Theorem 16.15.20AA7242F2CF85

Proof. For $0\leq t\leq 1$ consider the equation

and let $T\subset [0,1]$ be the set of $t$ for which it has a smooth real solution. For $t=0$ the solution is $a=0$, so $T\neq \varnothing $. We show that $T$ is open and closed in $[0,1]$; then $T=[0,1]$, since $[0,1]$ is connected, and the case $t=1$ gives existence. Uniqueness is proved at the end.

Step 1: a priori estimates. Let $a$ be a smooth solution for some $t\in [0,1]$, with metric $g$. In this step, constants depend only on $X$, $\omega $, $\kappa $, $\beta $ and $f$, but not on $t$ or on the solution. Since $\| tf\| _\infty \leq \| f\| _\infty $ and, by (16.15.1), $\sup _ XD_ h(tf)=t\sup _ XD_ hf\leq \sup _ XD_ hf$, lemma 16.15.7, applied to $tf$, gives $|a|\leq \| f\| _\infty $ and $C_0^{-1}h\leq g\leq C_0h$ with $C_0$ independent of $t$. Hence $La=-4\pi D_ ha=-4\pi (H-\operatorname {tr}_ hk)$ is bounded, and proposition 16.14.9 bounds $a$ in $W^{2,p}(X)$ for every finite $p$; by (16.14.3) and Morrey's inequality, this bounds $\| a\| _{C^1}$ and $\| D^2a\| _{L^ p}$ on compact subsets of charts. By lemma 16.15.13, the first derivatives of the $g_{i\bar j}$ are bounded on compact subsets of the charts of a fixed finite atlas.

Fix finitely many coordinate balls $U_\nu $ whose closures lie in chart domains, and concentric balls $V'_\nu \Subset V_\nu \Subset U_\nu $ such that the $V'_\nu $ cover $X$. In the coordinates of $U_\nu $, the function $a$ solves (16.15.18) with $\phi _0=\log \det k-\beta tf$, whose derivatives are bounded independently of $t$. By the preceding paragraph, on $U_\nu $ the matrix $G=g$ lies between two fixed positive multiples of the identity, it is Lipschitz with a fixed constant, so that we may take $\varpi (r)=Cr$, and $\| a\| _{C^1(U_\nu )}+\| D^2a\| _{L^ p(U_\nu )}$ is bounded for every finite $p$. By lemma 16.15.17, $\| a\| _{W^{\ell ,p}(V_\nu )}$ is bounded for all $\ell \geq 3$ and $p<\infty $, and by Morrey's inequality every derivative of $a$ is bounded on $V'_\nu $. Thus every $C^\ell $ norm of $a$, computed in these coordinates on the sets $V'_\nu $, is bounded independently of $t$.

Step 2: openness. Let $t_0\in T$, with smooth solution $a$ and metric $g$; thus $\kappa +\mathrm{dd}^{\mathrm{c}}a$ is a Kähler form. For $t$ near $t_0$ we look for a solution of the form $a+b$, and we find $b$ by the contraction mapping principle in a Sobolev space. Fix an integer $s>n$. We use the spaces $H^ r$ of section 10.2, defined with the coordinate balls $U_\nu $ of Step 1 and a partition of unity $(\chi _\nu )$ subordinate to them; by lemma 10.2.1(1), other choices give equivalent norms. For a smooth function $v$ with compact support in $U_\nu $, written in the coordinates of $U_\nu $, let $\| v\| _{\langle r\rangle }$ be defined by the formula of lemma 16.15.19 with $d=2n$, so that $\| u\| _{H^ r}^2=\sum _\nu \| \chi _\nu u\| _{\langle r\rangle }^2$ for smooth $u$. Choose smooth functions $\tilde\chi _\nu $, $\check\chi _\nu $ on $X$ with values in $[0,1]$ and compact supports in $U_\nu $ such that $\tilde\chi _\nu =1$ near $\operatorname{Supp}\chi _\nu $ and $\check\chi _\nu =1$ near $\operatorname{Supp}\tilde\chi _\nu $. We use three facts.

