Uniform smooth Monge–Ampère bounds by the Calabi third-order estimate

In complex dimension one the equation is exactly ${\mathrm{\Delta }}_0\varphi =\rho -1$, because a one-by-one Hermitian determinant is its sole entry. Its right side has zero $dV$ mean. The inverse in [F4], with identical reference and solving metric, bounds $\varphi$ in $C^{k+2,\alpha }$ by $\mathrm{\parallel }\rho -1{\mathrm{\parallel }}_{C^{k,\alpha }}$ for every $k\ge 0$. Differentiating $\rho =e^F$ expresses these norms in terms of $b$ and finitely many $M_r$. This proves all claimed bounds when $n=1$. In the remainder assume $n\ge 2$, so the real dimension $d=2n$ is greater than two.

We first derive, rather than import, the required fixed-background Poincaré and Sobolev constants. Write $d\nu =dV\mathrm{/}{V}_0$. For every smooth real zero-mean $u$, the operator $P=-{\mathrm{\Delta }}_0$ has scalar positive principal symbol, is formally symmetric, and has precisely constant kernel by Stokes and connectedness. Thus [F2] gives a smooth $v=Gu$ with $Pv=u$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_2$. Integration by parts yields $\mathrm{\parallel }u{\mathrm{\parallel }}_2^2={\int }_X⟨\mathrm{\partial }u,\mathrm{\partial }v{⟩}_{g}d\nu \le C\mathrm{\parallel }\mathrm{\partial }u{\mathrm{\parallel }}_2\mathrm{\parallel }u{\mathrm{\parallel }}_2$, proving $\mathrm{\parallel }u{\mathrm{\parallel }}_2\le C_P\mathrm{\parallel }\mathrm{\partial }u{\mathrm{\parallel }}_2$ (including $u=0$). For Sobolev, localize a smooth $w$ by a finite subordinate partition with compact support in real coordinate balls. Extend each localized function by zero to ${\mathbb{R}}^d$ and apply [F3] with $p=2<d$. Derivatives of the fixed cutoffs contribute at most $C\mathrm{\parallel }w{\mathrm{\parallel }}_2$; metric and density are bounded above and below on these finite compact charts. Summing the finitely many inequalities gives $\mathrm{\parallel }w{\mathrm{\parallel }}_{2\chi }^2\le C_S(\mathrm{\parallel }\mathrm{\partial }w{\mathrm{\parallel }}_2^2+\mathrm{\parallel }w{\mathrm{\parallel }}_2^2)$, with $\chi =n\mathrm{/}(n-1)>1$. Both $C_P$ and $C_S$ depend only on the fixed background. Approximation of the localized functions extends this inequality to the $W^{1,2}$ functions used below.

We supply the local metric algebra for the Laplacian estimate. Put $h_{i\bar{j}}=g_{i\bar{j}}+{\varphi }_{i\bar{j}}$, $U={\mathrm{tr}}_{g}h$, $W={\mathrm{tr}}_{h}g$, and ${\mathrm{\Delta }}_h=h^{i\bar{j}}{\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}$. Holomorphic normal coordinates for $g$ exist at each point: first make $g=I$ by a linear change and then use $z^k=w^k-\frac{1}{2}\mathrm{\Gamma }(g{)}_{ij}^k(0)w^i{w}^j$, where $\mathrm{\Gamma }(g{)}_{ij}^k=g^{k\bar{l}}{\mathrm{\partial }}_i{g}_{j\bar{l}}$ is symmetric in $i,j$ because $d{\omega }_0=0$. The identity Jacobian and [F5] make this a genuine coordinate change; direct differentiation makes $\mathrm{\partial }g=0$. A unitary linear change also diagonalizes $h$ there, with eigenvalues ${\lambda }_i>0$. Set $R_{i\bar{j}k\bar{l}}^g=-{\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}{g}_{k\bar{l}}$ at this normal point, agreeing with the curvature defined globally by $R(g{)}_{i\bar{j}k}^l=-{\mathrm{\partial }}_{\bar{j}}\mathrm{\Gamma }(g{)}_{ik}^l$. Twice differentiating $\log \det h=\log \det g+F$, and using the Kähler equalities ${\mathrm{\partial }}_i{h}_{k\bar{l}}={\mathrm{\partial }}_k{h}_{i\bar{l}}$, gives the exact scalar identity ${\mathrm{\Delta }}_{h}U={\mathrm{\Delta }}_{0}F+{\sum }_{i,k}{R}_{i\bar{i}k\bar{k}}^g({\lambda }_k\mathrm{/}{\lambda }_i-1)+Q$, where $Q={\sum }_{i,k,a}{\lambda }_i^{-1}{\lambda }_a^{-1}\mathrm{\mid }{\mathrm{\partial }}_i{h}_{k\bar{a}}{\mathrm{\mid }}^2$. To verify the terms, differentiating $g^{-1}$ contributes ${\sum }_{i,k}({\lambda }_k\mathrm{/}{\lambda }_i)R_{i\bar{i}k\bar{k}}^g$; commuting the two mixed derivatives of $h$ converts the other second-derivative term to ${\mathrm{\Delta }}_0\log \det h+Q$; and ${\mathrm{\Delta }}_0\log \det g=-{\sum }_{i,k}{R}_{i\bar{i}k\bar{k}}^g$. The derivative-of-inverse formula $\mathrm{\partial }{h}^{-1}=-h^{-1}(\mathrm{\partial }h)h^{-1}$ gives precisely the displayed nonnegative $Q$. Thus all signs and curvature terms are fixed locally.

