# Perelman's example on nonuniqueness of tangent cones at infinity

Perelman has an example on manifolds with nonunique tangent cones at infinity. The paper is here. It is a complete manifold with positive Ricci curvature, Euclidean volume growth, and quadratic curvature decay. The metric has the form $ds^2=dt^2+A(t)^2 dx^2+B^2 (t) dy^2+C^2 (t) dz^2$, with a particular choice of $A,B,C$ as functions of $t$ given in the paper linked above. While the $Ric>0$ is easier to verify, I have difficulty to understand why the tangent cones are not unique. My question is: what different cross sections (I mean the set $\{r=1\}$ on the tangent cones at infinity) they have if we pick different sequences $r_i \rightarrow +\infty$ in $(M,\frac{1}{r_i^2}g, p)$ to get the tangent cone. Since different tangent cones must have the same cone angle, those cross sections must have the same Hausdorff measure. In this sense it is harder for me to imagine why they could be different. Any help will be appreciated.

The unit spheres are isometric to Berger's spheres. They have the same volume, but not isometric.

• Oh, Now I can see it. Thank you very much! – Bo_Y Oct 14 '12 at 3:12