Ricci曲率下界条件下的加权体积单调性与等周比较
Weighted volume monotonicity and isoperimetric comparison under a lower Ricci curvature bound
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中文总结 AI 辅助
本文在Ricci曲率下有界的完备黎曼流形上，建立了加权Bishop-Gromov体积比较定理，证明了在权函数比非递减时加权体积比的单调性，并应用于推导加权等周不等式。
中文摘要 AI 辅助
我们建立了Ricci曲率下有界的完备黎曼流形上Bishop-Gromov体积比较定理的加权推广。给定一点$p$和正径向权函数$f$和$h$，我们考虑测地球的加权体积以及常截面曲率$k$的单连通空间形式中相应的模型加权体积。我们证明，在$h/f$非递减的假设下，这两个加权体积之比关于半径是非递增的。我们还刻画了等号情形，表明等号迫使权函数之比为常数，且相应的测地球与模型球等距。作为应用，我们推导出加权Bishop-Gromov型比较、环形比较不等式和加权体积加倍估计。我们进一步应用单调性公式获得测地球的加权面积-体积不等式和尖锐的加权等周型比较。
英文摘要
We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature $k$. We prove that, under the assumption that $h/f$ is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.