Set $u=\varphi$ and $\mathrm{\Theta }={\sum }_{j=0}^{n-1}{\omega }_{\varphi }^j\wedge {\omega }_0^{n-1-j}$. It is closed and positive, with $\mathrm{\Theta }\ge {\omega }_0^{n-1}$. The equation and Stokes give ${\int }_{X}u(1-\rho )dV={\int }_{X}i\mathrm{\partial }u\wedge \bar{\mathrm{\partial }}u\wedge \mathrm{\Theta }\ge n^{-1}{\int }_X\mathrm{\mid }\mathrm{\partial }u{\mathrm{\mid }}_g^{2}dV$. Hence $\mathrm{\parallel }\mathrm{\partial }u{\mathrm{\parallel }}_2^2\le C\mathrm{\parallel }u{\mathrm{\parallel }}_1$, with $C$ depending on $n$ and $\mathrm{\parallel }1-\rho {\mathrm{\parallel }}_{\mathrm{\infty }}\le 1+b$. Step 1.2 and $\mathrm{\parallel }u{\mathrm{\parallel }}_1\le \mathrm{\parallel }u{\mathrm{\parallel }}_2$ for the probability measure $d\nu$ imply $\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\le C\mathrm{\parallel }u{\mathrm{\parallel }}_2$, so $\mathrm{\parallel }u{\mathrm{\parallel }}_2$ is bounded independently of the solution. Here the signed integral is controlled by $\int \mathrm{\mid }u\mathrm{\mid }$, not by the vanishing integral $\int u$.

