AI 中文总结
研究推广Ricci-Yang-Mills流的耦合几何演化方程，推导曲率估计并发现类似Perelman熵的单调泛函，证明有限时间奇点处黎曼曲率强主导规范曲率，并给出S^4上SU(2)丛的收缩自相似解实例。
AI 中文摘要
我们研究一族耦合几何演化方程，这些方程描述闭流形上黎曼度量与非阿贝尔规范场的变形，该族方程推广了Ricci-Yang-Mills流。我们推导了内部曲率估计，找到了一个保持的积分曲率条件，并发现了一个与Perelman熵类似的尺度不变单调泛函。特别地，我们证明了在所有维度上，有限时间奇点处黎曼曲率对规范曲率的强主导性。我们还提供了一个在S^4上的SU(2)丛上的收缩自相似解的非平凡显式例子。
英文摘要
We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci--Yang--Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a $SU(2)$ bundle over $S^4$.
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