17.8 Energy along weak geodesics0753F06C3AFA

Proof. Fix a ball $B\Subset \Omega $ and a concentric ball $B_1$ with $B\Subset B_1\Subset \Omega $. Put $h_ j=\max \{ h,-j\} $ and let $\widehat h_ j$ be the function supplied by corollary 7.7.5 for $h_ j$ and $B$. These are locally bounded plurisubharmonic functions with $\widehat h_ j\geq h_ j\geq h$, equal to $h_ j$ outside $B$, with $(\mathrm{dd}^{\mathrm{c}}\widehat h_ j)^ d=0$ on $B$; they decrease with $j$, because $\widehat h_ j$ is increasing in $h_ j$. Their decreasing limit $v$ is plurisubharmonic by theorem 5.10.1, since it dominates $h$, and it equals $h$ outside $B$. Maximality gives $v\leq h$, hence $v=h$.

Fix $C>0$ and $\varepsilon >0$, and put

The functions $z_ j$ are plurisubharmonic, locally uniformly bounded (between $-C$ and $\widehat h_1$), and decrease to $z$, so they converge to $z$ locally in capacity on the bounded open set $B_1$ (proposition 7.5.6). The weights take values in $[0,1]$, and since $s\mapsto (s+C)/(s+C+\varepsilon )$ is Lipschitz with constant $\varepsilon ^{-1}$ on $[-C,\infty )$, we have $|a_ j-a|\leq \varepsilon ^{-1}|z_ j-z|$, so $a_ j\to a$ locally in capacity on $B_1$. The limit weight $a$ is quasi-continuous on $B_1$, because $z$ is (theorem 7.5.2) and $a$ is a continuous function of $z$. On $B$, the weight $a_ j$ vanishes where $\widehat h_ j\leq -C$, and on the plurifine open set $\{ \widehat h_ j>-C\} $ we have $z_ j=\widehat h_ j$; hence plurifine locality of Bedford–Taylor products (theorem 5.47.22) gives

The last assertion of theorem 7.5.8, applied on $B_1$ with $d$ factors $z_ j$ and the weights $a_ j$, gives

so $a(\mathrm{dd}^{\mathrm{c}}z)^ d=0$ on $B$. Since $a>0$ exactly on $\{ h>-C\} $ and $z=\max \{ h,-C\} $, the measure $\mathbf1_{\{ h>-C\} }(\mathrm{dd}^{\mathrm{c}}z)^ d$ vanishes on $B$. Taking $C=k$ for all integers $k$, the truncated products in the definition of $\langle (\mathrm{dd}^{\mathrm{c}}h)^ d\rangle $ all vanish on $B$; hence the non-pluripolar product is locally finite and zero on $B$. A cover of $\Omega $ by such balls proves the lemma. $\square$

Proof. Let $Z$ be either $X\times S$, on which the group $\Gamma =\mathbb R$ acts by the imaginary translations $(x,\zeta )\mapsto (x,\zeta +\mathrm iy)$, or $X\times A$ with $A=\{ \mathrm e^{-1}<|z|<1\} $, on which $\Gamma =\mathbb R$ acts by the rotations $(x,z)\mapsto (x,\mathrm e^{\mathrm iy}z)$, according to where $D$ lies. Put $\tau (x,\zeta )=\operatorname{Re}\zeta $ in the first case and $\tau (x,z)=-\log |z|$ in the second, and let $\theta _ Z$ be the pullback of $\theta $ to $Z$. The action of $\Gamma $ preserves $\theta _ Z$ and $\tau $, and the $\Gamma $-invariant functions on $Z$ are exactly the functions whose value at $(x,w)\in Z$ is $v_{\tau (x,w)}(x)$, for a family $(v_ t)_{0<t<1}$ of functions on $X$; by the discussion following definition 17.7.1, such a function is $\theta _ Z$-psh if and only if the complexification of $(v_ t)$ on $X\times S$ is $p^*\theta $-psh. Let $U$ be the $\Gamma $-invariant function on $Z$ defined by the weak geodesic $(G_ t)$; it is $\theta _ Z$-psh and bounded above by $\max \{ \sup _ Xu_0,\sup _ Xu_1\} $, since $G_ t\leq (1-t)u_0+tu_1$.