We next construct the full connection-difference calculation needed for Calabi's estimate. Let $D$ and $\mathrm{\nabla }$ be the connections with holomorphic coefficients $\mathrm{\Gamma }(h{)}_{ij}^k=h^{k\bar{l}}{\mathrm{\partial }}_i{h}_{j\bar{l}}$ and $\mathrm{\Gamma }(g{)}_{ij}^k=g^{k\bar{l}}{\mathrm{\partial }}_i{g}_{j\bar{l}}$, and antiholomorphic parts equal to $\bar{\mathrm{\partial }}$ on holomorphic tangent sections. Differentiating their metric matrices proves metric compatibility, and the Kähler equalities prove their torsion vanishes. Put $T_{ij}^k=\mathrm{\Gamma }(h{)}_{ij}^k-\mathrm{\Gamma }(g{)}_{ij}^k$, a tensor with two holomorphic covariant indices and one holomorphic contravariant index, and $S=\mathrm{\mid }T{\mathrm{\mid }}_h^2=h^{i\bar{a}}{h}^{j\bar{b}}{h}_{k\bar{l}}{T}_{ij}^k\overline{T_{ab}^l}$. Write $R^{\mathrm{\prime }}$ for the curvature of $h$ and $R$ for that of $g$, using the sign in step 1.3. Since $T$ has only holomorphic indices, ${\bar{D}}_{b}T={\mathrm{\partial }}_{\bar{b}}T$, and hence $({\bar{D}}_{b}T{)}_{ij}^k=R_{i\bar{b}j}^k-R_{i\bar{b}j}^{\mathrm{\prime }k}$. Also $[D_a,{\bar{D}}_b]$ acts on $T$ by $R_{a\bar{b}p}^{\mathrm{\prime }k}{T}_{ij}^p-R_{a\bar{b}i}^{\mathrm{\prime }p}{T}_{pj}^k-R_{a\bar{b}j}^{\mathrm{\prime }p}{T}_{ip}^k$, by applying the connection to each of its three indices. Its contraction with $h^{a\bar{b}}$ is a three-term action $\mathcal{R}T$ of the Ricci endomorphism. These commutator formulas follow directly by expanding $D_a={\mathrm{\partial }}_a+\mathrm{\Gamma }(h{)}_a$ on vector components and using the dual action on each covariant index. Finally ${\mathrm{\partial }}_a\mathrm{\Gamma }(h{)}_i-{\mathrm{\partial }}_i\mathrm{\Gamma }(h{)}_a+[\mathrm{\Gamma }(h{)}_a,\mathrm{\Gamma }(h{)}_i]=0$, by differentiating the displayed coefficients and using the derivative-of-inverse formula; thus $[D_a,D_i]=0$. Commuting this equality with ${\bar{D}}_b$ and using $\mathrm{\Gamma }(h{)}_{ai}^k=\mathrm{\Gamma }(h{)}_{ia}^k$ gives $D_a{R}_{i\bar{b}j}^{\mathrm{\prime }k}=D_i{R}_{a\bar{b}j}^{\mathrm{\prime }k}$. This is the exact Kähler Bianchi identity used below, derived from the connection matrices.

For $p\ge 2$, repeat the preceding integration with $u\mathrm{\mid }u{\mathrm{\mid }}^{p-2}$. It gives $\frac{4(p-1)}{n{p}^2}\mathrm{\parallel }\mathrm{\partial }(\mathrm{\mid }u{\mathrm{\mid }}^{p\mathrm{/}2}){\mathrm{\parallel }}_2^2\le \mathrm{\parallel }1-\rho {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_{p-1}^{p-1}$. At zeros or nonintegral exponents use the smooth multiplier $u(u^2+{\epsilon }^2{)}^{(p-2)\mathrm{/}2}$ and pass to the limit; its derivative is $(u^2+{\epsilon }^2{)}^{(p-4)\mathrm{/}2}((p-1)u^2+{\epsilon }^2)$, nonnegative, and convergence of the energy follows by dominated convergence for the fixed smooth bounded $u$ and its gradient. This justifies the formula also for the weak derivative of $\mathrm{\mid }u{\mathrm{\mid }}^{p\mathrm{/}2}$. Apply step 1.2 to this function to obtain $\mathrm{\parallel }u{\mathrm{\parallel }}_{p\chi }^p\le Cp\max (1,\mathrm{\parallel }u{\mathrm{\parallel }}_p^p)$, using $p^2\mathrm{/}(p-1)\le 2p$ and Hölder for the probability measure. Thus $A_{j+1}\le (C{p}_j{)}^{1\mathrm{/}{p}_j}{A}_j$ for $p_j=2{\chi }^j$ and $A_j=\max (1,\mathrm{\parallel }u{\mathrm{\parallel }}_{p_j})$. The sums ${\sum }_j{p}_j^{-1}$ and ${\sum }_j(\log p_j)\mathrm{/}{p}_j$ converge, so ${\sup }_j{A}_j\le C{A}_0$. A continuous function on this compact positive-density space satisfies ${\lim }_{p\to \mathrm{\infty }}\mathrm{\parallel }u{\mathrm{\parallel }}_p=\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$: every neighbourhood on which its modulus is close to its maximum has positive measure. Step 2.1 therefore gives a uniform $C^0$ bound on $\varphi$.

