Affiliation:
1. Department of Mathematics , Penn State University , 109 McAllister Building , University Park , PA 16802 , USA
Abstract
Abstract
We extend Vétois’ Obata-type argument and use it to identify a closed interval
I
n
I_{n}
,
n
≥
3
n\geq 3
, containing zero such that if
a
∈
I
n
a\in I_{n}
and
(
M
n
,
g
)
(M^{n},g)
is a compact conformally Einstein manifold with nonnegative scalar curvature and
Q
4
+
a
σ
2
Q_{4}+a\sigma_{2}
constant, then it is Einstein.
We also relax the scalar curvature assumption to the nonnegativity of the Yamabe constant under a more restrictive assumption on 𝑎.
Our results allow us to compute many Yamabe-type constants and prove sharp Sobolev inequalities on compact Einstein manifolds with nonnegative scalar curvature.
In particular, we show that compact locally symmetric Einstein four-manifolds with nonnegative scalar curvature extremize the functional determinant of the conformal Laplacian, partially answering a question of Branson and Ørsted.
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