Now $D\subset Z$ and $\mathrm{dd}^{\mathrm{c}}a=\theta _ Z$ on $D$, so that $h=U+a$ is plurisubharmonic on $D$; we show that it is maximal. Let $v$ be plurisubharmonic on $D$ with $v\leq h$ outside a compact set $K\subset D$. Every boundary point of $D$ in $Z$ has a neighborhood disjoint from $K$, on whose intersection with $D$ we have $v-a\leq U$; since $U$ is upper semicontinuous, the boundary condition of proposition 5.9.5 holds, and the function $\widetilde U$ equal to $\max \{ U,v-a\} $ on $D$ and to $U$ elsewhere is $\theta _ Z$-psh (apply the proposition in charts after adding smooth local potentials of $\theta _ Z$). Moreover $\widetilde U\geq U$, and $\widetilde U=U$ outside $K$. Since $v-a$ is bounded above on the compact set $K$, the function $\widetilde U$ is bounded above.

The functions $\widetilde U\circ \gamma $, $\gamma \in \Gamma $, are $\theta _ Z$-psh and uniformly bounded above, so their regularized upper envelope $\breve U$ is $\theta _ Z$-psh (theorem 5.10.2, applied in charts after adding smooth local potentials); it is $\Gamma $-invariant and satisfies $\breve U\geq \widetilde U$. The compact set $\tau (K)$ lies in an interval $[r_1,r_2]$ with $0<r_1\leq r_2<1$, so the $\Gamma $-orbit of $K$ lies in $\{ r_1\leq \tau \leq r_2\} $. Outside this set every $\widetilde U\circ \gamma $ equals $U\circ \gamma =U$, so $\breve U=U$ on the open set $\{ \tau <r_1\} \cup \{ \tau >r_2\} $, because $U$ is upper semicontinuous. Therefore $\breve U$ is defined by a subgeodesic whose slices coincide with $G_ t$ for $t<r_1$ and for $t>r_2$; in particular its endpoint potentials are $u_0$ and $u_1$. It is a competitor for the Perron segment between $u_0$ and $u_1$, so $\breve U\leq U$. Consequently $\widetilde U\leq U$, so $v-a\leq U$ on $D$, that is, $v\leq h$. This is maximality in the sense of definition 7.7.6. $\square$

Proof. Step 1: setting up a compact extension. Assume first that $u_0,u_1\in \mathcal E^\infty (X,\theta ,\phi )$, and write $G_ t=G_ t(u_0,u_1)$. Let $A=\{ \mathrm e^{-1}<|z|<1\} \subset \mathbb C\subset \mathbb P^1$, and let $U(x,z)=G_{-\log |z|}(x)$ be the complexification on $X\times A$. Choose a constant $C$ with $\phi -C\leq u_0,u_1\leq \phi +C$. Then $\phi -C$ is a competitor for the rooftop envelope $P_\theta (u_0,u_1)$, so the bounds of theorem 17.7.8 give $\phi -C\leq U\leq \phi +C$. Fix concentric annuli $A_0\Subset A_1\Subset A$. On $Y=X\times \mathbb P^1$, with projections $p_1\colon Y\to X$ and $p_2\colon Y\to \mathbb P^1$, let $\beta =p_2^*\omega _{\mathrm{FS}}$, where $\omega _{\mathrm{FS}}$ is the Fubini–Study form of mass one, and put

with a constant $B>0$ chosen below.

To glue $U$ to $\Phi $ we use a smooth function $q$ on $\mathbb P^1$ that is small on $A_0$ and large near the boundary of $A_1$. Write $A_0=\{ r_0<|z|<r_0'\} $ and $A_1=\{ r_1<|z|<r_1'\} $, so that $\mathrm e^{-1}<r_1<r_0<r_0'<r_1'<1$, and choose $\varepsilon >0$ with $r_1+\varepsilon <r_0$ and $r_0'<r_1'-\varepsilon $. Let $\chi _0$ be a smooth function on $\mathbb R$ such that $1-\chi _0$ is a smooth cutoff function for $[r_0,r_0']$ in $(r_1+\varepsilon ,r_1'-\varepsilon )$ (lemma 2.5.1(4)); thus $\chi _0$ takes values in $[0,1]$, vanishes on $[r_0,r_0']$, and equals $1$ outside $(r_1+\varepsilon ,r_1'-\varepsilon )$. The function $q(z)=(2C+4)\chi _0(|z|)-(C+2)$, with $q(\infty )=C+2$, is smooth on $\mathbb P^1$, because $|z|$ is smooth on $\mathbb C\setminus \{ 0\} $ and $q$ is constant near $0$ and near $\infty $. It equals $-C-2$ on $A_0$ and $C+2$ on a neighborhood of $\partial A_1$. Choose $B$ so large that $B\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q\geq 0$.

