24.2.5 Uniqueness and selectionDFEF338E1977

Where the density vanishes, the linearized equation degenerates, and the zero set can separate regions where it is elliptic. The question is whether the difference of two solutions can vary from one such region to another. Throughout this subsection $\omega $ is a Hermitian form on $X$, $f$ is smooth and nonnegative, and potentials are normalized by $\sup _ Xu=0$.

On $\mathbb P^3$, theorem 14.26.1 gives a nonclosed Hermitian form and a smooth nonnegative density with two distinct smooth normalized solutions; there the density vanishes on an open set. The construction of theorem 14.27.1 has uncountably many smooth normalized solutions for a density that is positive off two disjoint real hypersurfaces, and after a modification of the background, theorem 14.28.1 achieves exactly quadratic vanishing on those hypersurfaces. Thus neither a zero set of zero volume nor vanishing of finite order guarantees uniqueness. In all these examples the background is a Hermitian form with bounded mass; the degeneracy responsible for nonuniqueness lies in the density and in the solution forms.

These examples restrict any geometric use of the equation. A construction assigning a canonical potential to arbitrary nonnegative data must specify a selection procedure and prove that it is well defined; existence in the minimal ordinary singularity type does not provide one. The scalar, which is unique, and for strictly positive data the corresponding variational volume, remain meaningful without uniqueness of the potential. In low dimension uniqueness holds under additional conditions. On a compact Hermitian surface, 14.23.7 proves uniqueness for $C^2$ potentials when the common density is continuous and positive on a dense set, with no differential condition on the metric; in dimension three, 14.23.1 uses instead a sign condition on $\mathrm{dd}^{\mathrm{c}}\omega $. The two arguments rest on different integration identities: the cofactor identity (14.22.14) and a signed estimate for the quadratic difference.

When $\{ f>0\} $ is connected and dense, theorem 14.24.1 shows that two bounded normalized solutions with the same positive scalar coincide; the theorem applies to the measure $cf\, \mathrm{d}V$ after the smooth positive ratio $\mathrm{d}V/\omega ^ n$ is absorbed into the density. In all the counterexamples above whose zero set consists of hypersurfaces, including the real-analytic example of 14.29.1, the positivity set has three connected components. None of them contradicts 24.7.1, which concerns strictly positive densities $\mathrm{e}^ F$ with $F$ continuous.

For a fixed nonzero density $f\in L^ p(\mathrm{d}V)$, $p>1$, the exponential equation does select a solution. By theorem 11.44.2, a weakly compact set of norming probabilities determines the limit: as the exponent tends to zero, the suitably shifted exponential solutions converge to the greatest subsolution of the critical equation that satisfies the integral normalizations given by these probabilities. In a Hermitian background the convergence is uniform; in an arbitrary smooth big background it holds in $L^1(X)$ and locally uniformly off the pole set of the strict barrier. The selected limit inherits the symmetries of the data (proposition 15.20.1), and in the projective example theorem 15.20.4 identifies it with the explicit symmetric $C^{1,1}$ solution of proposition 14.26.14. Selection thus coexists with nonuniqueness of the normalized critical equation. For the merely integrable radial densities on zero-lower-mass products, theorem 15.36.5 determines the whole supremum-normalized limit, even with joint variation of the density parameter and exponent. Scalar centering has a finite limit exactly below the moment boundary of proposition 15.36.8; at the boundary and above it diverges. For logarithmic product models with positive residual, theorem 11.45.9 also gives selection under independent variation of the reference coefficients, background, density and exponent; centering follows each varying critical scalar. Selection for arbitrary integrable data and general varying singular reference types remains open.

Dual attainment gives a different criterion. In dimension at least two, suppose that $f\neq 0$, that $b_\omega (f)>0$, and that $D_{f^{1/n}}$ has a smooth strictly positive maximizing Gauduchon current. Then the normalized threshold solution is unique, and the whole family of normalized solutions with positive exponent converges uniformly to it (corollary 12.10.12, corollary 12.10.13). The density of example 12.10.15 satisfies this condition although it vanishes on an open set. The size of the zero set alone therefore does not decide uniqueness; the dual condition also depends on the background.

For prescribed radial nonlogarithmic types, theorem 11.46.8 gives a different kind of regularity: a relative density error $O(r^\delta )$ controls the residual after the singularity is subtracted. The critical $C^1$ and $C^2$ thresholds are attained and are sharp by proposition 11.46.17. Positive cumulative central mass allows smooth crossing of density zeros, and proposition 11.46.20 gives an open zero annulus with a globally smooth residual. The product construction corollary 11.46.22 transfers this regularity under a zero-lower-mass factor hypothesis. These are radial criteria; they do not imply regularity or uniqueness for general nonsymmetric data.