Boundary expansions of complete conformal metrics with negative Ricci curvatures

Abstract

We study the boundary behaviors of a complete conformal metric which solves the $\sigma _k$-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also $C^1$ and $C^2$ estimates for this metric multiplied by the square of the distance in a small neighborhood of the boundary.

5. Appendix

Principle coordinates are discussed in details in [13]. In this appendix, we collect several results from [13] most relevant to our study.

Let $\Omega \subset {\mathbb {R}}^n$ be a bounded domain with a smooth boundary $\partial \Omega$ and d be the distance function to $\partial \Omega$. Set $\Gamma _\mu =\{x\in {\overline{\Omega }}|d(x)<\mu \}$ for sufficiently small $\mu$.

According to Lemma 14.16 [13], for each point $x\in \Gamma _\mu ,$ there exists a unique point $y=y(x)\in \partial \Omega$ such that $|x-y(x)|=d(x).$ The points x and y are related by $x=y(x)+\nu (y(x))d(x),$ where $\nu$ is the unit inner normal vector to $\partial \Omega$.

Next, let $x_0\in \Gamma _\mu$ and $y_0\in \partial \Omega$ be related by $|x_0-y_0|=d(x_0).$ According to Lemma 14.17 [13], in terms of a principle coordinate system at $y_0,$ we have

$$\begin{aligned} D^2d(x_0)=\text {diag}\Big (\frac{-\kappa _1}{1-\kappa _1d},\cdots , \frac{-\kappa _{n-1}}{1-\kappa _{n-1}d},0\Big ), \end{aligned}$$

(5.1)

where $\kappa _1,\cdots , \kappa _{n-1}$ are the principle curvatures of $\partial \Omega$ at $y_0$. By the proof of Lemma 14.17 [13], we also have

$$\begin{aligned} Dd(x_0)=\nu (y_0)=(0,0,\cdots ,1). \end{aligned}$$

(5.2)