Then $\Phi +q$ is $\Theta $-psh, and so is $U$ on $X\times A_1$, since $\beta \geq 0$. Define $W=\max \{ U,\Phi +q\} $ on $X\times A_1$ and $W=\Phi +q$ on $Y\setminus (X\times A_1)$. Near $X\times \partial A_1$ we have $\Phi +q=\Phi +C+2>U$ outside the pole set of $\Phi $, on which both functions are $-\infty $; hence $W=\Phi +q$ on a collar inside $X\times A_1$. Thus, near every point of $Y$, $W$ is either the maximum of two $\Theta $-psh functions or equal to $\Phi +q$, and $W$ is $\Theta $-psh on $Y$. Moreover $W-\Phi $ is bounded, and $W=U$ on $X\times A_0$, since there $\Phi +q=\Phi -C-2<U$ outside the pole set of $\Phi $.

Step 2: a current on $\mathbb P^1$. Put $T=\Theta +\mathrm{dd}^{\mathrm{c}}W$ and $R=\Theta +\mathrm{dd}^{\mathrm{c}}\Phi $, and define the current of order zero on $\mathbb P^1$

Let $\chi $ be a smooth function on $\mathbb P^1$. By theorem 17.2.5 on the compact Kähler manifold $Y$, applied to the bounded quasi-psh differences $W-\Phi $ and $\chi \circ p_2$, we may move $\mathrm{dd}^{\mathrm{c}}$ from $\chi \circ p_2$ to $W-\Phi $. By (17.2.1), $\mathrm{dd}^{\mathrm{c}}(W-\Phi )\wedge \langle T^ j\wedge R^{n-j}\rangle =\langle T^{j+1}\wedge R^{n-j}\rangle -\langle T^ j\wedge R^{n-j+1}\rangle $, and the sum over $j$ telescopes. Therefore

On the other hand, corollary 13.6.1, with $Z=\mathbb P^1$ and $\gamma =B\omega _{\mathrm{FS}}$, shows that $H$ is given by the function $z\mapsto E_\theta ^\phi (W(\cdot ,z))$. In particular, on $A_0$, $H$ is the function $z\mapsto E_\theta ^\phi (G_{-\log |z|})$.

Step 3: $\mathrm{dd}^{\mathrm{c}}H$ vanishes on $A_0$. On $X\times A_0$ put $L=p_1^*\theta +\mathrm{dd}^{\mathrm{c}}U\geq 0$, so that $T=L+B\beta $ there. Locally $T$ has a plurisubharmonic potential which is the sum of a potential of $L$ and a smooth potential of $B\beta $. The product $\langle T^{n+1}\rangle $ is finite by theorem 13.5.4, so multilinearity (proposition 13.2.6), compatibility with smooth factors (proposition 13.4.3) and $\beta \wedge \beta =0$ give, on $X\times A_0$,

with all terms locally finite. Every point of $X\times A_0$ has a coordinate ball $D\subset X\times A_0$ on which $p_1^*\theta =\mathrm{dd}^{\mathrm{c}}a$ for a smooth function $a$ (lemma 6.14.2); there $L=\mathrm{dd}^{\mathrm{c}}(U+a)$, and lemma 17.8.2, lemma 17.8.1 give $\langle L^{n+1}\rangle =0$ on $D$. Hence $\langle L^{n+1}\rangle =0$ on $X\times A_0$, so that $\langle T^{n+1}\rangle =(n+1)B\beta \wedge \langle T^ n\rangle $ on $X\times A_0$. Every slice $W(\cdot ,z)$ has bounded difference from $\phi $ and hence mass $m$, so corollary 13.6.1 with weight one gives $(p_2)_*(\beta \wedge \langle T^ n\rangle )=m\omega _{\mathrm{FS}}$.

