The quaternionic contact Yamabe problem and the Yamabe constant of the qc spheres

A note on existence for the fractional Yamabe problem

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500450 ◽

2016 ◽

Vol 19 (04) ◽

pp. 1650045

Author(s):

Meng Wang

Keyword(s):

Yamabe Problem ◽

Yamabe Constant

The fractional Yamabe problem was proposed by González and Qing in [Fractional conformal Laplacians and fractional Yamabe problems, Anal. PDE 6(7) (2013) 1535–1576]. One of their results is that if the fractional Yamabe constant satisfies [Formula: see text], then the fractional Yamabe problem is solvable for [Formula: see text]. Using the method from Brezis–Lieb, we give a new, and shorter proof, of this statement.

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An Obata-type theorem on compact Einstein manifolds with boundary

Geometriae Dedicata ◽

10.1007/s10711-021-00598-y ◽

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 ¯ ] ) .

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