A twisted Chern–Ricci flow is the equation $\partial _ t\omega =-\operatorname {Ric}(\omega )+\zeta $, where the twist $\zeta $ is a fixed smooth closed real $(1,1)$-form. Diameter bounds along such flows cannot depend only on the maximal time and on the initial Monge–Ampère density: the twist and the length of the time interval enter the geometry, even for flat Kähler metrics.
Example 12.34.14.706929790BE5 Let $E=\mathbb C/(\mathbb Z+\mathrm i\mathbb Z)$, let $X=E^ n$ with $n\geq 2$, and let $\eta _ j=(\mathrm i/2)\mathrm dz_ j\wedge \mathrm d\bar z_ j$ be the flat forms pulled back from the factors. Put $\omega _0=\sum _{j=1}^ n\eta _ j$. All the flows below start at $\omega _0$, with potential zero and initial density one relative to $\omega _0^ n$. In particular, hypotheses on the initial data expressed through a decomposition $\mathrm e^{\psi ^+-\psi ^-}$ of the density, with quasi-plurisubharmonic functions $\psi ^\pm $ subject to fixed $L^ p$ bounds, hold uniformly for all these flows, with $\psi ^+=\psi ^-=0$.
For each $M>0$, the twist $\zeta _ M=M\eta _1-\eta _2$ gives the explicit solution
\[ \omega _ M(t)=(1+Mt)\eta _1+(1-t)\eta _2+ \sum _{j=3}^ n\eta _ j, \qquad 0\leq t<1, \]
of $\partial _ t\omega =-\operatorname {Ric}(\omega )+\zeta _ M$. Its maximal smooth existence time is exactly one, while
\[ \operatorname {diam}(X,\omega _ M(1/2)) \geq \sqrt{1+M/2}\, \operatorname {diam}(E,\eta _1). \]
For the single fixed twist $\zeta =\omega _0$, the solution is $(1+t)\omega _0$ for all $t\geq 0$, and its diameter grows as $\sqrt{1+t}\, \operatorname {diam}(X,\omega _0)$.
Proof.
Each of these metrics has constant coefficients, so its Chern–Ricci form vanishes; differentiation in $t$ proves the flow equations, and positivity of the coefficients gives the stated existence intervals. In the first family, pair a hypothetical positive representative at a time $t\geq 1$ with the closed positive form $\eta _1\wedge \eta _3\wedge \cdots \wedge \eta _ n$, where the empty product is $1$ when $n=2$. The potential term integrates to zero, and the pairing is a positive fixed multiple of $1-t$, so it cannot be strictly positive. Integrating the flow equation shows that every smooth solution differs from the affine background by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, so no smooth positive continuation beyond $t=1$ is possible.
Projection to the first elliptic factor does not increase lengths once the metric of that factor is scaled by $1+Mt$. Taking two points that differ only in the first factor proves the lower bound for the diameter. For the second flow, every path length is multiplied by $\sqrt{1+t}$, which proves the exact formula for the diameter.
$\square$
On a fixed finite time interval, a common Dini modulus for the potentials and a uniform bound for the affine backgrounds do give control of distances and diameters, by corollary 5.39.8. The example shows why such bounds must be imposed when the twist varies or the time interval grows. It does not concern the limit metric at the maximal time of an untwisted flow.
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