Let $\omega$ be a fixed Kähler form on $M$, and $$ \mathscr{H}_{\omega}=\{v\in C^{\infty}(M)\mid \omega_v:=\omega+\sqrt{-1}\partial\overline{\partial}v>0\} $$
Now Aubin-Yau functional on $\mathscr{H}_{\omega}$ is defined as $$ I(u_0,u_1)=\frac{1}{V}\int_M(u_0-u_1)(\omega_{u_1}^m-\omega_{u_0}^m),\quad u_0,u_1\in\mathscr{H}_{\omega} $$ where $V=\int_M\omega^m$.
Now I'm going to show for all $u_0,u_1\in\mathscr{H}_\omega$:
- We have $$ I(u_0,u_1)\ge0 $$
- If $$ I(u_0,u_1)=0 $$ and $$ \int_Mu_0\omega^n=\int_Mu_1\omega^n=0 $$ then $u_0=u_1$.
For (1) I want to write $$ I(u_0,u_1)=\int_M(u_0-u_1)(\omega_{u_1}-\omega_{u_0})\wedge(\omega_{u_1}^{m-1}+\dots+\omega_{u_0}^{m-1}) $$ and it's clear $$ (\omega_{u_1}^{m-1}+\dots+\omega_{u_0}^{m-1})>0 $$
But I don't know how to deal with these. Could anyone give some hints about these two problems? Thanks in advance
Note that $\omega_{u_1}-\omega_{u_0}$ is $\sqrt{-1}\partial\bar{\partial}(u_1-u_0)$, so you can integrate by parts to get $$I(u_0,u_1)=\int_X(u_0-u_1)\sqrt{-1}\partial\bar{\partial}(u_1-u_0)\wedge(\omega_{u_1}^{m-1}+\dots+\omega_{u_0}^{m-1})=\int_X\sqrt{-1}\partial(u_1-u_0)\wedge\bar{\partial}(u_1-u_0)\wedge(\omega_{u_1}^{m-1}+\dots+\omega_{u_0}^{m-1}).$$ This has the shape $\int_X\sqrt{-1}\partial f\wedge\bar{\partial}f\wedge\Omega$ for a positive $n-1$ form $\Omega$, so it is non-negative. It vanishes exactly when $f$ is constant, and this implies both claims.