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

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 ${\text{Ss}}^n$. 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 ${\text{Ss}}^n$. These examples show that the collection of ancient Yamabe flows on ${\text{Ss}}^n$ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on ${\text{Ss}}^2$, all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on ${\mathbb{R}}^n$, 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 ${\mathbb{R}}^n$. 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.