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

我们在具有光滑边界\（\ partial W \）的紧致Einstein n流形\（（W，\ bar {g}）\）上展示一种Obata型定理。假设边界\（\ partial W \）在\（（W，\ bar {g}）\）中最小。如果\（（\ partial W，\ bar {g} | _ {\ partial W}）\）与\（（S ^ {n-1}，g_S）\）没有保形地同构，则对于任何爱因斯坦度量\ （\ check {g} \ in [\ bar {g}] \）中的边界条件最小，直到重新缩放为止，我们都有\（\ check {g} = \ bar {g} \）。在这里，\（g_S \）和\（[\ bar {g}] \）分别表示\（（n-1）\）上的标准舍入度量-sphere \（S ^ {n-1} \）和保形类\（\ bar {g} \）。此外，如果我们假设\（\ partial W \ subset（W，\ bar {g}）\）完全是测地线，那么对于\（（W，\ partial的相对Yamabe常数），我们还显示了Gursky-Han型不等式W，[\ bar {g}]）\）。