Theorem 12.34.4B6FA0B70406A

Proof. Call a time $t\geq 0$ admissible if $\theta _ t$ has a smooth positive representative. The admissible times form an interval containing zero, since convex combinations of positive representatives at two times are positive representatives at the intermediate times, and the set is open because positivity is an open condition. We show that the flow exists on $[0,t_1)$ for every admissible $t_1>0$ and that it cannot exist beyond $\tau $.

Step 1: the scalar equation. Fix an admissible $t_1>0$, and choose a smooth real function $a$ with $\theta _{t_1}+\mathrm{dd}^{\mathrm{c}}a>0$. The reference forms

are uniformly positive, with all derivatives uniformly bounded. Put

Then $\dot\gamma _ t=-\operatorname {Ric}(\Omega )$, where the Ricci form of a volume form is defined by the same local expression. Consider the scalar equation

Applying $\mathrm{dd}^{\mathrm{c}}$ shows that $\omega (t)=\gamma _ t+\mathrm{dd}^{\mathrm{c}}u(t)$ solves the metric flow. Conversely, for a solution $\omega (t)$ of the metric flow, the function $u(t)=2\pi \int _0^ t\log (\omega (s)^ n/\Omega )\, \mathrm ds$ satisfies $\omega (t)=\gamma _ t+\mathrm{dd}^{\mathrm{c}}u(t)$ and solves (12.34.5). The linearization of (12.34.5) is $\partial _ t-2\pi \operatorname {tr}_{\omega (t)}\mathrm{dd}^{\mathrm{c}}$, a strictly parabolic operator, so the classical short-time existence theory for smooth strictly parabolic equations gives a solution. The difference of two solutions satisfies a linear parabolic equation whose principal coefficients, the averages of the inverse metrics along the segment between the two solutions, are positive; the maximum principle therefore proves uniqueness.

Suppose that the maximal time $t_*$ of this solution satisfies $t_*<t_1$. We derive estimates independent of $t<t_*$ that allow us to continue the solution.

Step 2: bounds for $u$ and $\dot u$. For a large constant $C$, the functions $Ct$ and $-Ct$ are a supersolution and a subsolution of (12.34.5), respectively: at a spatial extremum of $u$ the complex Hessian has the appropriate sign, and $\gamma _ t^ n/\Omega $ is bounded above and away from zero. The maximum principle gives $|u|\leq Ct_1$.

Set $v=\dot u$ and $\mathcal L=2\pi \operatorname {tr}_{\omega (t)}\mathrm{dd}^{\mathrm{c}}$. Differentiation gives

Since $\gamma _ t$ is affine in time, these identities imply

By the maximum principle, the first expression is bounded above by its initial value zero, and the second is bounded below by $\min _ Xt_1v(0)$. The bound for $u$ and $t_1-t\geq t_1-t_*>0$ give a lower bound for $v$. The first inequality gives an upper bound for $v$ on $[t_*/2,t_*)$, and the smooth solution bounds it on $[0,t_*/2]$. In particular, $\omega (t)^ n/\Omega =\mathrm{e}^{v/(2\pi )}$ is bounded above and away from zero.

Step 3: a parabolic trace inequality. We next bound the trace, keeping track of the time derivative of the reference metric. In the coordinates of lemma 14.5.1, write $g(t)$ for the matrix of $\gamma _ t$ and, as in that lemma, $h=g+(u_{i\bar j})$ for the matrix of $\omega (t)$. Put

so that $\log (\det h/\det g)=v/(2\pi )+F$. The family $g(t)$ has uniformly bounded Chern torsion and curvature, so at each fixed time lemma 14.5.1 applies with uniform constants. Direct differentiation of $S$ gives

In $(\partial _ t-\mathcal L)\log S$, the term $\Delta _ gv/S$ cancels the corresponding term in the lower bound for $\mathcal L\log S$ given by the lemma. With $\tau _ k=\sum _ iT_{ik}^ i$ the torsion trace of $g(t)$, as in the lemma, we obtain on $\{ S\geq 1\} $

Indeed, $\Delta _ gF$ and $\operatorname {tr}_ g\dot g$ are uniformly bounded, $S^{-1}\leq B/n^2$, and $-Cg\leq \dot g\leq Cg$ bounds the last quotient above. These bounds account for all the terms coming from the time dependence of the reference metric.

Step 4: the trace bound. Choose a constant $C_0$ such that $1\leq s=u+C_0\leq D$ on the whole time interval, and maximize

on $X\times [0,t_0]$, where $t_0<t_*$ and the constant $A\geq 1$ will be fixed independently of $t_0$. If the maximum is attained at time zero, the initial data bound $Q$. At a maximum with positive time, $\partial \log S=(A+s^{-2})\partial u$. Since $|\tau |_ h^2\leq CB$, Young's inequality bounds the absolute value of the first term on the right of (12.34.6) by

If $S^2\leq (A+1)^2D^3$, the trace is already bounded at the maximum. Otherwise the last coefficient is at most $C$, and

Choosing $A$ with $2\pi A>C+1$ bounds $B$ at the maximum. In terms of the eigenvalues, $S\leq n(\det h/\det g)B^{n-1}$ for $n\geq 2$, and $S=\det h/\det g$ for $n=1$, so the determinant bound of Step 2 bounds $S$ at the maximum in every case. The bound for $u$ transfers this estimate from $Q$ to $S$ everywhere, uniformly in $t_0$, and the lower bound for the determinant then bounds the smallest eigenvalue away from zero. Thus $\omega (t)$ and $\gamma _ t$ are uniformly equivalent for $t<t_*$.

Step 5: higher estimates and continuation. Uniform ellipticity now gives the higher estimates by classical local parabolic regularity. In a coordinate ball, write the logarithmic determinant as a concave function of the complex part of the real Hessian. On the compact set of positive matrices just obtained, its supporting affine functions have uniformly positive and bounded slopes, and their infimum extends the operator to a concave, uniformly elliptic function of all real symmetric matrices. The dependence on space and time enters only through the smooth family $g(t)$ and the fixed density $\Omega $. The parabolic Evans–Krylov estimate gives uniform $C^{2+\alpha ,1+\alpha /2}$ bounds away from the initial time, and the smooth solution covers a fixed initial interval. Differentiating the equation and applying parabolic Schauder estimates then bounds all spatial and time derivatives uniformly. This is the parabolic counterpart of the local argument used in corollary 14.5.8.

The bounds for the time derivatives show that $u(t)$ converges in every spatial $C^ k$ norm as $t\uparrow t_*$. By uniform ellipticity, the smooth limit still has a strictly positive solution form, and short-time existence with this limit as initial datum extends the solution beyond $t_*$, a contradiction. Hence the flow exists up to every admissible time $t_1$, and by uniqueness these solutions patch together on $[0,\tau )$.

Step 6: maximality. Integrating the metric flow gives

Hence every time at which a smooth positive solution exists is admissible, which proves maximality. $\square$