In this article, we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature of unitary volume, for simplicity, CPE metrics. Here, we prove that a CPE metric admitting a non‐trivial closed conformal vector field must be isometric to a round sphere metric, which provides a partial answer to the CPE conjecture.

中文翻译：

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临界点方程和闭合共形矢量场

在本文中，为了简单起见，我们研究CPE度量，将总标量曲率函数的临界点限制为具有恒定单位体积标量曲率的度量空间。在这里，我们证明了允许非平凡的封闭共形矢量场的CPE度量必须与圆球度量相等，这为CPE猜想提供了部分答案。