The Abbena-Thurston Manifold as a Critical Point

1996 ◽  
Vol 39 (3) ◽  
pp. 352-359 ◽  
Author(s):  
Joon-Sik Park ◽  
Won Tae Oh
Keyword(s):  
Scalar Curvature ◽  
Critical Point ◽  
Ricci Operator

AbstractThe Abbena-Thurston manifold (M,g) is a critical point of the functional where Q is the Ricci operator and R is the scalar curvature, and then the index of I(g) and also the index of — I(g) are positive at (M,g).

A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion

1992 ◽  
Vol 35 (4) ◽  
pp. 455-462 ◽  
Author(s):  
D. E. Blair ◽  
D. Perrone
Keyword(s):  
Scalar Curvature ◽  
Critical Point ◽  
Integral Functional ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Contact Condition ◽  
Contact Metric Manifolds ◽  
Point Condition ◽  
Webster Scalar Curvature ◽  
Group Of Isometries

AbstractChern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature.

scholarly journals CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

2012 ◽  
Vol 49 (3) ◽  
pp. 655-667 ◽  
Author(s):  
Jeong-Wook Chang ◽  
Seung-Su Hwang ◽  
Gab-Jin Yun
Keyword(s):  
Scalar Curvature ◽  
Critical Point ◽  
Total Scalar Curvature

scholarly journals Multiplicity of Radial Ground States for the Scalar Curvature Equation Without Reciprocal Symmetry

2020 ◽  
Author(s):  
Francesca Dalbono ◽  
Matteo Franca ◽  
Andrea Sfecci
Keyword(s):  
Dynamical System ◽  
Scalar Curvature ◽  
Critical Point ◽  
Ground State ◽  
Ground States ◽  
Fast Decay ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
Reciprocal Symmetry ◽  
Unique Critical Point

Abstract We study existence and multiplicity of positive ground states for the scalar curvature equation $$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$ Δ u + K ( | x | ) u n + 2 n - 2 = 0 , x ∈ R n , n > 2 , when the function $$K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+$$ K : R + → R + is bounded above and below by two positive constants, i.e. $$0<\underline{K} \le K(r) \le \overline{K}$$ 0 < K ̲ ≤ K ( r ) ≤ K ¯ for every $$r > 0$$ r > 0 , it is decreasing in $$(0,{{{\mathcal {R}}}})$$ ( 0 , R ) and increasing in $$({{{\mathcal {R}}}},+\infty )$$ ( R , + ∞ ) for a certain $${{{\mathcal {R}}}}>0$$ R > 0 . We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio $$\overline{K}/\underline{K}$$ K ¯ / K ̲ which guarantees the existence of a large number of ground states with fast decay, i.e. such that $$u(|x|) \sim |x|^{2-n}$$ u ( | x | ) ∼ | x | 2 - n as $$|x| \rightarrow +\infty $$ | x | → + ∞ , which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique.

scholarly journals Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

2018 ◽  
Vol 2020 (19) ◽  
pp. 6539-6568
Author(s):  
Norihisa Ikoma ◽  
Andrea Malchiodi ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Critical Point ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Multiplicity Result ◽  
Round Sphere ◽  
Willmore Energy ◽  
Nondegenerate Critical Point ◽  
Three Dimensional Space

Abstract Let $(M,g)$ be a three-dimensional Riemannian manifold. The goal of the paper is to show that if $P_{0}\in M$ is a nondegenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than $32\pi $; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass.

scholarly journals Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold

2019 ◽  
Vol 11 (1) ◽  
pp. 59-69 ◽  
Author(s):  
A. Ghosh
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Ricci Soliton ◽  
Kenmotsu Manifold ◽  
Curvature Invariant ◽  
Ricci Operator ◽  
Reeb Vector ◽  
Reeb Vector Field

First, we prove that if the Reeb vector field $\xi$ of a Kenmotsu manifold $M$ leaves the Ricci operator $Q$ invariant, then $M$ is Einstein. Next, we study Kenmotsu manifold whose metric represents a Ricci soliton and prove that it is expanding. Moreover, the soliton is trivial (Einstein) if either (i) $V$ is a contact vector field, or (ii) the Reeb vector field $\xi$ leaves the scalar curvature invariant. Finally, it is shown that if the metric of a Kenmotsu manifold represents a gradient Ricci almost soliton, then it is $\eta$-Einstein and the soliton is expanding. We also exhibited some examples of Kenmotsu manifold that admit Ricci almost solitons.

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