Pome

Proposition 11.23.16621E75C5A9F4

Proposition 11.23.16.621E75C5A9F4 For the paths (11.23.15),

\[ D'_\varepsilon (0) =4\varepsilon \left(\frac K3-1\right)+o(\varepsilon ) \quad \text{as }\varepsilon \downarrow 0. \]

In particular, this derivative is positive for sufficiently small $\varepsilon >0$. Any two logarithmic Monge–Ampère flows with target $\nu _ K$ and these initial potentials have the same positive initial derivative of their total-variation distance.

Proof. For a real function $h$ depending only on $x_1,x_2$, the coefficient matrix of $\mathrm{dd}^{\mathrm{c}}h$ with respect to the forms $(\mathrm i/2)\mathrm dz_ j\wedge \mathrm d\bar z_ k$ is $(4\pi )^{-1}(h_{x_ jx_ k})$. Put $R_\varepsilon (t)=\mathrm d\mu _\varepsilon (t)/\mathrm d\mu $ and $R_0(t)=\mathrm d\mu _0(t)/\mathrm d\mu $. Differentiating the determinant at the diagonal initial matrices gives

\[ \begin{split} \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0} =\frac1{4\pi }\bigl (& B_\varepsilon \partial _{x_1}^2\log A_\varepsilon +A_\varepsilon \partial _{x_2}^2\log B_\varepsilon \\ & -K(B_\varepsilon -1)\partial _{x_1}^2P -K(A_\varepsilon -1)\partial _{x_2}^2P\bigr ). \end{split} \]

Since $\partial _{x_ j}^2P=-4\pi ^2P$, Taylor expansion in $\varepsilon $ yields the uniform estimate

\[ \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0} =\pi ^2\varepsilon S(1-KP)+O(\varepsilon ^2). \]

At $t=0$, the difference of the densities is

\[ q_\varepsilon =r_\varepsilon -1 =-\pi \varepsilon S+\pi ^2\varepsilon ^2P. \]

For each fixed $x_2$, its coefficient of $c_1$ is $-\pi \varepsilon (1-\pi \varepsilon c_2)\ne 0$, so as a function of $x_1$ it has at most two zeros in a period, and its zero set has measure zero. Differentiation under the integral sign, justified by the uniform bound on the derivatives of the smooth densities, gives

\[ D'_\varepsilon (0)=\frac12\int _ X\operatorname {sgn}(q_\varepsilon ) \left.\partial _ t(R_\varepsilon -R_0)\right|_{t=0}\mu . \]

The set $\{ S=0\} $ also has measure zero, and $\operatorname {sgn}(q_\varepsilon )\to -\operatorname {sgn}(S)$ off this set. Dominated convergence and lemma 11.23.14 give

\[ \lim _{\varepsilon \downarrow 0}\frac{D'_\varepsilon (0)}\varepsilon =\frac{\pi ^2}2\int _ X|S|(KP-1)\mu =4\left(\frac K3-1\right). \]

Every flow with the stated initial data has the velocities (11.23.15) at $t=0$, so its initial density derivatives, and hence the derivative of the distance, coincide with this computation. $\square$

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