17.16 Perron interpolation and the geodesic equationE4ED60C8BCA5

Proof. Affine functions of $\operatorname{Re}z$ have zero complex Hessian, and a positive affine change of time is induced by a holomorphic affine change of the strip coordinate. Convex combinations preserve the inequality between currents, and the change $(\theta ,u_ t)\mapsto (\theta +\mathrm{dd}^{\mathrm{c}}h,u_ t-h)$ preserves the current itself. Maxima are covered by lemma 5.24.1. $\square$

Proof. For a compact interval $J\Subset I$, choose $a'<\inf J\leq \sup J<b'$ in $I$. By lemma 5.41.1, the path lies below the chord through $u_{a'}$ and $u_{b'}$, hence below a constant on $J$, and by proposition 5.41.3 the function $\inf _{a'<t<b'}u_ t$ is an integrable lower bound. For almost every $x$, the function $t\mapsto u_ t(x)$ is finite and convex, hence continuous, and dominated convergence proves continuity in $L^1$ on $J$.

For the chord bound, choose sequences of times tending to $a$ and $b$ along which the assumed $L^1$ convergence holds almost everywhere. Applying convexity between these times and passing to the limit proves (17.16.4) almost everywhere in $x$, for each $t$, and hence everywhere by lemma 5.24.3.

For the joint upper limit at $a$, note that in (17.16.4) the coefficient of $u_ a$ tends to one, while the term involving $u_ b$ has a vanishing coefficient and a uniform upper bound. Upper semicontinuity of $u_ a$ gives the assertion when $u_ a(x)$ is finite. At a pole of $u_ a$, bound $u_ a$ near $x$ above by an arbitrary real number, take the joint upper limit, and let that number tend to $-\infty $. The argument at $b$ is the same. $\square$

Proof. Convexity in $t$ and the upper limits at the endpoints bound every competitor by $(1-t)u_0+t u_1$. This bound is jointly upper semicontinuous on the interior and locally bounded above, so it persists after regularization. The local envelope theorem of section 5.40 shows that $U$ is quasi-plurisubharmonic with $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}U\geq 0$, and $U$ is still independent of $\Im z$. Since it dominates an admissible competitor, each slice belongs to $\mathrm{PSH}(X,\theta )$. The chord bound gives the joint upper limits at the endpoints, by the argument at the end of the proof of proposition 17.16.3. The envelope therefore belongs to its defining family and is its largest member.

If a competitor $w_ t$ runs from $u_0$ to $u_1$, then

In particular, almost everywhere,

which proves $u_ t^{\mathrm P}\to u_0$ in $L^1$ as $t\downarrow 0$. The argument at $t=1$ is the same.

For $0\leq a<b\leq 1$, let $v_ t$ be the Perron envelope on $(a,b)$ between $u_ a^{\mathrm P}$ and $u_ b^{\mathrm P}$. The original path restricted to $(a,b)$ runs between these endpoints and is admissible by proposition 17.16.3. Thus $v_ t$ also runs between them and dominates $u_ t^{\mathrm P}$. The joint upper limits of $v_ t$ at $a$ and $b$ allow us to glue its complexification to $U$ outside the smaller strip, by lemma 5.24.2; if $a=0$ or $b=1$, the same upper limits show that the glued family is still admissible for $u_0,u_1$. The resulting competitor cannot exceed $U$, which proves $v_ t=u_ t^{\mathrm P}$ on $(a,b)$. $\square$

Proof. Consider the barrier

It is a subgeodesic by proposition 17.16.2, and the bounded difference gives $w_0=u_0$ and $w_1=u_1$. Upper semicontinuity gives the joint upper limits at the endpoints. Off the common poles, $|w_ t-u_0|\leq Ct$ and $|w_ t-u_1|\leq C(1-t)$, so $w_ t$ runs from $u_0$ to $u_1$, and proposition 17.16.6 applies.

The lower bounds follow from $w_ t\leq u_ t^{\mathrm P}$, and the upper bounds from the chord estimate; together they give the asserted singularity type. Off the common poles, put $f(t)=u_ t^{\mathrm P}(x)$. For $0<s<t<1$, monotonicity of the slopes of secants of the convex function $f$ gives

The two-sided estimates at the endpoints cover the cases $s=0$ and $t=1$, which proves the Lipschitz bound on $[0,1]$. $\square$

Proof. The Schur complement of $H$ in the block matrix is $(\ddot u-\xi ^*H^{-1}\xi )/4$. Since $H>0$, the block matrix is semipositive exactly when this complement is nonnegative, and its determinant, which is $\det H$ times the complement, vanishes exactly when the complement vanishes.

