Symmetry Breaking and Moduli for Ancient Yamabe Flows on the Sphere

Abstract

For every $n\ge 3$, we construct compact Type II ancient Yamabe flows in the round conformal class of the unit $n$-sphere $\Ss^n$. After stereographic projection and rescaling of time and conformal factor, the solutions are asymptotic backward in time to finite superpositions of Aubin--Talenti bubbles with mutually comparable scales. For integers $h\ge 1$ and $k\ge 2$, the limiting centers form $h$ coaxial regular $k$-gons; when $h\ge 2$, their distinct layer heights may be prescribed arbitrarily in the interval $(-1,1)$. If $hk\ge 3$, the resulting flows are not geometrically equivalent to any rotationally symmetric ancient flow. For each flow, there is a fixed point of $\Ss^n$ at which the Ricci tensor has both positive and negative eigenvalues on every time slice. The corresponding volume-normalized Yamabe flows are also ancient. After rescaling each time slice to a fixed volume, the metrics converge to a wedge of $hk$ isometric round $n$-spheres in the Gromov--Hausdorff topology, as time tends to $-\infty$. For fixed $h,k\ge 2$, the concentration configurations admit an explicit $(h-1)$-parameter description up to Möbius transformations. Consequently, the construction yields geometrically inequivalent flows indexed by a nonempty open subset of ${\mathbb{R}}^{h-1}$, all with the same fixed-volume backward Gromov--Hausdorff limit. The proof uses inner--outer gluing for the quasilinear parabolic equation in $n$-dimensional space that exhibits Kelvin symmetry. Weighted Euclidean and spherical Hölder estimates control the nonlinear terms and, together with parabolic regularity, yield smooth extension of the conformal metrics across the north pole of $\Ss^n$.

arXiv commentv2: Title and exposition revised; proofs clarified, with a more detailed treatment of the backward geometric limits and the classification of concentration configurations up to Möbius transformations