Theorem 14.21.36EB399397505

of 14.21.3. We first locate the extrema. Fix $s>0$ and consider a maximum point of

For smooth data, the complex Hessian of this function is nonpositive at a maximum point, that is,

Since $sC>0$, this gives $B>0$, and since $A^ n=B^ n$, also $A>0$. In a $B$-unitary frame put $J=B^{-1/2}AB^{-1/2}$, $K=B^{-1/2}CB^{-1/2}$. Since $\det J=1$, arithmetic–geometric mean and (14.21.11) give

Consequently $\operatorname {tr}K\leq n$, $B\geq C/n$, and $\det K\leq (\operatorname {tr}K/n)^ n\leq 1$. In particular $B^ n\geq C^ n$.

For $C^{1,1}$ potentials, fix a maximum point $p$ of (14.21.10) and apply 14.21.5 on a coordinate ball, with radii tending to zero and errors $\delta _ j\downarrow 0$. For the set of full measure in that lemma, take the points where both Hessians are defined and semipositive and have the common determinant given by (14.21.1). With $\beta _0=\mathrm{dd}^{\mathrm{c}}|z|^2$, the resulting points $x_ j\to p$ satisfy

for large $j$. The matrix argument just given yields $f(x_ j)\mathrm{d}V_{x_ j}\geq C_ j(x_ j)^ n$. Since $f$, $\mathrm{d}V$ and $C$ are continuous, letting $j\to \infty $ gives $p\in P_\chi $; no continuity of the Hessians is needed. Thus every maximum point of the perturbed function lies in $P_\chi $. The set $P_\chi $ is closed, by continuity of $f$, $\mathrm{d}V$ and $C$, hence compact, and it is contained in $U$ because $C^ n>0$. Letting $s\downarrow 0$, a limit point of these maximum points is a maximum point of $w$ lying in $P_\chi $. Applying the same argument to $-w$ gives a minimum point of $w$ in $P_\chi $ as well. In particular, $P_\chi $ is nonempty.

On a relatively compact ball in $U$, the determinants of $A$ and $B$ have a positive lower bound, while their coefficients have an essential upper bound. Every eigenvalue is therefore bounded below by the determinant lower bound divided by a fixed power of the upper bound. Thus $U\subset E_{\rm unif}$.

By (14.20.1), $w$ solves a linear equation in nondivergence form with measurable coefficients, whose principal quadratic form is a fixed positive multiple of $Q$. On each relatively compact coordinate ball in $E_{\rm unif}$, these coefficients are uniformly elliptic. 14.21.6, applied to $w$ and to $-w$, makes $w$ constant on the component of $E_{\rm unif}$ containing the maximum point found above, and constant on the component containing the minimum point found above. Continuity extends the two constant values to the closures, proving (14.21.4). Distinct extrema force disjoint closures.

The same argument applies on $U$ itself, since $U$ contains the extremal points found in $P_\chi $ and the cofactor is locally uniformly elliptic on $U\subset E_{\rm unif}$. This proves the assertions with $U$ in place of $E_{\rm unif}$.

The component criteria are immediate except for the dense two-component case. If an open dense subset of connected $X$ has at most two components with disjoint closures, those closures are disjoint nonempty closed sets whose union is $X$, contradicting connectedness. Finally, when $Z=\{ f=0\} $ is a connected embedded real hypersurface, flattening charts assign to each point of $Z$ the unordered pair of components containing its two local sides. This pair is locally constant along $Z$, so connectedness makes it independent of the point; the two labels may coincide or be exchanged along a loop. Every component of $X\setminus Z$ has a nonempty boundary in $Z$, since otherwise it would be both open and closed in connected $X$. Near such a boundary point it contains a local side. Thus the pair accounts for every component, and $U$ has at most two components. It is dense because a hypersurface has empty interior. This proves the last assertion without choosing a global side. $\square$