25.8 Smooth and continuous equationsE1BC130DDD5C

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On a Hermitian manifold, the smooth Monge–Ampère equation determines a normalization constant together with the potential. For $n\geq 2$, the existence theorem theorem 14.2.1 is Tosatti and Weinkove's [Corollary 1, TW10]. Their equation, for a smooth function $F$ prescribing the logarithm of the density, is written with $\mathrm i\partial \bar\partial \varphi $; the substitution $u=2\pi \varphi $ gives the convention of this book, and a prescribed smooth volume form $\mathrm dV$ is absorbed by replacing $F$ with $F+\log (\mathrm dV/\omega ^ n)$. The concavity argument with Gauduchon metrics for the normalization constant in theorem 14.8.1 also appears in Collins's [Theorem 2.13, Collins21].

The proof of the uniform estimate for the potential in section 14.1 by weighted integration follows Cherrier and Tosatti–Weinkove [Cherrier87, TW10]: a finite recurrence absorbs the torsion terms before the Sobolev iteration. The calculation of the torsion terms in the second-order estimate of section 14.1 goes back to Cherrier; its form with completed squares and the maximum-principle argument follow [Sections 3 and 9, TW15].

The interpolation in lemma 14.5.7 is the mechanism of Jiaogen Zhang and Xi Zhang's [v1, Proposition 4.1 and its proof, p. 16, ZhangHessianInterpolation26]. The fixed scalar operator is uniformly elliptic for a Hermitian metric as well, so the linear implication here does not require closedness. Their nonlinear Hessian application assumes a Kähler metric and a small product of its negative bisectional-curvature bound with the solution's oscillation. Its argument for data close to one in $L^1$ uses a stability estimate with $q>n/k$ and a uniform $L^ q$ bound; the smallness threshold depends on that bound. The interpolation lemma supplies neither this nonlinear trace estimate nor an arbitrary-$L^1$ stability criterion.

Prescribing the volume form of a Gauduchon metric leads to a different equation, which contains first-order torsion terms. The existence theorem theorem 14.10.1 is due to Székelyhidi, Tosatti and Weinkove [Theorem 1.4, STW17], again with the potential multiplied by $2\pi $. The bound for the potential in section 14.11 follows Székelyhidi's [Propositions 10–11 and Remark 12, Sze18], specialized to the determinant of the Gauduchon equation, and the second-order argument in section 14.14 follows [Section 3, STW17]. The identities (14.13.4), taken from [Section 4, STW17], explain the structure of the first-order perturbation.

The gradient estimate combines a blow-up argument with a Liouville theorem. In (14.16.3), where $A_ j$ are the rescaling factors of the blow-up, the first-order term is kept before rescaling and tends to zero at the rate $A_ j^{-1}$, as in [Section 6, Sze18]. The Euclidean argument in section 14.15 specializes [Theorem 20, Sze18] to the Gauduchon cone. Once the equation is uniformly elliptic, lemma 14.17.1 uses the Evans–Krylov theorem for variable coefficients of Tosatti, Wang, Weinkove and Yang [Definition 3.2 and Theorem 3.4, TWWY15].

For densities in $L^ p$, the existence of continuous solutions of the equation with a normalization constant (theorem 14.19.2) and of the exponential equation (14.19.4) is due respectively to Kołodziej and Nguyen [Theorem 0.1, KN15] and to Nguyen [Theorem 0.1, Nguyen16]. Their operator is twice the one used here: for the exponent $\lambda >0$ in (14.19.4), the conversion is $u=2\varphi $, where $\varphi $ is their solution, and the exponent in their equation is $2\lambda $. These results require neither bounded mass nor positive lower volume. Compactness comes from local capacity estimates: the arguments of [Section 5, KN15] and [Section 6, Dinew16] lead to theorem 14.19.21, which is the statement of [Corollary 5.10, KN15]. A common bound for the densities is essential here; the $L^1$ compactness of normalized potentials alone does not suffice.

For measures satisfying a power bound in capacity, Kołodziej and Nguyen [Proposition 2.4, pp. 2954–2956, KNHolder18] give stability measured against the Monge–Ampère measure of the solution. The thin-sublevel estimate and backwards iteration of lemma 14.19.14, lemma 14.19.16 follow the capacity argument in [v2, Theorem 5.3 and Lemma 5.4, pp. 10–12, KN15], using their modified comparison theorem [v2, Theorem 2.3, pp. 5–6, KN15]. The geometric iteration in the book covers every power $1+\alpha $, $\alpha >0$, including the total-capacity endpoint. Choosing the sublevel height proportional to the cube of the comparison reserve gives the exponent $\alpha /[3n(1+\alpha )+\alpha ]$ in proposition 14.19.23; the square-capacity case has exponent $1/(6n+1)$. The measure can be singular. The preprint [v2, p. 1, KN15] uses $\mathrm{dd}^{\mathrm{c}}=\mathrm i\partial \bar\partial /\pi $, twice our operator; its backgrounds become $2\omega $ with the same potentials, and its top products acquire the factor $2^ n$.

