On a Yamabe Type Problem in Finsler Geometry

2017 ◽  
Vol 60 (2) ◽  
pp. 253-268
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Constant Scalar Curvature ◽  
Conformal Invariants ◽  
Type Problem ◽  
Conformal Class ◽  
Standard Sphere ◽  
Yamabe Constant ◽  
Cartan Torsion

AbstractIn this paper, a newnotion of scalar curvature for a Finslermetric F is introduced, and two conformal invariants Y(M, F) and C(M, F) are deûned. We prove that there exists a Finslermetric with constant scalar curvature in the conformal class of F if the Cartan torsion of F is suõciently small and Y(M, F)C(M, F) < Y(Sn) where Y(Sn) is the Yamabe constant of the standard sphere.

scholarly journals Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

2019 ◽  
Vol 21 (03) ◽  
pp. 1850021 ◽  
Author(s):  
Xuezhang Chen ◽  
Liming Sun
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Compact Manifold ◽  
Constant Scalar Curvature ◽  
Riemannian Metric ◽  
Conformal Metrics ◽  
Nonzero Constant ◽  
Dimension Formula ◽  
Yamabe Constant ◽  
Boundary Mean Curvature

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

CONFORMAL TRANSFORMATIONS OF COMPACT SELF-DUAL MANIFOLDS

1994 ◽  
Vol 05 (01) ◽  
pp. 125-140 ◽  
Author(s):  
Y. S. POON
Keyword(s):  
Scalar Curvature ◽  
Symmetry Group ◽  
Positive Constant ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Topological Classification ◽  
Conformal Transformations ◽  
Conformal Class ◽  
Zero Scalar Curvature

We prove that when the dimension of the group of conformal transformations of a compact self-dual manifold is at least three, the conformal class contains either a metric with positive constant scalar curvature or a metric with zero scalar curvature. This result is combined with a topological classification of 4-manifolds to provide a complete geometrical classification of the compact self-dual manifolds whose symmetry group is at least three-dimensional.

scholarly journals Killing fields generated by multiple solutions to the Fischer–Marsden equation II

2016 ◽  
Vol 27 (10) ◽  
pp. 1650080
Author(s):  
Daniel Guan ◽  
Paul Cernéa
Keyword(s):  
Scalar Curvature ◽  
Multiple Solutions ◽  
Vector Fields ◽  
Killing Vector ◽  
Solution Space ◽  
Constant Scalar Curvature ◽  
Standard Sphere ◽  
Dimension Formula ◽  
Killing Fields ◽  
Sectional Curvatures

In the process of finding Einstein metrics in dimension [Formula: see text], we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space [Formula: see text] can be at most [Formula: see text], with equality implying that [Formula: see text] is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor [Formula: see text]. In this second paper, we apply our results in the first paper to show that either [Formula: see text] is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most [Formula: see text].

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

scholarly journals An Obata-type theorem on compact Einstein manifolds with boundary

2021 ◽  
Author(s):  
Kazuo Akutagawa
Keyword(s):  
Boundary Condition ◽  
Smooth Boundary ◽  
Type Theorem ◽  
Type Inequality ◽  
Einstein Metric ◽  
Manifolds With Boundary ◽  
Einstein Manifolds ◽  
Conformal Class ◽  
Totally Geodesic ◽  
Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

Topological tools for the prescribed scalar curvature problem on S n

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Dina Abuzaid ◽  
Randa Ben Mahmoud ◽  
Hichem Chtioui ◽  
Afef Rigane
Keyword(s):  
Scalar Curvature ◽  
Conformal Metrics ◽  
Standard Sphere ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Hopf Formula ◽  
Curvature Problem ◽  
Formula Type

AbstractIn this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].

Generalized -Einstein Real Hypersurfaces in and

2020 ◽  
Vol 63 (4) ◽  
pp. 909-920
Author(s):  
Yaning Wang
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Coefficient ◽  
Real Hypersurface ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Real Hypersurfaces

AbstractIn this paper we obtain some new characterizations of pseudo-Einstein real hypersurfaces in $\mathbb{C}P^{2}$ and $\mathbb{C}H^{2}$. More precisely, we prove that a real hypersurface in $\mathbb{C}P^{2}$ or $\mathbb{C}H^{2}$ with constant mean curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein. We prove that a real hypersurface in $\mathbb{C}P^{2}$ with constant scalar curvature is generalized ${\mathcal{D}}$-Einstein with constant coefficient if and only if it is pseudo-Einstein.

A Note on Randers Metrics of Scalar Flag Curvature

2012 ◽  
Vol 55 (3) ◽  
pp. 474-486 ◽  
Author(s):  
Bin Chen ◽  
Lili Zhao
Keyword(s):  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Flag Curvature ◽  
Projectively Flat ◽  
Randers Metrics

AbstractSome families of Randers metrics of scalar flag curvature are studied in this paper. Explicit examples that are neither locally projectively flat nor of isotropic S-curvature are given. Certain Randers metrics with Einstein α are considered and proved to be complex. Three dimensional Randers manifolds, with α having constant scalar curvature, are studied.

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