Let $B$ bound the absolute background curvature components in unitary frames and let $C$ bound $\mathrm{\mid }{\mathrm{\Delta }}_{0}F\mathrm{\mid }+n^{2}B$, uniformly from $M_2$ and the fixed geometry. The identity in step 1.3 gives ${\mathrm{\Delta }}_{h}U\ge Q-BUW-C$. Moreover $\mathrm{\mid }\mathrm{\partial }U{\mathrm{\mid }}_h^2={\sum }_i{\lambda }_i^{-1}\mathrm{\mid }{\sum }_k{\mathrm{\partial }}_i{h}_{k\bar{k}}{\mathrm{\mid }}^2\le U{\sum }_{i,k}{\lambda }_i^{-1}{\lambda }_k^{-1}\mathrm{\mid }{\mathrm{\partial }}_i{h}_{k\bar{k}}{\mathrm{\mid }}^2\le UQ$, by weighted Cauchy–Schwarz. Hence ${\mathrm{\Delta }}_h\log U\ge -BW-C\mathrm{/}U\ge -(B+C\mathrm{/}{n}^2)W$, since $UW\ge n^2$. Choose $A=B+C\mathrm{/}{n}^2+1$. Since ${\mathrm{\Delta }}_h\varphi =n-W$, the function $H=\log U-A\varphi$ satisfies ${\mathrm{\Delta }}_{h}H\ge W-An$. At its maximum $p$, $W(p)\le An$, so each ${\lambda }_i(p)\ge (An{)}^{-1}$, while ${\prod }_i{\lambda }_i=\rho \le b$ gives $U(p)\le nb(An{)}^{n-1}$. For arbitrary $x$, maximality gives $U(x)\le U(p)e^{A(\varphi (x)-\varphi (p))}$, bounded by step 3.1. Finally ${\lambda }_i\le U$ and ${\prod }_i{\lambda }_i\ge a$ give ${\lambda }_i\ge a\mathrm{/}{U}^{n-1}$. We have obtained ${\mathrm{\Lambda }}^{-1}g\le h\le \mathrm{\Lambda }g$ for a uniform $\mathrm{\Lambda }$, with no background curvature-sign assumption.

Define $Z=h^{a\bar{b}}{D}_a{\bar{D}}_{b}T$. Step 2.2 and $Dh=0$ give the tensor identity $Z_{ij}^k=h^{a\bar{b}}{D}_a{R}_{i\bar{b}j}^k-D_i({\mathrm{Ric}}^{\mathrm{\prime }}{}_j^k)$. Here the contraction $h^{a\bar{b}}{R}_{a\bar{b}j}^{\mathrm{\prime }k}=h^{k\bar{l}}{\mathrm{Ric}}_{j\bar{l}}^{\mathrm{\prime }}$ is checked at an $h$-normal point: ${\mathrm{\partial }}_a{\mathrm{\partial }}_{\bar{a}}{h}_{j\bar{k}}={\mathrm{\partial }}_j{\mathrm{\partial }}_{\bar{k}}{h}_{a\bar{a}}$ by the two Kähler derivative symmetries, so both contractions equal $-{\mathrm{\partial }}_j{\mathrm{\partial }}_{\bar{k}}\log \det h$ after raising the output index. First metric derivatives vanish there; since the equality is tensorial it holds in every coordinate system. The Monge–Ampère equation supplies the fixed Ricci covariant tensor $q_{j\bar{l}}=\mathrm{Ric}(g{)}_{j\bar{l}}-F_{j\bar{l}}={\mathrm{Ric}}_{j\bar{l}}^{\mathrm{\prime }}$. It and its $\mathrm{\nabla }$-derivative are uniformly bounded by the background and $M_3$; no positivity of $q$ is required. For the first term of $Z$, $D_{a}R={\mathrm{\nabla }}_{a}R+T\ast R$: explicitly the correction is $T_{ap}^k{R}_{i\bar{b}j}^p-T_{ai}^p{R}_{p\bar{b}j}^k-T_{aj}^p{R}_{i\bar{b}p}^k$. For the second, $D_i({\mathrm{Ric}}^{\mathrm{\prime }}{}_j^k)=h^{k\bar{l}}({\mathrm{\nabla }}_i{q}_{j\bar{l}}-T_{ij}^p{q}_{p\bar{l}})$. Metric equivalence from step 4.1 therefore gives $\mathrm{\mid }Z{\mathrm{\mid }}_h\le C(1+\sqrt{S})$ and $\mathrm{\mid }\mathcal{R}T{\mathrm{\mid }}_h\le C\sqrt{S}$, where the latter uses the bounded Ricci endomorphism, not the full unknown curvature $R^{\mathrm{\prime }}$. This closes every coefficient in these identities in terms of quantities already bounded.

