质量、容函数与质容比
Mass, capacitary functions, and the mass-to-capacity ratio
AI总结：
该研究探讨非负标量曲率渐近平坦3维流形的ADM质量、调和函数与边界容量的关联，推导单调量、几何不等式与质容比恒等式，给出正质量定理新证明，并研究质容比约束下流形的性质。
摘要：
我们研究了ADM质量、在无穷远处趋于零的正调和函数，以及具有非负标量曲率的渐近平坦3维流形边界的容量之间的联系。首先，我们给出了通过调和函数表征ADM质量的新公式。随后，若基础流形具有简单拓扑，我们推导出一族单调量与几何不等式。作为直接应用，我们得到了3维黎曼正质量定理的若干新证明。其中一种证明引出了新的充分条件，可通过分隔边界与无穷远的区域的C⁰几何来推导质量的正性。这类充分条件的一个特例表明：若包围边界的区域体积相对较小，则质量为正。作为进一步的应用，我们得到了质容比（mass-to-capacity ratio）的积分恒等式。我们还将这些不等式推广为旋转对称球面外的空间Schwarzschild流形上的等式。此外，我们证明质容比始终以1减去边界的归一化Willmore泛函的平方根为下界。受上述发现的启发，我们对满足质容比约束的流形展开了研究。我们指出，这类流形满足改进的不等式，其质量存在仅依赖于边界数据的上界，不存在包围边界的闭合极小曲面，且这些流形包含Bartnik拟局域质量语境下的静态延拓。
英文摘要：
We study connections among the ADM mass, positive harmonic functions tending to zero at infinity, and the capacity of the boundary of asymptotically flat $3$-manifolds with nonnegative scalar curvature. First we give new formulae that detect the ADM mass via harmonic functions. Then we derive a family of monotone quantities and geometric inequalities if the underlying manifold has simple topology. As an immediate application, we observe several additional proofs of the $3$-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions that imply positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such sufficient conditions shows, if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on spatial Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by our findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio. We point out such manifolds satisfy improved inequalities, their mass has an upper bound depending only on the boundary data, there are no closed minimal surfaces enclosing the boundary, and these manifolds include static extensions in the context of the Bartnik quasi-local mass.
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