1. Since $s>n$, lemma 10.2.1(3) gives continuous inclusions $H^{s+2}\subset C^2(X)$ and $H^ s\subset C^0(X)$.

2. For a smooth function $\psi $ with compact support in $U_\nu $, multiplication by $\psi $ is bounded for $\| \cdot \| _{\langle s\rangle }$, by lemma 16.15.19; and $\| \partial _\mu v\| _{\langle r\rangle }\leq 2\pi \| v\| _{\langle r+1\rangle }$ for real coordinate derivatives, since $\widehat{\partial _\mu v}=2\pi \mathrm i\xi _\mu \widehat v$.

3. $\| \check\chi _\nu b\| _{\langle s+2\rangle }\leq C\| b\| _{H^{s+2}}$ for smooth $b$. Indeed, $\Sigma =\sum _\iota \check\chi _\iota \geq 1$, because $\check\chi _\iota =1$ wherever $\chi _\iota >0$; so the functions $\check\chi _\iota /\Sigma $ form a partition of unity subordinate to the same charts, and by lemma 10.2.1(1) the norm of $H^{s+2}$ computed with them is equivalent to the given one. Moreover $\check\chi _\nu b=\eta _\nu \cdot (\check\chi _\nu /\Sigma )b$, where $\eta _\nu $ is a smooth function with compact support in $U_\nu $ equal to $\Sigma $ near $\operatorname{Supp}\check\chi _\nu $, and multiplication by $\eta _\nu $ is bounded for $\| \cdot \| _{\langle s+2\rangle }$ by lemma 16.15.19.

For a real function $b\in C^2(X)$, let $Y$ be the tensor $Y^ m{}_ j=g^{m\bar l}b_{j\bar l}$; its powers $(Y^ q)^ m{}_ j=Y^ m{}_{a_1}Y^{a_1}{}_{a_2}\cdots Y^{a_{q-1}}{}_ j$ are tensors, so the traces $\operatorname {tr}(Y^ q)$ are functions on $X$. In a chart, with the matrices $G=(g_{i\bar j})$ and $B=(b_{i\bar j})$, the traces of the powers of $Y$ are those of $BG^{-1}$, which is similar to the Hermitian matrix $G^{-1/2}BG^{-1/2}$; let $\lambda _1,\ldots ,\lambda _ n$ be their real eigenvalues. Then $\det (G+B)=\det G\prod _ i(1+\lambda _ i)$ and $\operatorname {tr}Y=\sum _ i\lambda _ i=g^{i\bar j}b_{i\bar j}=D_ gb$. Since $|\lambda _ i|\leq \| G^{-1}\| \, |B|$, and $G$ is bounded below on compact subsets of the charts of a finite atlas, (i) gives $r_0>0$ such that $\| b\| _{H^{s+2}}\leq r_0$ implies $|\lambda _ i|\leq 1/2$; then $G+B>0$, that is, $\kappa +\mathrm{dd}^{\mathrm{c}}(a+b)>0$, and

Put $\mathcal N_ t(a+b)=\log (\det (G+B)/\det k)-\beta (a+b-tf)$, so that $a+b$ solves the equation for $t$ exactly when $\mathcal N_ t(a+b)=0$. Since $\log (\det G/\det k)=\beta (a-t_0f)$,

