Pome

Proposition 18.15.1C083614DAC5B

Proposition 18.15.1.C083614DAC5B For $F\in C^\infty (M,\mathbb R)$, the function

\[ \widehat F(y)=\min _{z\in Z\setminus D}\{ F(y,z)-\psi (z)\} \qquad (y\in C) \]

is finite and continuous. The two envelopes satisfy

95879292B693 18.15.2
\begin{equation} \begin{split} P_{\Theta ,F}[u]& =\Psi +P_{\eta ,\widehat F}[w]\circ p,\\ P_{\Theta ,F}[u]_{\mathcal I} & =\Psi +P_{\eta ,\widehat F}[w]_{\mathcal I}\circ p. \end{split} \end{equation}

Proof. Apply proposition 18.14.1 to $p:M\to C$, with $H=C\times D$, $\beta =\eta $ and $\Psi =\psi \circ r$. The local logarithmic form of $\psi $ defines a smooth metric on $\mathcal O_ M(H)$ with curvature $r^*\kappa $. Thus $\Theta +\mathrm{dd}^{\mathrm{c}}\Psi =p^*\eta +[H]$, and $H$ is smooth and transverse to the fibers. The induced fiber minimum is the function $\widehat F$ above. For $F=0$, the normalization $\sup _ Z\psi =0$ gives $\widehat0=0$. $\square$

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