# Generalized stereographic projection

Let $$(M^n,g)$$ be a closed (compact, without boundary) Riemannian manifold and let $$p\in M$$. Let $$\square: =c\Delta+S$$ be the conformal Laplacian of $$(M,g)$$. Here $$c=4\frac{n-1}{n-2}$$ is a constant, $$\Delta$$ is the Laplacian and $$S$$ is the scalar curvature defined by $$g$$. Lastly, let $$G_p:M\setminus \{p\}\to \mathbb{R}$$ be the Green function of $$\square$$, so $$\square\, G_p=0$$ everywhere on $$M\setminus \{p\}$$. See here for a discussion of the conformal Laplacian $$\square$$ and its Green function $$G_p$$.

If $$(M,g)$$ has positive scalar curvature, then the generalized stereographic projection of $$(M,g)$$ is defined to be the Riemannian manifold $$(\hat{M},\hat{g})$$, where $$\hat{M}=M\setminus \{p\} \qquad \text{and} \qquad \hat{g}=G_p^{k-2}g,$$ and $$k=2n/(n-2)$$ is a constant. This is well defined when $$(M,g)$$ has positive scalar curvature, since in this case the Green function $$G_p$$ is strictly positive.

Question 1: If $$n\geq 4$$, is the Riemannian metric $$\hat{g}$$ on $$\hat{M}$$ complete?

I suspect this is the case as Lee and Parker prove that metric $$\hat{g}$$ is asymptotically flat of order at least 2.

My second question is related to the curvature properties of $$(\hat{M},\hat{g})$$. Note that $$(\hat{M},\hat{g})$$ has zero scalar curvature, as follows from the fact that $$\square\, G_p=0$$.

Question 2: How is the Ricci curvature of $$\hat{g}$$ related to the Ricci curvature of $$g$$? More specifically, if $$g$$ has Ricci curvature bounded above or below, does this give bounds on the Ricci curvature of $$\hat{g}$$?

• A naive question : is classical stereographic projection from the sphere minus one of its pole to a plane really a particular case of what you explain ? – Jean Marie Jan 12 at 22:40
• You can establish an asymptotic estimate for $G_p$ near $p,$ which in particular lets you bound it from below by $r^k$ for some constant $k(n).$ (You should definitely be able to find this in Lee-Parker.) You can then verify completeness of the metric by using this bound to show that the length of any curve approaching $p$ is infinite. – Anthony Carapetis Jan 13 at 7:08
• @JeanMarie: this is explained very nicely in Section 6 of Lee and Parker's paper (see hyperlink in my question). – rpf Jan 16 at 15:30