To rewrite the equation in terms of forms, note that

Multiplying the second identity by $\eta _ t^ n>0$ gives (17.16.10). $\square$

Proof. Vary the path as $u_ t+s v_ t$, where $v_ t$ is smooth in $(x,t)$ and $v_0=v_1=0$, and put $f=\dot u_ t$. Differentiating the action at $s=0$, using $\partial _ s(\omega _{u_ t+sv_ t}^ n)|_{s=0}=n\mathrm{dd}^{\mathrm{c}}v_ t\wedge T_ t$, gives

We integrate by parts in $t$ in the first term, using $\partial _ t(\eta _ t^ n)=n\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$, and on $X$ in the second, using lemma 11.1.2 with the whole form $f^2T_ t$. Thus

The product rule gives

After the term $nf\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$ cancels, the integrand is $v_ t$ times the difference between the right-hand and left-hand sides of (17.16.13). Since $\eta _ t>0$ on the compact set $X\times [0,1]$, every smooth $v_ t$ gives an admissible variation for small $|s|$, and the fundamental lemma of the calculus of variations gives both implications. $\square$

Proof. The path is admissible and equals the chord between its endpoints, so by the chord bound no admissible competitor can be larger; it is therefore the Perron segment. Its acceleration and the spatial derivatives of its velocity vanish, which proves (17.16.10). In (17.16.13), the only remaining term is $\frac n2a^2\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})$, which vanishes exactly under the stated condition. $\square$

Proof. Write $\alpha =\alpha _ n$ and $\beta =\beta _ n$. Locally $\beta =(\mathrm i/(2\pi ))\partial r\wedge \bar\partial r$, and $\partial r$ has a nonzero vertical component. Every vector on $Y\times S$ therefore has a lift annihilated by $\partial r$. Testing the current on these lifts removes $\beta $; integrating the resulting distributional inequality against a nonnegative compact fiber density gives positivity on the base. Conversely, pullback of a positive base current plus $\beta $ is positive. An invariant genuine potential descends canonically by restricting to local holomorphic sections, so the argument also applies to singular potentials.

For bounded $v$, the local product with smooth corrections from section 13.14 gives

All other terms vanish by $\beta ^2=0$ and the dimension of $Y$. Apply this identity to $\max (v,-j)$ and restrict to $\{ v>-j\} $. Increasing finite-level limits give (17.16.17), without assigning mass to the pole set. The identity $\int _{\pi ^{-1}(y)}\beta =2$ from subsection 23.1.32 gives (17.16.18).

The full Perron competitor family is preserved by $E$. Its supremum and upper semicontinuous regularization are invariant. The chord bound and a pulled-back base competitor make this envelope a genuine subgeodesic by proposition 17.16.6. Positivity descent makes its descended envelope a base competitor with the prescribed joint boundary upper bounds. Conversely, every base competitor pulls back to a full competitor. The two envelopes consequently agree; in submersion charts, upper semicontinuous regularization commutes with pullback. $\square$

Proof. Fiber integration gives the exact restricted metric

The Kähler completion theorem therefore gives the stated completion and CAT$(0)$ property after multiplying base distances by $\sqrt{2n}$. If $\widetilde d_ D$ denotes the completed metric written with operator $\mathrm i\partial \bar\partial $ and background $\kappa $, the normalization is explicitly

because $\kappa +\mathrm i\partial \bar\partial (v/(2\pi ))=\kappa +\mathrm{dd}^{\mathrm{c}}v$. The preceding proposition identifies its unique geodesic with the full Hermitian Perron envelope. The base capacity endpoint limits give $L^1(Y)$ convergence. Since

the same holds on $X$; every smooth positive $\mathrm dV$ has bounded density ratio with $\omega ^ n$ on compact $X$. For smooth $v$, the constant translation is a base geodesic with speed $|c|$. Smooth metric approximation and translation by the same constant extend the distance formula to every $v\in \mathcal E^2$. $\square$

Proof. The length inequality and Cauchy–Schwarz give

Equality holds for the constant-speed geodesic. Equality for a connecting curve forces constant speed and equality between length and endpoint distance, hence forces a geodesic; uniqueness follows from the CAT$(0)$ property.

For a smooth basic path $\pi ^*v_ s$, put $h=v_{s+\delta }-v_ s$. On its short base Perron segment, parametrized by $0\leq \tau \leq 1$, the chord bounds above, and

bounds below. Indeed, the spatial forms along the chord have a uniform positive lower bound, $\mathrm dh=O(\delta )$, and proposition 17.16.9 makes the displayed path a subgeodesic for a fixed sufficiently large $C$. The right initial velocity of the Perron segment is therefore $h+O(\delta ^2)$ uniformly. The classical smooth-endpoint Kähler distance formula gives

Integration proves $\mathcal A_ B=\mathscr S$ along this path.

