Proposition 14.2.4366AA2E61158

Proof. At a maximum of $u-v$, one has $\omega _ u\leq \omega _ v$, so the equations give $\mathrm{e}^ b\leq \mathrm{e}^ c$. At a minimum one obtains $\mathrm{e}^ b\geq \mathrm{e}^ c$. Therefore $b=c$.

Put $h=u-v$ and $\eta _ t=\omega _ v+t\mathrm{dd}^{\mathrm{c}}h=(1-t)\omega _ v+t\omega _ u$ for $0\leq t\leq 1$. These forms are strictly positive. Subtracting the logarithms of the two equations and writing the difference as the integral of its derivative along the segment, we obtain

The right-hand side is a linear second-order operator applied to $h$. Its coefficients are smooth, its principal part is uniformly elliptic on the compact manifold, and it has no zeroth-order term. The strong maximum principle therefore implies that $h$ is constant. $\square$