# Is the conformal compactification of $M \setminus \{ p \}$ unique?

Let $$(M,c)$$ be a compact conformal manifold and $$p \in M$$. $$M$$ is a conformal compactification of $$M \setminus \{ p \}$$, because the embedding $$M \setminus \{p\} \hookrightarrow M$$ is an isometry. Is this conformal compactification unique?

I believe I saw a preprint on the Arxiv that gave a positive answer to this question around a month ago, but I can no longer find it despite spending a long time on it. The reason I am interested in this, is that I am wondering whether the initial statement can be proven using the idea from M Eastwood: Uniqueness of the stereographic embedding.

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Theorem 1.4 in C. Frances' preprint "Rigidity at the boundary for conformal structures and other Cartan geometries" asserts that the geodesic compactification is unique (up to conformal diffeomorphism) in this situation. More generally, he proves uniqueness when one removes a closed set of Hausdorff dimension strictly less than $$\dim M - 1$$.