For $M$ a Riemannian manifold, with Riemannian metric $g$ and $x$ a point in M, what is the meaning of "$g$ on $M\backslash\{x\}$ has an 'asymptotically flat end at $x$'."? (See this paper on page 16, line 4).

What are some basic properties of such metrics?


The sentence you wrote in your OP makes no sense.

The sentence which you intend to quote reads:

[T]he metric $g_{v_\infty}$, on $M\setminus \{x_\infty\}$ has an 'asymptotically flat end at $x_\infty$'.

Now, $g_{v_\infty}$ is not a Riemannian metric on $M$. The metric $g_{v_\infty}$ is constructed from $g$ with a conformal weight given by $v_\infty$, which by definition only exists on $M\setminus\{x_\infty\}$ as the function $v_\infty$ blows up at $x_\infty$. Hence $g_{v_\infty}$ is a Riemannian metric only on $M\setminus \{x_\infty\}$.

Therefore, the correct interpretation of the sentence you quoted is that

  1. The (non-compact) Riemannian manifold $(M\setminus \{x_\infty\}, g_{v_\infty})$ is asymptotically flat
  2. It only has one end, which is a neighborhood of $x_\infty$ in the following sense. There exists a neighborhood $V$ of $x_\infty$ (in the topology of the smooth compact manifold $M$) such that $M\setminus V$ is compact (automatically true) and that there exists a diffeomorphism of $V\setminus \{x\}$ with $\mathbb{R}^d \setminus \overline{B(1,0)}$ such that the usual asymptotic flatness decay condition for the metric $g_{v_\infty}$ holds.

In other words, the point $x_\infty$ is "the point at infinity" for its corresponding asymptotic flat end, or that one can regard $(M,g)$ as a one-point conformal compactification of $(M\setminus \{x_\infty\},g_{v_\infty})$

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