Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere

Abstract

For every $n \ge 3$, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round $n$-sphere $\Ss^n$. These families are pairwise distinct up to conformal equivalence, and no member is conformally equivalent to a rotationally symmetric solution. At every negative time, the Ricci curvature tensor of each solution is indefinite at some point. Moreover, the associated backward limit space is a wedge sum of finitely many isometric copies of $\Ss^n$. These examples show that the collection of ancient Yamabe flows on $\Ss^n$ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on $\Ss^2$, all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on $\R^n$, whose positive entire solutions are only the standard bubbles. The construction uses a non-radial inner--outer gluing scheme. After stereographic projection, we reformulate the flow as a conformally invariant parabolic problem on $\R^n$. By exploiting Kelvin invariance and switching between the Euclidean and spherical formulations as needed, we control the non-radial modes directly without reducing the problem to one space dimension. Weighted Hölder estimates provide the pointwise control needed to establish the Type II behavior, the Ricci-sign property, conformal inequivalence, and the description of the backward limits in a straightforward manner.