14.17 Existence for the Gauduchon ansatz6904C64D5831

Proof. Work with $v=u/(2\pi )$ and write $Q=Q_ u$ for the matrix in (14.13.1). By proposition 14.11.1, theorem 14.14.1, proposition 14.16.1, the function $v$, its gradient and its complex Hessian are uniformly bounded, and so are the constants $b$. Hence the eigenvalues of $Q$ are uniformly bounded above, and by the equation its determinant is uniformly bounded below. Every eigenvalue is therefore also bounded below by a positive constant. Choose fixed numbers $0<a<b_0$ such that $aI\leq Q\leq b_0I$ in unitary frames for every solution.

The function $\Delta _\gamma v=\operatorname {tr}_\gamma (\mathrm i\partial \bar\partial v)$ is uniformly bounded, and $\Delta _\gamma $ is a fixed uniformly elliptic operator. Its interior $W^{2,p}$ estimates, applied on a finite cover, bound $v$ in $W^{2,p}$ for every finite $p$. Choose $p>2n$ and $0<\alpha <1-2n/p$; Sobolev embedding gives a uniform $C^{1,\alpha }$ bound. Thus the matrix $R(x)=Q_0(x)+Z_ x(\partial v(x))$ is uniformly bounded in $C^\alpha $.

On a coordinate ball, let $\mathcal H_ x(A)$ denote the Hermitian matrix of $\mathrm i\partial \bar\partial v$ at $x$ in a fixed smooth unitary frame, expressed as a function of the real Hessian $A=D^2_{\mathbb R}v(x)$. It is a real linear function of $A$ depending smoothly on $x$. If a real symmetric matrix $N$ is nonnegative, then $\mathcal H_ x(N)\geq 0$, and its trace is comparable to $\operatorname {tr}_{\mathbb R}N$ with uniform positive constants on the ball. This follows by evaluating on a vector and its rotation by $\mathrm i$, or from the coordinate formula for $v_{i\bar j}$.

Freeze the function $\partial v(x)$ already obtained in $R(x)$, and define for all real symmetric matrices $A$

The infimum is over a fixed compact set of positive Hermitian matrices, and each expression in braces is an affine function of $A$, so $\widetilde\Phi $ is concave in $A$. If $R(x)+P_ n(\mathcal H_ x(A))$ lies in that compact set, then $\widetilde\Phi (A,x)$ equals its logarithmic determinant: each expression in braces is a tangent affine function of the concave logarithmic determinant, and the tangent at that matrix itself attains equality.

For $N\geq 0$, each of these affine functions increases by

when $A$ is replaced by $A+N$. Since $P_ n$ preserves positivity and trace, this increment lies between fixed positive multiples of $\operatorname {tr}_{\mathbb R}N$, uniformly in $B$ and $x$. Taking infima preserves both inequalities, so $\widetilde\Phi $ is uniformly elliptic on all real symmetric matrices.

Smoothness of $\mathcal H_ x$ and the uniform $C^\alpha $ bound for $R$ also give

These are the hypotheses of the local Evans–Krylov estimate for concave uniformly elliptic equations with Hölder dependence on $x$, in the formulation given in section 25.8. The value $\widetilde\Phi (0,x)$ is uniformly bounded, by the compactness of the set of matrices $B$ and the bound for $R$. By the choice of $a$ and $b_0$, the given smooth function solves

on the ball. The Evans–Krylov estimate, applied on smaller balls of a finite cover, yields a uniform $C^{2,\beta }$ bound for some $\beta >0$.

Differentiating the original matrix equation gives linear uniformly elliptic equations for the first derivatives of $v$, whose coefficients, including the first-order coefficients coming from $Z$, are now Hölder continuous. Local Schauder estimates give $C^{3,\beta }$ bounds, and differentiation and induction give all higher bounds. The finite coordinate cover makes the bounds global. The only regularity results used here are the standard $W^{2,p}$ estimates for the Laplacian, the Evans–Krylov estimate for concave equations, and the Schauder estimates. $\square$

of theorem 14.10.1. Let

We use the continuity method along the path $F_ t$. At $t=0$, the pair $u=0$, $b=0$ solves the equation, since $\eta _0=\gamma _0$. During the continuity argument we normalize potentials by $\int _ Xu\, \gamma ^ n=0$; adding a constant does not affect the equation, and at the end we impose $\sup _ Xu=0$ instead.

For openness, the derivative of the equation with respect to the mean-zero potential and the scalar is $(w,a)\mapsto \mathcal L_ uw-a$. Given a right-hand side $q$, integration against the adjoint probability measure of proposition 14.12.1 forces $a=-\int _ Xq\, \mathrm{d}\mu _ u^\gamma $. The Fredholm alternative then gives a unique mean-zero solution $w$, and Schauder estimates bound this inverse between Hölder spaces. Since positivity of $\Gamma _ u$ is an open condition, the implicit function theorem gives solutions for all nearby parameters.

For closedness, note that the functions $F_ t$ are uniformly bounded in every $C^ k$ norm. After supremum normalization, proposition 14.11.1 bounds $u$ and $b$, the second-order and rescaling estimates bound the gradient and the complex Hessian, and lemma 14.17.1 gives all higher bounds. Subtracting the mean preserves all derivative bounds and changes the potentials by uniformly bounded constants. A subsequence of mean-zero solutions along a convergent sequence of parameters therefore converges smoothly to a solution at the limiting parameter, and the bounds $aI\leq Q\leq b_0I$ preserve strict positivity in the limit. Thus the set of solvable parameters is nonempty, open and closed, hence equal to $[0,1]$.

Finally, consider two solutions $(u,b)$ and $(w,c)$ for the same $F$. At a maximum of $u-w$ their gradients agree and their complex Hessians are ordered, so the first-order terms are equal there and, since $P_ n$ preserves the order, $Q_ u\leq Q_ w$. The equation then gives $b\leq c$, and at a minimum the reverse inequality gives $b\geq c$. Hence $b=c$. Along the segment between $u$ and $w$, the matrices $Q$ remain positive because the ansatz is affine in the potential. Integrating the derivative of the logarithmic determinant along that segment produces a uniformly elliptic scalar operator with first-order terms and no zeroth-order term that annihilates $u-w$. The maximum principle shows that $u-w$ is constant, and the supremum normalization makes the constant zero. $\square$