与$Q$-曲率相关的几何流

中文摘要 AI 辅助

Branson的$Q$-曲率是共形几何中标量曲率的自然四阶对应物。类似于从线性化标量曲率恢复Ricci张量的经典方法，Lin和Yuan通过线性化$Q$-曲率的正式伴随引入了对称$2$-张量$J_g$，并建立了其关于Bach张量和辅助曲率项的显式分解。在本文中，我们研究在维数$n \ge 4$的闭黎曼流形上相关的四阶几何流$\partial_t g = -2J_g$。我们利用双调和DeTurck规范建立了短时存在性和唯一性，并在曲率界下导出了全局和局部积分光滑估计。在维数四中，我们证明了在一致Sobolev不等式和在整个最大存在区间内总$L^2$曲率能量足够小的假设下，长时间存在性。在一致曲率和Sobolev界下，$J$的$L^2$-范数的时间可积性意味着在固定坐标下光滑收敛到$J$-平坦度量。显式的Einstein和乘积解说明了演化过程，并表明仅靠小曲率能量并不能防止退化。正的初始Yamabe常数和正的总$Q$-曲率在整个四维光滑流中给出一个一致的正Yamabe下界，而在更高维数中，总$Q$-曲率即使在平坦度量附近也可能减小。最后，我们证明了每个闭四维$J$-孤立子都是平凡的，其度量是Bach平坦且具有常数$Q$-曲率，其孤立子向量场是Killing的。

英文摘要

Branson's $Q$-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor $J_g$ through the formal adjoint of the linearized $Q$-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow $\partial_t g = -2J_g$ on closed Riemannian manifolds of dimension $n \ge 4$. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total $L^2$ curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the $L^2$-norm of $J$ implies smooth convergence in fixed coordinates to a $J$-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total $Q$-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total $Q$-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional $J$-soliton is trivial, its metric is Bach-flat with constant $Q$-curvature, and its soliton vector field is Killing.

发表机构

• Xiamen University of Technology（厦门理工学院）
• Sun Yat-sen University（中山大学）

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