2 Preliminaries

All manifolds and Riemannian metrics are assumed to be of class $ C^\infty \hskip -1.5pt$ . By definition, a manifold is connected. We use the symbol $ \delta $ for divergence.

On a manifold with a torsion-free connection $ \nabla \hskip -.7pt$ , the Ricci tensor $ \mathrm {r}$ satisfies the Bochner identity $ \mathrm {r}(\cdot ,v)=\hskip .7pt\delta \hskip 1.2pt\nabla \hskip -.7pt v -\hskip .7pt d\hskip .7pt[\delta \hskip .7pt v]$ , where $ v$ is any vector field. Its coordinate form $ R_{jk} v^{k}=v^{k}{}_{,\,jk}-v^{k}{}_{,\,kj}$ arises via contraction from the Ricci identity $ v^{l}{}_{,\hskip .7pt jk}-v^{l}{}_{,\hskip .7pt kj}=R_{jkq}{}^l v^q$ . (We use the sign convention for $ R$ such that $ R_{jk} = R_{jqk}{}^q\hskip -.7pt$ .) Applied to the gradient $ v$ of a function $ \tau\kern-0.8pt$ on a Riemannian manifold, this yields

(2-1) $$ \begin{align} \delta[\nabla d\tau\kern-0.8pt]=\mathrm{r}(v,\cdot) +dY\quad\text{with } v=\nabla\tau\kern-0.8pt\quad\text{and}\quad Y=\Delta\tau\kern-0.8pt. \end{align} $$

Also, given a function $ \tau\kern-0.8pt$ on a Riemannian manifold,

(2-2) $$ \begin{align} 2[\nabla d\tau\kern-0.8pt](v,\cdot)=dQ,\quad\text{where } v=\nabla\hskip-.7pt\tau\kern-0.8pt\quad\mathrm{and}\quad Q=g(v,v), \end{align} $$

as one sees noting that, in local coordinates, $ (\tau\kern-0.8pt\hskip -1.5pt_{\hskip -.7pt,\hskip .7pt k}\tau\kern-0.8pt^{,\hskip .4pt k})_{,\hskip .4pt j} =2\tau\kern-0.8pt\hskip -1.5pt_{\hskip -.7pt,\hskip .7pt kj}\tau\kern-0.8pt^{,\hskip .4pt k}$ . We can rewrite the relations (2-1)–(2-2) using the interior product $ \imath _v$ , and then they read

(2-3) $$ \begin{align} \mathrm{(a)}\quad \delta\hskip.4pt[\nabla d\tau\kern-0.8pt]=\imath_v\mathrm{r}\hskip.7pt\, +\hskip.7pt\,dY,\quad\mathrm{(b)}\quad 2\imath_v[\nabla d\tau\kern-0.8pt]=dQ\quad\text{with ({1\mbox{-}4}).} \end{align} $$

Finally, for the Ricci tensor $ \mathrm {r}$ and scalar curvature $ \mathrm {s}$ of any Riemannian metric,

(2-4) $$ \begin{align} 2\hskip.7pt\delta\mathrm{r}=d\hskip.4pt\mathrm{s}, \end{align} $$

which is known as the Bianchi identity for the Ricci tensor. Its coordinate form $ 2g^{kl}R_{j k,l}=s_{\hskip -.7pt,\hskip .7pt j}$ is immediate if one transvects with (‘multiplies’ by) $ g^{kl}$ the equality $ R_{j kl}{}^q{}_{,\hskip .7pt q} =R_{jl,k}-R_{kl,j}$ obtained by contracting the second Bianchi identity.

The harmonic-flow condition for a vector field $ v$ on a Riemannian manifold $ (M\hskip -.7pt,g)$ , meaning that the flow of $ v$ consists of (local) harmonic diffeomorphisms, is known [Reference Nouhaud19] to be equivalent to the equation

(2-5) $$ \begin{align} g(\Delta v,\cdot )=-\mathrm{r}\hskip.4pt(v,\cdot), \end{align} $$

the vector field $ \Delta v$ having the local components $ [\Delta v]^j = v^{j,k}{}_k$ . See also [Reference Dodson, Trinidad Pérez and Vázquez-Abal14, Theorem 3.1]. When $ v=\nabla \hskip -.7pt\tau\kern-0.8pt$ is the gradient of a function $ \tau\kern-0.8pt:M\to \mathrm {I\!R}$ ,

(2-6) $$ \begin{align} \text{the harmonic-flow condition ({2\mbox{-}5}) amounts to ({1\mbox{-}3}).} \end{align} $$

In fact, by (2-1), $ 2\mathrm {r}\hskip .7pt(v,\cdot )+\hskip .7pt dY = \delta \hskip .4pt[\nabla d\tau\kern-0.8pt] +\mathrm {r}\hskip .7pt(v,\cdot )=g(\Delta v,\cdot ) +\mathrm {r}\hskip .4pt(v,\cdot )$ , as $ [\Delta v]_j=v_{j,k}{}^k = \tau\kern-0.8pt\hskip -1.5pt_{,jk}{}^k\hskip -.7pt =\tau\kern-0.8pt\hskip -1.5pt_{,kj}{}^k = \tau\kern-0.8pt^{,kj}{}_k=(\delta \hskip .4pt[\nabla d\tau\kern-0.8pt])\hskip -.4pt_j$ .

