论文标题
边界的总平均曲率和非负标量曲率填充
Total mean curvature of the boundary and nonnegative scalar curvature fill-ins
论文作者
论文摘要
在本文的第一部分中，我们证明了任意边界指标对带有边界的紧凑型歧管的正标度曲率（PSC）度量的可扩展性，这完全解决了由于gromov引起的开放问题（请参阅问题\ ref Ref {Extensive1}）。然后，我们引入一个不变的填充（请参阅定义\ ref {fillininvariant}），并讨论其与渐近平坦（AF）和渐近双曲线（AH）歧管的正质量定理的关系。此外，我们证明了AH流形的正质量定理意味着对AF流形。最后，我们给出了一些填充不变的估计值，这些估计为Gromov在\ cite {gro19}中提出的猜想提供了一些部分肯定的答案（请参见Cunjecture \ ref cef {conj0}和Cunixute \ ref Ref {conj1}下面）
In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question \ref{extension1}). Then we introduce a fill-in invariant (see Definition \ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give some estimates for the fill-in invariant, which provide some partially affirmative answers to Gromov's conjectures formulated in \cite{Gro19} (see Conjecture \ref{conj0} and Conjecture \ref{conj1} below)
