Corollary 14.5.836B42C74EE0B

Proof. Theorem 14.5.5 gives uniform ellipticity and a bound for $\Delta _ gu=S-n$. The linear elliptic $W^{2,p}$ estimates for the fixed operator $\Delta _ g$ first bound $u$ in $C^{1,\alpha }$ for every $0<\alpha <1$. The local Evans–Krylov estimate for the concave logarithmic determinant then gives a uniform $C^{2,\beta }$ bound for some $0<\beta <1$. To apply it in its real form, project real symmetric Hessians to their complex Hermitian part and extend the logarithmic determinant concavely from the compact set of positive matrices given by the preceding theorem. The extension is uniformly elliptic on real symmetric matrices because the projection preserves positivity and the trace, so the usual Evans–Krylov theorem applies. Finally, differentiating the equation gives linear uniformly elliptic equations with Hölder coefficients, and Schauder estimates and induction give all higher bounds. No regularity results beyond these standard local elliptic estimates are needed. $\square$