[论文解读] Rigidity of Critical Point Metrics under some Ricci curvature constraints

Tongzhu Li, Junlong Yu|arXiv (Cornell University)|Mar 11, 2026

Geometric Analysis and Curvature Flows被引用 0

一句话总结

论文在一定的 Ricci 曲率约束下证明了 CPE（临界点方程）猜想成立，表明当 traceless Ricci 张量满足给定条件（包括常数范数或三次迹不等式，以及对 3D 的细化）时，CPE 度量必为爱因斯坦（球面）。

ABSTRACT

A critical point metric is a critical point of the total scalar curvature functional restricted to the space of constant scalar curvature metrics on a closed manifold with unit volume. It was conjectured in 1980's that every critical point metric must be Einstein. In this paper, we prove that this conjecture is true if the norm of the traceless Ricci operator $|\widetilde{Ric}|$ is constant. For $3$-dimensional case, we prove that the conjecture is true, if the traceless Ricci operator satisfies $tr((\widetilde{Ric})^3)\geq -\frac{R}{12}|\widetilde{Ric}|^2$, where $R$ denotes the scalar curvature. where R denotes the scalar curvature.

研究动机与目标

• Motivate and study the CPE conjecture: to determine when critical point metrics of the total scalar curvature functional are Einstein.
• Derive integral identities related to the traceless Ricci tensor on CPE manifolds to establish rigidity under curvature constraints.
• Show that constant norm of the traceless Ricci tensor forces the metric to be Einstein; extend results to 3D under trace inequalities.

提出的方法

• Use the Euler-Lagrange framework for the total scalar curvature functional restricted to unit-volume metrics with constant scalar curvature.
• Derive and manipulate integral identities involving the traceless Ricci tensor 过滤 and the potential function f via divergence theorem.
• Employ expansions of the Riemann curvature tensor, Weyl tensor (where applicable), and Ricci identities to obtain key propositions (e.g., Propositions 2.1-2.3)
• In 3D, use special Ricci tensor decompositions and associated identities to obtain further rigidity results (Propositions 3.1-3.2).
• Apply Lagrange multipliers-inspired bounds (Lemma 3.1) to relate cubic and quadratic traces of 过滤 Ricci, yielding Theorems 1.4-1.6.

实验结果

研究问题

• RQ1 Under what Ricci curvature constraints does a CPE metric become Einstein (spherical)?
• RQ2 Does a constant norm of the traceless Ricci operator force rigidity toward spherical geometry?
• RQ3 What 3D curvature conditions (in terms of tr((过滤 t Ric)3) and |过滤 t Ric|) ensure CPE metrics are spherical?
• RQ4 How do integral identities involving the potential function f and 过滤 t Ric control rigidity on CPE manifolds?

主要发现

• If the squared norm of the traceless Ricci tensor |过滤 r Ric|^2 is constant on a CPE metric, then 过滤 r Ric = 0 and the metric is spherical (Theorem 1.3).
• For any closed CPE metric, if there exists k>=0 with \u000260\na (1+f)^{2k} 过滤 r Ric(\nabla f,\nabla f) dv_g \n 0, then the metric is spherical (Theorem 1.2).
• Corollary: If 过滤 r Ric(\nabla f,\nabla f) dv_g \n 0, the metric is spherical (Corollary 1.1).
• In 3D, if tr((过滤 r Ric)^3) \n \n \n -R/12 |过滤 r Ric|^2, then 过滤 r Ric = 0 and the manifold is isometric to S^3 (Theorem 1.4).
• Theorem 1.5 yields rigidity under |过滤 r Ric|^2 <= R^2/24, implying 过滤 r Ric = 0 and spherical geometry.
• Theorem 1.6 provides a 3D rigidity condition with -5R/24 |过滤 r Ric|^2 <= tr((过滤 r Ric)^3) <= 0, concluding 过滤 r Ric = 0 and spherical geometry.