Corollary 15.20.2D0124640DBA7

Proof. Use the radial coordinate $r$, rank-one forms $P,Q,R$, Fubini–Study form $\omega _ X$, and scalar functions $F,x,y,z$ from 14.26.2, 14.26.7. Set $w=F(r)$, $\omega _0=xP+yQ+zR$, $H=\mathrm{dd}^{\mathrm{c}}w/2$, $A_0=\omega _0-H$ and $B_0=\omega _0+H$ as in 14.26.9. The Hermitian form is $\omega =\omega _0+\mathrm{dd}^{\mathrm{c}}\rho $, with $\rho =\varepsilon (r^2-1)\kappa (r)$ from 14.26.10. Choose the cutoff $\kappa $ even; all its required support and plateau properties are preserved by this choice. Then $\rho $ is even, while $w=F(r)$ is odd. The involution sends $r$ to $-r$, interchanges $P,Q$, and fixes $R$. The coefficient identities $x(-r)=y(r)$ and $z(-r)=z(r)$ show that $\iota ^*\omega _0=\omega _0$. Thus $\iota ^*\omega =\omega $, and $\iota ^*\omega _ X=\omega _ X$.

Since $\iota ^*H=-H$, the involution interchanges $A_0$ and $B_0$. Their cubes are equal, so $f\omega _ X^3$ is invariant. It also interchanges $\widetilde u=-w/2-\rho $ and $\widetilde v=w/2-\rho $. These functions therefore have the same supremum, and the normalized solutions $u,v$ are interchanged. They are distinct, so neither is invariant.

Apply proposition 15.20.1 to the group generated by $\iota $. Here $c=1$, as the two constructed solutions already have that scalar. An invariant cluster solution $a$ cannot equal either $u$ or $v$. Its bounded equation with smooth density gives Hölder regularity by theorem 15.5.1. This gives a third solution. Its regularity is determined in subsection 15.20.3, subsection 15.20.7: it is $C^{1,1}$ but not $C^2$. $\square$