For stationarity, let $L(w)$ be the first variation at a smooth basic Perron path. Uniform spatial positivity allows every smooth endpoint-zero variation. Group invariance of the action and of the path gives $L(g^*w)=L(w)$ for $g\in E$. Average over Haar probability measure to obtain

for a smooth endpoint-zero base variation $q$. Fiber integration reduces this to $2n$ times the closed-base first variation. With $\eta _ t=\kappa +\mathrm{dd}^{\mathrm{c}}v_ t$, integration by parts gives that variation as

The bracket vanishes by the smooth base homogeneous equation. Thus $L(w)=0$ for every variation on $X$. $\square$

Proof. The constant path is basic, equals the chord, and is the full Perron segment by corollary 17.16.14. Its stationarity also follows from proposition 17.16.22. Let $w(t,r)$ be smooth, two-periodic in $r$, and zero at $t=0,1$. The paths $ct+\varepsilon w$ have positive spatial forms for sufficiently small $|\varepsilon |$. Expanding (17.16.24) to second order gives

The integral of $w_ rw_{rr}$ over a period vanishes, while $\int _0^1\langle w_ tw_{rr}\rangle \mathrm dt =-\frac12[\langle w_ r^2\rangle ]_0^1=0$. Completing the square gives the Hessian

For a positive integer $k$, take

Then $w_{k,t}+dcw_{k,r}=\pi \cos (\pi t)\cos (\pi k(r-dct))$, and (17.16.26) is

It is negative for all sufficiently large $k$. Distinct modes and their derivatives are orthogonal under the $r$ average, so the quadratic form is negative on every nonzero finite combination of these modes. Each combination gives a positive path for sufficiently small variation parameter and has action strictly below $nc^2$. Its raw length satisfies $L^2\leq 2\mathscr S$ on the unit interval, so it is strictly below $\sqrt{2n}|c|$. $\square$

Proof. On the strip, the holomorphic map $F([z],s)=[\mathrm e^{a_0s/2}z_0:\cdots :\mathrm e^{a_{n-1}s/2}z_{n-1}]$ satisfies $\operatorname {pr}_ Y^*\kappa +\mathrm{dd}^{\mathrm{c}}V=F^*\kappa $. The imaginary-time phases are unitary, so $V$ depends only on $\operatorname{Re}s$. This form has rank $d$, strictly positive spatial slices, and zero $(d+1)$st power. The classical bounded Kähler Dirichlet uniqueness identifies it with the base Perron segment; proposition 17.16.16, proposition 17.16.22 give the full segment and its stationarity.

For $x_ j=|z_ j|^2/\sum |z_ j|^2$, the initial velocity is $\sum _ ja_ jx_ j$. A standard complex Gaussian vector has independent exponential squared coordinates, and its projective direction has distribution $\kappa ^ d$. Dividing those coordinates by their sum gives the uniform simplex distribution, with moments

The smooth distance formula and (17.16.20) give the claimed squared distance; the action is half of it. $\square$

Proof. In the chart $z_0=1$, write $s=\log |z'|^2$, $L=\log (1+\mathrm e^ s)$, $P=\mathrm e^ s/(1+\mathrm e^ s)$, and $T=1+L-s\geq 1$. The local potential $\Phi =L+1-T^ a$ has derivative

Both $A$ and $A'$ are positive, so the radial complex Hessian is positive. Its bounded-above local potential extends plurisubharmonically across $o$ by lemma 5.24.7. The function extends smoothly across the hyperplane at infinity. Its bounded plurisubharmonic truncations decrease to a nontrivial canonical potential: near $o$, $|v_ a|=O((1-s)^ a)$ is integrable against radial smooth volume $O(\mathrm e^{ds})\, \mathrm ds$.

The radial mass flux is $Q(s)=A(s)^ d$. Its sphere coefficient is one, since $\mathrm d^{\mathrm c}\log |z'|^2$ integrates to one on the Hopf circle and $\mathrm{dd}^{\mathrm{c}}\log |z'|^2$ induces mass-one Fubini–Study on $\mathbb P^{d-1}$. Stokes on an annulus gives mass $Q(s_2)-Q(s_1)$. Since $Q(-\infty )=0$ and $Q(+\infty )=1$, finite-level locality and annular exhaustion give non-pluripolar mass one. The pole has no atom, and smoothness gives no mass on the hyperplane at infinity.

As $s\to -\infty $,

The squared energy therefore converges exactly when $2a-d(1-a)<0$. At equality the integrand has order $T^{-1}$, giving logarithmic divergence. All other compact regions have smooth potentials and measures. Fiber integration gives the pullback assertions. Finally, $v_ a-v_ b=T^ b-T^ a\to +\infty $ at $o$, so mutual bounded comparison fails. Both points lie in the completed metric space and are distinct, giving finite positive distance. $\square$

Proof. For finite $C$, the obstacle $\pi ^*\min (v+C,0)$ and the full competitor family are invariant under $E$. The same argument as in proposition 17.16.16 gives

Increasing limits and their upper semicontinuous regularizations commute with pullback in submersion charts, proving the model identity. The classical full-mass characterization on $Y$ gives $P_\kappa [v]=0$ for every $v\in \mathcal E^2$, hence the assertion for $\mathcal B_2$.

Pullback preserves parameterwise concavity, order, ordinary-model normalization and cutoff values of a base model test curve. It commutes with pointwise infima, and the submersion property makes it commute with upper semicontinuous regularization of suprema. Thus it commutes with both Legendre transforms. Finally, the Perron descent identifies every finite ray segment with the full Hermitian segment; its finite-energy endpoints and theorem 17.16.19, proposition 17.16.22 give the geodesic and minimizing assertions. $\square$