We estimate $\mathcal Q$ in $H^ s$. Let $b$ be smooth and real with $\| b\| _{H^{s+2}}\leq r_0$. In the chart $U_\nu $ let $Y_\nu =\tilde\chi _\nu Y$, a matrix of smooth functions with compact support, and $\| Y_\nu \| _{\langle s\rangle }=\max _{m,j}\| \tilde\chi _\nu Y^ m{}_ j\| _{\langle s\rangle }$. Since $\tilde\chi _\nu Y^ m{}_ j=(\tilde\chi _\nu g^{m\bar l})\, \partial _ j\partial _{\bar l}(\check\chi _\nu b)$, facts (ii) and (iii) give $\| Y_\nu \| _{\langle s\rangle }\leq C_ Y\| b\| _{H^{s+2}}$. For matrices $M$, $N$ of smooth compactly supported functions, lemma 16.15.19 gives $\| (MN)_{mj}\| _{\langle s\rangle }\leq nC_ s\max \| M_{\cdot \cdot }\| _{\langle s\rangle }\max \| N_{\cdot \cdot }\| _{\langle s\rangle }$, and therefore

where $C_\chi $ bounds multiplication by $\chi _\nu $. Since $Y_\nu =Y$ on $\operatorname{Supp}\chi _\nu $, the function $\chi _\nu \mathcal Q(b)$ is the sum of the series $\sum _{q\geq 2}\frac{(-1)^{q+1}}q\chi _\nu \operatorname {tr}(Y_\nu ^ q)$, which converges uniformly, with supports in $\operatorname{Supp}\chi _\nu $. Decreasing $r_0$, we may assume that $nC_ sC_ Yr_0\leq 1/2$. Then the series converges absolutely for $\| \cdot \| _{\langle s\rangle }$; since its partial sums converge in $L^2$, their Fourier transforms converge in $L^2$ (theorem 2.9.4), a subsequence converges almost everywhere, and Fatou's lemma gives

Summing over $\nu $, we obtain $\| \mathcal Q(b)\| _{H^ s}\leq C_ Q\| b\| _{H^{s+2}}^2$. In the same way, using $Y^ q-Y'^ q=\sum _{i=0}^{q-1}Y^ i(Y-Y')Y'^{q-1-i}$, we obtain

for two such functions $b$, $b'$.

Both estimates extend to real elements of $H^{s+2}$. Let $H^{s+2}_{\mathbb R}$ be the closed subspace of real elements of $H^{s+2}$. Smooth real functions are dense in it, since taking real parts does not increase the norms: $|\widehat{\chi \operatorname{Re}v}(\xi )|^2\leq \frac12(|\widehat{\chi v}(\xi )|^2+|\widehat{\chi v}(-\xi )|^2)$ for real $\chi $. If $b\in H^{s+2}_{\mathbb R}$ with $\| b\| _{H^{s+2}}<r_0$, and smooth real $b_ j$ converge to $b$ in $H^{s+2}$, with norms less than $r_0$, then the $\mathcal Q(b_ j)$ form a Cauchy sequence in $H^ s$; by (i), $b_ j\to b$ in $C^2$, so $\mathcal Q(b_ j)\to \mathcal Q(b)$ uniformly, and, again by (i), the limit in $H^ s$ is $\mathcal Q(b)$. Hence the two estimates for $\mathcal Q$ hold on the ball $\| b\| _{H^{s+2}}<r_0$ of $H^{s+2}_{\mathbb R}$.

The operator $\mathcal D\colon H^{s+2}\to H^ s$ is invertible, with bounded inverse. Indeed, $\kappa +\mathrm{dd}^{\mathrm{c}}a=\frac{\mathrm i}{2\pi }g_{i\bar j}\, \mathrm dz_ i\wedge \mathrm d\bar z_ j$ is a Kähler form, and by (10.2.14), applied with $\omega '=\kappa +\mathrm{dd}^{\mathrm{c}}a$, we have $D_ g-\beta =-(4\pi )^{-1}(L_{\omega '}+4\pi \beta )$. Since $\omega $ is an arbitrary Kähler form on $X$, lemma 10.2.5(3) holds for $\omega '$; applied to functions, on which the Hodge Laplacian of $\omega '$ is $L_{\omega '}$, with $c=4\pi \beta >0$, it shows that $D_ g-\beta $ maps $H^{s+2}$ isomorphically onto $H^ s$, because these spaces do not depend on the Kähler form (lemma 10.2.1(4)). Let $M_{\mathcal D}$ be the norm of its inverse. Since $\mathcal D$ is a real operator, its inverse maps real elements to real elements. For $b\in H^{s+2}$, the element $\mathcal Db$ of $H^ s$ is the continuous function $D_ gb-\beta b$, by approximation with smooth functions and (i).

