Lemma 16.20.1BB171C00B874

Proof. If $u$ is identically $-\infty $, the constants $\phi _ j=-j$ will do, so we assume that $u$ takes a finite value somewhere. Fix a distance $d$ defining the topology of the compact manifold $X$, and put $M=\max _ Xu<\infty $. For each positive integer $j$ define

the maximum is attained because the function in braces is upper semicontinuous in $y$. The $v_ j$ are finite and $j$-Lipschitz: a point where $u$ is finite bounds the maximum from below, and the triangle inequality bounds their differences. They decrease in $j$ and dominate $u$ (take $y=x$). If $u(x)$ is finite, a maximizing point $y_ j$ satisfies $jd(x,y_ j)\leq M-u(x)$, so $y_ j\to x$, and since $v_ j(x)\leq u(y_ j)$, upper semicontinuity gives $v_ j(x)\downarrow u(x)$. If $u(x)=-\infty $, then for every $N$ there is a small ball about $x$ on which $u<-N$, while outside this ball $u(y)-jd(x,y)\leq M-jd(x,y)<-N$ for large $j$. Thus $v_ j(x)\downarrow u(x)$ in this case as well.

The $v_ j$ are only Lipschitz, so we smooth them while keeping monotonicity. Set $\phi _0=M+1$ and $\eta _ j=4^{-j}$, and suppose inductively that $\phi _{j-1}\geq u+\eta _{j-1}$. The continuous function

is at least $u+2\eta _ j$, since $\eta _{j-1}=4\eta _ j$. A continuous function on $X$ can be approximated uniformly by smooth functions: convolution in a chart converges uniformly on compact subsets of the chart (lemma 2.5.1(1)), and a finite smooth partition of unity subordinate to charts (proposition 5.2.6) sums these errors with total weight one. Choose a smooth real $\phi _ j$ with $\| \phi _ j-q_ j\| _\infty <\eta _ j$. Then

These bounds continue the induction, show that the sequence $\phi _ j$ decreases, and, since $v_ j+3\eta _ j\downarrow u$, show that $\phi _ j\downarrow u$, also at the points where $u=-\infty $. $\square$