Lemma 23.1.44EF821157486E

Proof. Set $m(x)=\int _ EV\, \mathrm dA_ E$ and $e(x)=\int _ E|V_ w|^2\, \mathrm dA_ E$. Periodic integration by parts gives

The mean of every fiber derivative vanishes. Expanding $\Delta _ V$ yields

Thus $f(x)=\log (1+\mathrm e^ x)+m(x)-e(x)/2$ is convex.

Both $m$ and $e$ are smooth global functions on the projective base: local fiber coordinates change by holomorphic translations, preserving fiber Haar area, the vertical derivative and the Dirichlet integral. They are $U(2)$-invariant. Smoothness and radial invariance at $\xi =0$ give $m',e'\to 0$ at the negative end. At the positive end use a normal coordinate $z_0/z_1$ to the hyperplane $\{ z_0=0\} $ on the base. The invariant base functions are constant along this hyperplane and smooth across it; their $x$ derivatives tend to zero. This does not require that the pointwise potential be constant on its elliptic fibers at that end. Consequently $f'(-\infty )=0$ and $f'(+\infty )=1$. Integrating the nonnegative $f''$ proves the flux identity.

For each fixed $w$, $R_ V\geq 0$ implies $V_{xx}\geq -1$ on all of $\mathbb R$. Hence for every real $h$,

Taking $h=V_ x(x,w)$ and using $V\leq 0$ proves the derivative bound. Finally $\mu \leq 1$ proves the bound for $A_ V$. $\square$