First let me introduce some definitions.

  • By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in $\mathbb{C}$. Two germs are considered the same if they coincide on some punctured neighborhood of $0$.

  • A germ $g$ is said to be complete if $\lim_{z\rightarrow 0}d(z, z_0)=+\infty$ for any $z_0\in U$, where $d(\cdot,\cdot)$ is the distance induced by $g$.

  • To germs are said to be bi-Lipschitz if their proportion is bounded from above and below by positive constants (on some punctured neighborhood of $0$).

A lemma in a paper of Benoist and Hulin (Lemma 5.2 therein) implies the following highly nontrivial result:

Theorem. All germs with pinched negative curvature (i.e. with curvature bounded from above and below by negative constants) are bi-Lipschitz to each other.

One can prove this theorem by adapting the arguments in that paper: first extend germs to complete conformal metrics on $\mathbb{C}\setminus\{0\}$ with bounded curvature, then apply Generalized Maximum Principle of Omori-Yau. This proof seems somewhat unnatural, since it uses a strong global theorem while the problem is purely local.

Questions. Does there exist similar results in the literature about bi-Lipschitz classification of complete germs? Is there a more direct proof of the above theorem?


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