The solutions $b\in H^{s+2}_{\mathbb R}$ of $\mathcal N_ t(a+b)=0$ with $\| b\| _{H^{s+2}}<r_0$ are the fixed points of

Choose $0<r<r_0$ with $4M_{\mathcal D}C_ Qr\leq 1$, and suppose that $|t-t_0|\leq \delta $, where $2M_{\mathcal D}\beta \delta \| f\| _{H^ s}\leq r$. On the closed ball $\overline B_ r$ of radius $r$ in $H^{s+2}_{\mathbb R}$ we then have $\| \Phi _ t(b)\| _{H^{s+2}}\leq M_{\mathcal D}(\beta \delta \| f\| _{H^ s}+C_ Qr^2)\leq r/2+r/4$ and $\| \Phi _ t(b)-\Phi _ t(b')\| _{H^{s+2}}\leq 2M_{\mathcal D}C_ Qr\| b-b'\| _{H^{s+2}}\leq \frac12\| b-b'\| _{H^{s+2}}$. The iterates $b^{(0)}=0$, $b^{(j+1)}=\Phi _ t(b^{(j)})$ therefore stay in $\overline B_ r$, their successive differences decrease geometrically, so they form a Cauchy sequence, and their limit $b$ is a fixed point of $\Phi _ t$, by the Lipschitz bound. Thus $\mathcal Db=-\beta (t-t_0)f-\mathcal Q(b)$ in $H^ s$, hence pointwise by (i), that is, $\mathcal N_ t(a+b)=0$. The function $u=a+b$ is of class $C^2$, satisfies $\kappa +\mathrm{dd}^{\mathrm{c}}u>0$, and in every chart it solves (16.15.18) with $\phi _0=\log \det k-\beta tf$. By lemma 16.15.17, $u$ is smooth. Hence $T$ contains $[t_0-\delta ,t_0+\delta ]\cap [0,1]$, and $T$ is open in $[0,1]$.

Step 3: closedness. Let $t_ j\in T$ converge to $t_*\in [0,1]$, with solutions $a_ j$. By Step 1, every $C^\ell $ norm of $a_ j$ on the compact sets $\overline{V'_\nu }$ is bounded independently of $j$. Hence, for every $\ell $, the derivatives of order at most $\ell $ of the $a_ j$ are uniformly bounded on each $\overline{V'_\nu }$, and they are equicontinuous there by the mean value theorem, since the derivatives of order $\ell +1$ are bounded and the balls $V'_\nu $ are convex. By the Arzelà–Ascoli theorem (proposition 2.7.4) and a diagonal argument over $\ell $ and $\nu $, a subsequence converges, together with all its derivatives, uniformly on each $\overline{V'_\nu }$. The limit $a_*$ is smooth, and its derivatives are the limits of the derivatives of the $a_ j$, because uniform convergence of functions and of their first derivatives on a ball implies that the limit is continuously differentiable with the limit derivatives. The uniform lower bound $g_ j\geq C_0^{-1}h$ of Step 1 gives $\kappa +\mathrm{dd}^{\mathrm{c}}a_*>0$, and $a_*$ solves the equation for $t_*$. Thus $t_*\in T$, and $T$ is closed.

Step 4: uniqueness. Let $a$ and $a'$ be smooth solutions of (16.15.8). At a maximum point of $a-a'$ the complex Hessian of $a-a'$ is negative semidefinite, so $0<k+(a_{i\bar j})\leq k+(a'_{i\bar j})$, and the equations give $\mathrm e^{\beta (a-f)}\leq \mathrm e^{\beta (a'-f)}$ there. Hence $a-a'\leq 0$ at that point, and therefore everywhere. Interchanging $a$ and $a'$ proves that $a=a'$. $\square$