The threshold equation in theorem 15.33.1 combines this iteration with bounded-density solvability and the full Gauduchon formula. Local $L^1$ compactness uses Demailly's Propositions 4.21 and 5.9 [Chapter I, pp. 36–37 and 40, DemCADG]. The weighted local product estimate is Nguyen's [v1, Proposition 1.1, pp. 5–6, Nguyen16]; uniform continuity of bounded products is Bedford and Taylor's [Corollary 2.8 and Theorem 2.10, p. 10, BT82]. For a Hermitian background, the local closed product must still be expanded with every smooth correction factor, as in proposition 5.28.2. The logarithmic construction in theorem 15.33.6 uses the exponential volume–capacity estimate (5.29.3) and gives a datum outside every higher $L^ p$ class with an exact continuous threshold solution. The added capacity hypothesis leaves the general root-tail and exact-equation questions open.

For a continuous solution whose measure is dominated by a Hölder reference, theorem 15.5.4 combines the weighted Chern estimate with the square-capacity stability. The coordinated transform parameter and comparison reserve keep the minimizing radii comparable to the smoothing radius. The resulting exponent is sufficient for the range theorem; no optimal exponent is asserted. The argument assumes continuity of the solution.

The smoothness theorem theorem 14.19.32 is the strictly positive, globally smooth-data case of Badiane, Lu and Zeriahi's [v1, Theorem 1.1, p. 2; Section 4.2, p. 17, BLZBoundedSmooth26]. Their auxiliary second-order estimate is Proposition 4.1, pp. 13–17. The square of the prescribed potential in lemma 14.19.31 absorbs the derivatives of the frozen density, and bounded exponential comparison identifies the uniform limit with the original solution. The argument then uses the concave uniformly elliptic extension of the logarithmic determinant and the Evans–Krylov theorem for variable coefficients [Definition 3.2, Theorem 3.4, and the paragraph following it, TWWY15]. Only the strictly positive Hermitian background and global smooth positive density are treated here; the primary paper also considers degenerate backgrounds and data with a singular regular locus.

The exponential term gives a comparison principle even when the densities have no positive lower bound; lemma 14.19.28 recovers [Lemma 2.3, Nguyen16]. Letting the exponent tend to zero then identifies the normalization constant through (14.19.35), as in [Corollary 0.2, Nguyen16]. This proves uniqueness of the constant, but not uniqueness of the limiting potential. The weighted mean (14.19.27), taken from [Proposition 2.5, Nguyen16], controls the normalization; here its lower bound follows directly from the estimate for sublevel sets.

Equations with prescribed singularities require relative envelopes and separate assumptions on the reference potential and on the ambient metric. The results of Alehyane–Lu–Salouf and of Xuan Li [ALS26, XL26] are therefore treated separately in section 15.1, section 15.37, section 10.31; their sources are discussed in section 25.12.

For fully nonlinear concave equations, the uniform estimate under a quantitative subsolution condition is due to Guo and Phong [v1, Theorem 3.1 and its proof, pp. 8–11, GP24]. The strict asymptotic gap of an admissible reference, the matrix bound, the linear stability estimate and the uniform compactness in subsection 14.19.42 follow Guo and Song [v1, Lemma 2.1, Proposition 2.8 and Lemma 2.9, pp. 4–11, GS26]. In the book, the zero-order estimate uses the real contact-set construction of proposition 14.11.1, and theorem 14.19.19 bounds both the auxiliary potential and its normalization constant; with these inputs the constants do not depend on a positive lower bound for the right-hand sides. Writing the potential of these sources as $u/(2\pi )$ converts their convention for smooth curvature forms into the normalization of this book.

The gradient estimate in subsection 14.19.47 is Guo and Song's [v1, Theorem 1.1 and Proposition 3.1 with Lemmas 3.2–3.6, pp. 11–17, GS26]; the earlier alternative for convex level sets is Székelyhidi's [arXiv v3, Propositions 5–6, pp. 5–7, Sze18]. The tangent-hyperplane argument in lemma 14.19.48 uses the common strict gap to keep the constants uniform at every positive level. The differentiation is carried out for $U=u/(2\pi )$, where $u$ is the solution, and the final finite net of smooth comparison functions is provided by corollary 14.19.46.