An Obata-type theorem on compact Einstein manifolds with boundary

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