On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang [ 43] in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

中文翻译：

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p-格林函数的单调性

在具有非负标量曲率和消失的第二同调性的完整 $p$-非抛物线 $3$ 维流形上，我们为 $p$-Green 函数沿 $1<p<3 的水平集建立了锐单调性公式$。这可以被视为 Munteanu-Wang [43] 在 $p=2$ 情况下最近结果的概括。当 $1<p\leq 2$ 时，不对 $p$-Green 函数做出平滑假设。一些刚性结果也得到了证明。