scholarly journals Equivariant Yamabe problem with boundary

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
Pak Tung Ho ◽  
Jinwoo Shin
Keyword(s):  
Scalar Curvature ◽  
Isometry Group ◽  
Yamabe Problem ◽  
Conformal Class ◽  
Invariant Metric

AbstractAs a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we study the equivariant Yamabe problem with boundary.

On the conformal scalar curvature equations on Finsler manifolds

2016 ◽  
Vol 14 (01) ◽  
pp. 1750008
Author(s):  
Neda Shojaee ◽  
Morteza MirMohammad Rezaii
Keyword(s):  
Scalar Curvature ◽  
Finsler Geometry ◽  
Yamabe Problem ◽  
Finsler Manifolds ◽  
Yamabe Flow ◽  
Conformal Deformations

In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

Chern-Yamabe problem and Chern-Yamabe soliton

2021 ◽  
Vol 32 (03) ◽  
pp. 2150016
Author(s):  
Pak Tung Ho ◽  
Jinwoo Shin
Keyword(s):  
Scalar Curvature ◽  
Complex Manifold ◽  
The Other ◽  
Compact Complex Manifold ◽  
Yamabe Problem ◽  
Conformal Metric ◽  
Hermitian Metric ◽  
Dimension Formula ◽  
Complex Dimension ◽  
Yamabe Soliton

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

scholarly journals On the Yamabe problem and the scalar curvature problems under boundary conditions

2002 ◽  
Vol 322 (4) ◽  
pp. 667-699 ◽  
Author(s):  
Antonio Ambrosetti ◽  
YanYan Li ◽  
Andrea Malchiodi
Keyword(s):  
Boundary Conditions ◽  
Scalar Curvature ◽  
Yamabe Problem

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 Scalar curvature and the multiconformal class of a direct product Riemannian manifold

2021 ◽  
Author(s):  
Saskia Roos ◽  
Nobuhiko Otoba
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Direct Product ◽  
Warped Product ◽  
Riemannian Metrics ◽  
Positive Function ◽  
Positive Scalar Curvature ◽  
Conformal Class ◽  
Positive Scalar ◽  
Technical Assumption

AbstractFor a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots \times (M_l, g_l)$$ ( M , g ) = ( M 1 , g 1 ) × ⋯ × ( M l , g l ) , we define its multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] as the totality $$\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}$$ { f 1 2 g 1 ⊕ ⋯ ⊕ f l 2 g l } of all Riemannian metrics obtained from multiplying the metric $$g_i$$ g i of each factor $$M_i$$ M i by a positive function $$f_i$$ f i on the total space M. A multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] contains not only all warped product type deformations of g but also the whole conformal class $$[\tilde{g}]$$ [ g ~ ] of every $$\tilde{g}\in [\![ g ]\!]$$ g ~ ∈ [ [ g ] ] . In this article, we prove that $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of positive scalar curvature if and only if the conformal class of some factor $$(M_i, g_i)$$ ( M i , g i ) does, under the technical assumption $$\dim M_i\ge 2$$ dim M i ≥ 2 . We also show that, even in the case where every factor $$(M_i, g_i)$$ ( M i , g i ) has positive scalar curvature, $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of scalar curvature constantly equal to $$-1$$ - 1 and with arbitrarily large volume, provided $$l\ge 2$$ l ≥ 2 and $$\dim M\ge 3$$ dim M ≥ 3 .

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 On isospectral compactness in conformal class for 4-manifolds

2019 ◽  
Vol 21 (05) ◽  
pp. 1850041 ◽  
Author(s):  
Xianfu Liu ◽  
Zuoqin Wang
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Weyl Curvature ◽  
Yamabe Invariant ◽  
Weyl Curvature Tensor

Let [Formula: see text] be a closed 4-manifold with positive Yamabe invariant and with [Formula: see text]-small Weyl curvature tensor. Let [Formula: see text] be any metric in the conformal class of [Formula: see text] whose scalar curvature is [Formula: see text]-close to a constant. We prove that the set of Riemannian metrics in the conformal class [Formula: see text] that are isospectral to [Formula: see text] is compact in the [Formula: see text] topology.

scholarly journals A calculus for conformal hypersurfaces and new higher Willmore energy functionals

2020 ◽  
Vol 20 (1) ◽  
pp. 29-60 ◽  
Author(s):  
A. Rod Gover ◽  
Andrew Waldron
Keyword(s):  
Differential Operators ◽  
Yamabe Problem ◽  
Conformal Class ◽  
Energy Functions ◽  
Energy Functionals ◽  
Conformally Compact ◽  
Higher Dimensional ◽  
Invariant Differential ◽  
The Right ◽  
Willmore Energy

AbstractThe invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].

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