The current $R$ is treated in the same way. We have $R=p_1^*\theta _\phi +B\beta $, and $\langle (p_1^*\theta _\phi )^{n+1}\rangle =0$. Indeed, the local potentials of $p_1^*\theta _\phi $ do not depend on the variable in $\mathbb P^1$, and so neither do their bounded truncations; each truncation is the decreasing limit of smooth plurisubharmonic functions of the $n$ variables of $X$ (proposition 5.9.2(2)), whose $(n+1)$st powers of $\mathrm{dd}^{\mathrm{c}}$ vanish, so the bounded products of the truncations vanish by theorem 7.3.7, and with them the truncated products defining $\langle (p_1^*\theta _\phi )^{n+1}\rangle $. Hence $\langle R^{n+1}\rangle =(n+1)B\beta \wedge \langle R^ n\rangle $, and the same fiber formula, applied to $\Phi $ in place of $W$, gives $(p_2)_*\langle R^{n+1}\rangle =(n+1)Bm\omega _{\mathrm{FS}}$. Hence the right side of (17.8.4) vanishes on $A_0$, that is, $\mathrm{dd}^{\mathrm{c}}H=0$ on $A_0$.

Step 4: the energy is affine. The function $\tau \mapsto E_\theta ^\phi (G_\tau )$ is locally Lipschitz on $(0,1)$. Indeed, let $0<\delta <\frac12$. On the common finite locus the convex functions $\tau \mapsto G_\tau (x)-\phi (x)$ take values in $[-C,C]$, so by lemma 17.7.2(4) their chord slopes on $[\delta ,1-\delta ]$ are bounded by $2C/\delta $. Thus $G_ s-(2C/\delta )|s-\tau |\leq G_\tau \leq G_ s+(2C/\delta )|s-\tau |$ for $s,\tau \in [\delta ,1-\delta ]$ on the common finite locus, hence everywhere, and monotonicity and the shift formula give $|E_\theta ^\phi (G_\tau )-E_\theta ^\phi (G_ s)|\leq (2Cm/\delta )|s-\tau |$.

Writing $z=\xi +\mathrm i\eta $, we have $\mathrm{dd}^{\mathrm{c}}f=(4\pi )^{-1}\Delta f\, \mathrm d\xi \wedge \mathrm d\eta $ for smooth functions $f$, and hence for distributions, so $\Delta H=0$ on $A_0$ by Step 3, and Weyl's lemma (theorem 3.1.11) shows that $H$ is represented on $A_0$ by a harmonic function. By Step 2, $H$ is also represented on $A_0$ by the function $z\mapsto E_\theta ^\phi (G_{-\log |z|})$, which is continuous by the Lipschitz bound just proved. Two locally integrable functions defining the same distribution agree almost everywhere (proposition 2.5.3(4)), and two continuous functions that agree almost everywhere agree everywhere, so this function is harmonic on $A_0$. A radial harmonic function on an annulus is an affine function of $\log |z|$: if $f(z)=g(\log |z|)$ is smooth, then $g$ is smooth and the Laplacian in polar coordinates gives $\Delta f=|z|^{-2}g''(\log |z|)$, so $g''=0$. Since $A_0$ was arbitrary, $\tau \mapsto E_\theta ^\phi (G_\tau )$ is affine on $(0,1)$.

Finally, $|u_1-u_0|\leq 2C$ on the common finite locus, so proposition 17.7.9(5) gives $|G_\tau -u_0|\leq 2C\tau $ and $|G_\tau -u_1|\leq 2C(1-\tau )$ there. By the Lipschitz bound of proposition 17.3.9, the energies converge to $E_\theta ^\phi (u_0)$ and $E_\theta ^\phi (u_1)$ as $\tau \to 0$ and $\tau \to 1$. This proves the theorem for $u_0,u_1\in \mathcal E^\infty (X,\theta ,\phi )$.

Step 5: finite energy. Let $u_0,u_1\in \mathcal E^1(X,\theta ,\phi )$. Adding the same constant $c$ to both endpoints adds $c$ to $G_ t$, by proposition 17.7.9(2), and adds $cm$ to both sides of the asserted identity. We may therefore assume $u_0,u_1\leq \phi $. Put $u_{j,k}=\max \{ u_ j,\phi -k\} $ for $j=0,1$. By proposition 17.7.9(3), $G_ t(u_{0,k},u_{1,k})\downarrow G_ t(u_0,u_1)$ as $k\to \infty $, and the limit has finite energy by theorem 17.7.8. The identity holds for $u_{0,k},u_{1,k}$ by the preceding steps, and proposition 17.3.13, applied at the endpoints and at each interior slice, passes it to the limit. $\square$