $-\Delta_{\phi}+cR$ under  the Ricci-Bourguignon flow, where  $\Delta_{\phi}$  is the Witten-Laplacian operator and $R$ is the scalar curvature. In the final  we show that some quantities dependent to the eigenvalues of  the geometric operator are  nondecreasing along the Ricci-Bourguignon flow on  closed manifolds  with nonnegative curvature.

New Results About the Lambda Constant and Ground States of the 𝑊-Functional

2020 ◽  
Vol 20 (3) ◽  
pp. 651-661
Author(s):  
Li Ma
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Variational Formulation ◽  
Compact Riemannian Manifold ◽  
Ground States ◽  
The Other ◽  
Concentration Compactness ◽  
Flat Manifold ◽  
Formula One ◽  
Compactness Method

AbstractIn this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold {(M,g)}. One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold {(M,g)}, if the scalar curvature s of {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., {s=0} in M. We also introduce a new constant {d(M,g)}. For the existence of the ground states, we use Lions’ concentration-compactness method.

scholarly journals Isometric immersion of a compact Riemannian manifold into a Euclidean space

1992 ◽  
Vol 46 (2) ◽  
pp. 177-178 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Isometric Immersion ◽  
Compact Riemannian Manifold

We show that an isometric immersion of an n−dimensional compact Riemannian manifold of non-negative Ricci curvature with scalar curvature always less than n(n−1)λ−2 into a Euclidean space of dimension n + 1 can never be contained in a ball of radius λ.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

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