PomeFB6C2382E066Lemma 12.7.2.FB6C2382E066 For every positive $(n-1,n-1)$-current $T$ of finite mass on $X$,
In particular, $\mathfrak D_ F$ is concave, positively homogeneous and weakly upper semicontinuous on $\mathscr G_\omega $, and its value does not depend on the auxiliary frame or on the metric used to express the matrix density.
Proof. Let $I$ be the identity matrix of size $n$. For any positive semidefinite Hermitian matrix $P$ of that size,
The inequality $\leq $ is the arithmetic–geometric mean inequality. For the reverse inequality, take, for real $\varepsilon >0$,
Its determinant is one. If $P$ is positive definite, the trace expression $\frac1n\operatorname {tr}(Q_\varepsilon P)$ converges to $(\det P)^{1/n}$ as $\varepsilon \downarrow 0$. If $P$ is singular, it converges to zero: each ratio of an eigenvalue of $P$ to that eigenvalue plus $\varepsilon $ is bounded, while the prefactor tends to zero. When $\operatorname {tr}P=n$, the trace expression is at most $1+\varepsilon $.
We apply this formula to $P_ T$ and multiply by $f_ F$. Every continuous positive $\beta $ in the asserted infimum has matrix $f_ FQ$ with $\det Q=1$, so it gives an upper bound for $\mathfrak D_ F(T)$. Conversely, for fixed $\varepsilon >0$ the measurable field $Q_\varepsilon $ formed with $P=P_ T$ is bounded above and below by positive constants. It can be approximated in $L^1(\sigma _ T)$ by continuous positive definite Hermitian fields with bounds of the same type. One way to construct them is to apply Lusin's theorem in finitely many bundle charts, extend the continuous fields from the resulting compact sets, and glue with a partition of unity; spectral truncation to a fixed positive interval preserves the bounds. Finally we divide each approximating field by the $n$th root of its determinant. This normalization is uniformly continuous on fields with spectrum in a fixed compact positive interval, so the normalized fields still converge in $L^1(\sigma _ T)$ and have determinant one. Their trace integrals converge to that of $Q_\varepsilon $.
Now let $\varepsilon \downarrow 0$. The bound $1+\varepsilon $ permits dominated convergence, and we obtain equality of the infimum and the integral. For a current of arbitrary positive finite trace mass we use homogeneity, and the zero current is immediate. The infimum is taken over functionals $T\mapsto \int _ X\beta \wedge T$ that are linear and weakly continuous, which proves concavity and upper semicontinuity. It also expresses $\mathfrak D_ F(T)$ without a matrix frame or an auxiliary metric, which proves the final assertion. $\square$
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