Differentiate the norm defining $S$, using $Dh=0$. At an $h$-normal point the product rule gives ${\mathrm{\Delta }}_{h}S=\mathrm{\mid }DT{\mathrm{\mid }}_h^2+\mathrm{\mid }\bar{D}T{\mathrm{\mid }}_h^2+2\mathrm{Re}⟨Z,T{⟩}_h-\mathrm{Re}⟨\mathcal{R}T,T{⟩}_h$: the two second-derivative terms have orders $D_a{\bar{D}}_b$ and ${\bar{D}}_b{D}_a$, whose difference is the commutator in step 2.2. All first-derivative terms are exactly the two displayed nonnegative squared norms. Thus step 5.1 gives ${\mathrm{\Delta }}_{h}S\ge -C(S+1)$. At the $g$-normal diagonal points of step 1.3 one has $S={\sum }_{i,k,a}({\lambda }_i{\lambda }_k{\lambda }_a{)}^{-1}\mathrm{\mid }{\mathrm{\partial }}_i{h}_{k\bar{a}}{\mathrm{\mid }}^2$, whereas the positive term $Q$ there is the same sum multiplied termwise by ${\lambda }_k$. Uniform metric equivalence consequently gives $Q\ge {\mathrm{\Lambda }}^{-1}S$. The trace identity of step 1.3 now gives ${\mathrm{\Delta }}_{h}U\ge {\mathrm{\Lambda }}^{-1}S-C$, since its curvature and $F$ terms are uniformly bounded under metric equivalence. Choose $A_1=\mathrm{\Lambda }(C+1)$. Then ${\mathrm{\Delta }}_h(S+A_{1}U)\ge S-C^{\mathrm{\prime }}$ with fixed $C^{\mathrm{\prime }}$. At a maximum point of this function, $S\le C^{\mathrm{\prime }}$, and boundedness of $U$ makes its maximum bounded. Since $S\le S+A_{1}U$, $S$ is uniformly bounded everywhere. Finally ${\mathrm{\nabla }}_i{h}_{j\bar{k}}=h_{l\bar{k}}{T}_{ij}^l$, obtained by subtracting the two metric-compatible connections; hence $\mathrm{\parallel }\mathrm{\nabla }h{\mathrm{\parallel }}_{C^0(g)}$ is bounded. This proves the required third-order metric estimate by an actual Bochner calculation, not an additional analytic assumption.

The last estimate makes $U$ uniformly Lipschitz in the fixed atlas, and therefore uniformly $C^{0,\alpha }$ for every fixed $\alpha <1$. The background equation ${\mathrm{\Delta }}_0\varphi =U-n$, the $C^0$ bound and the global linear Schauder estimate of [F4] yield a uniform $C^{2,\alpha }$ bound for $\varphi$. In a holomorphic chart, differentiating the determinant equation gives $h^{i\bar{j}}{\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}({\mathrm{\partial }}_c\varphi )={\mathrm{\partial }}_c(F+\log \det g)-h^{i\bar{j}}{\mathrm{\partial }}_c{g}_{i\bar{j}}$. Its coefficients are uniformly elliptic and $C^{0,\alpha }$ by steps 4.1 and 6.1, its source has a uniform $C^{0,\alpha }$ bound, and ${\mathrm{\partial }}_c\varphi$ has bounded supremum from $C^{2,\alpha }$. Apply the actual local estimate [F4], separately to real and imaginary parts, to bound $\varphi$ in $C^{3,\alpha }$ on inner charts. For higher derivatives, differentiate this equation successively. In the commutator with the principal coefficients at least one derivative hits $h^{-1}$; derivative-of-inverse expresses each such coefficient as a finite sum of products of copies of $h^{-1}$ and derivatives of $h$. Thus at each stage every commutator contains only derivatives already bounded at the preceding stage, while the differentiated $F$ is bounded by the given $M_r$. The higher-order/difference-quotient passage explicitly proved in [F4] on nested balls, or direct differentiation here since the solution is smooth, yields the next uniform $C^{m,\alpha }$ bound. A finite cover makes it global. Every step uses only finitely many prescribed derivatives of $F$. Along a smooth positive path, the positive lower bound and every such derivative bound are uniform by compactness of $X\times [0,1]$. This proves the entire Statement, including every integer $C^m$ estimate needed for continuity-path closure.