https://neckpinch.com/authors/seunghyeok-kim/New arXiv papers by Seunghyeok Kim, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2606.22355/arxiv:2606.22355Sun, 21 Jun 2026 00:00:00 GMTHaixia Chen, Seunghyeok Kim, Monica Musso — For every n ≥ 3, we construct uncountably many families of type II ancient solutions to the Yamabe flow on the unit round n-sphere Ssⁿ. 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 Ssⁿ. These examples show that the collection of ancient Yamabe flows on Ssⁿ has a much richer structure than suggested by two natural comparison problems: the compact ancient Ricci flows on Ss², all of which are known to be rotationally symmetric, and the elliptic Yamabe equation on Rⁿ, 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 Rⁿ. 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.Ricci flowYamabe flowAncient solutionsSingularitiesRotational symmetry & warped products