粘接Bartnik扩张，Bartnik质量的连续性和定义的等价性
Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions
摘要：
在Bartnik质量的上下文中，有两个根本不同的概念，一些紧黎曼流形$(Ω,γ)$的扩展与边界。在一种情况下，扩展被视为没有边界的流形，其中$(Ω,γ)$等距嵌入，在另一种情况下，扩展被视为具有边界的流形，其中边界数据由$\partial Ω$确定。我们给出了一种凸性条件，在此条件下，我们可以说这两种类型的扩展确实产生相同的Bartnik质量值。在同样的假设下，我们证明了Bartnik质量不断变化的边界数据。这也提供了一种方法来使用恒定中曲率（CMC）Bartnik数据的Bartnik质量估计值，以获得非CMC Bartnik数据的Bartnik质量估计值。这些结果的关键思想是一种方法，用于将给定的Bartnik数据的Bartnik扩展粘合到附近的其他Bartnik数据。
英文摘要：
In the context of the Bartnik mass, there are two fundamentally different notions of an extension of some compact Riemannian manifold $(Ω,γ)$ with boundary. In one case, the extension is taken to be a manifold without boundary in which $(Ω,γ)$ embeds isometrically, and in the other case the extension is taken to be a manifold with boundary where the boundary data is determined by $\partial Ω$. We give a type of convexity condition under which we can say both of these types of extensions indeed yield the same value for the Bartnik mass. Under the same hypotheses we prove that the Bartnik mass varies continuously with respect to the boundary data. This also provides a method to use estimates for the Bartnik mass of constant mean curvature (CMC) Bartnik data, to obtain estimates for the Bartnik mass of non-CMC Bartnik data. The key idea for these results is a method for gluing Bartnik extensions of given Bartnik data to other nearby Bartnik data.
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