Furthermore (see, for example, [Reference Derdzinski and Maschler11, Lemma 5.2]), on a Kähler manifold $ (M\hskip -.7pt,g)$ ,

(2-7) $$ \begin{align} \begin{array}{l} \text{conditions\ ({1\mbox{-}1})--({1\mbox{-}2})\ imply\ real}\text{-}\mathrm{holomorphicity\ of\ the}\\ \mathrm{gradient\ } v \hskip-.4pt = \hskip-1.5pt \ \nabla\tau\kern-0.8pt\mathrm{, \ while \ } J \hskip-.4ptv\mathrm{\ is \ then \ a \ holomorphic \ Killing}\\ \text{field, due to the resulting Hermitian symmetry of } \nabla d\tau\kern-0.8pt. \end{array} \end{align} $$

Since holomorphic mappings between Kähler manifolds are harmonic, every holomorphic vector field on a Kähler manifold satisfies (2-5); see also [Reference Dodson, Trinidad Pérez and Vázquez-Abal14, Remark 3.2]. Now, (1-5) follows from (2-6). In other words, as observed by Calabi [Reference Calabi and Yau5], on Kähler manifolds, one has

(2-8) $$ \begin{align} ({1\mbox{-}3}), \text{with } ({1\mbox{-}4}) \text{ for all real-holomorphic gradients } v=\nabla\hskip-.7pt\tau\kern-0.8pt\hskip.4pt. \end{align} $$

Given a tensor field $ \theta $ on a manifold $ M\hskip -.7pt$ , we say that a point $ x\in M$ is $ \theta $ -generic if $ x$ has a neighborhood on which either $ \theta =0$ identically, or $ \theta \ne 0$ everywhere. Such points clearly form a dense open subset of $ M\hskip -.7pt$ .

3 Ricci–Hessian equations

As a consequence of (1-1), for the scalar curvature $ \mathrm {s}$ , with (1-4),

(3-1) $$ \begin{align} n\hskip.4pt\sigma=Y\alpha + \hskip.7pt\mathrm{s},\quad\text{where } n=\dim M. \end{align} $$

Applying $ 2\hskip .4pt\imath _v$ or $ 2\hskip .4pt\delta $ to (1-1), we obtain, from (2-3)–(2-4) and (1-4),

(3-2) $$ \begin{align} \begin{array}{rl} \mathrm{(i)}& \alpha\hskip.7pt dQ + 2\hskip.4pt\mathrm{r}\hskip.4pt(v,\cdot)=2\sigma\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, +\,2\alpha\hskip.7pt[\mathrm{r}\hskip.4pt(v,\cdot)\,+\hskip.7pt\,dY]\,+\hskip.7pt\,d\hskip.7pt\mathrm{s}\, =\,2\hskip.7pt d\sigma. \end{array} \end{align} $$

In the case where (1-1)–(1-2) hold along with (1-3), one may rewrite (3-2) as

(3-3) $$ \begin{align} \begin{array}{rl} \mathrm{(i)}& \alpha\hskip.7pt dQ\,\hskip.4pt-\,dY = 2\sigma\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, +\,\alpha\hskip.7pt dY\hskip.7pt+\hskip.7pt\,d\hskip.7pt\mathrm{s}=2\hskip.7pt d\sigma, \end{array} \end{align} $$

which, in view of (2-2) and (3-1), amounts to nothing other than

(3-4) $$ \begin{align} \begin{array}{rl} \mathrm{(i)}& d(Q\alpha - Y)=(Q\alpha'\hskip.7pt+\,2\sigma)\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& d\hskip.4pt[Q\alpha'\hskip.7pt+\,(n-2)\hskip.4pt\sigma]=(Q\alpha"\hskip.7pt+\,Y\alpha')\hskip.7pt d\tau\kern-0.8pt, \end{array} \end{align} $$

as the assumption, in (1-2), that $\alpha $ is a $ C^\infty \hskip -1.5pt$ function of $ \tau\kern-0.8pt$ allows us to write

(3-5) $$ \begin{align} d\alpha=\alpha'\hskip-.7pt d\tau\kern-0.8pt,\quad\nabla\hskip-1.5pt\alpha\, =\,\alpha'\hskip-.7ptv,\quad 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, =\alpha'\hskip-.4pt dQ,\quad\mathrm{where\ }\alpha' = \hskip.7pt d\alpha/d\tau\kern-0.8pt, \end{align} $$

since (2-2) gives $ 2[\nabla d\tau\kern-0.8pt](\nabla \hskip -1.5pt\alpha ,\cdot ) =2\alpha '[\nabla d\tau\kern-0.8pt](v,\cdot )=\alpha '\hskip -.4pt dQ$ . Due to (3-4), conditions (1-1)–(1-3) imply that, locally, at points at which $ d\tau\kern-0.8pt\ne 0$ ,

(3-6) $$ \begin{align} \begin{array}{l} Q\alpha\hskip.7pt-\hskip.7pt Y\hskip.7pt\mathrm{\ and\ }\hskip.7pt\,Q\alpha' \ + \hskip.7pt(n-2)\hskip.4pt\,\sigma \ \mathrm{\ are\ functions\ of\ } \ \tau\kern-0.8pt\mathrm{,\ with}\\ \mathrm{the\ respective\ } \tau\kern-0.8pt \text{-derivatives } Q\alpha'+\hskip.7pt2\sigma\,\mathrm{\ and\ }\,Q\alpha"+\,Y\alpha',\\ \mathrm{which,\hskip-1.5pt\ consequently\hskip-1.5pt,\hskip-1.5pt\ must\hskip-.4pt\ themselves\hskip-.4pt\ be\hskip-.4pt\ functions\hskip-.4pt\ of\hskip-.4pt\ }\hskip.7pt\tau\kern-0.8pt\hskip-.7pt. \end{array} \end{align} $$

Assuming only (1-1), for $ n=\dim M\hskip -.7pt$ , with the aid of (3-1), we rewrite (3-2) as

$$ \begin{align*} \begin{array}{l} n\hskip.4pt[\alpha\hskip.7pt dQ + 2\hskip.4pt\mathrm{r}\hskip.4pt(v,\cdot)]\, -\,2(Y\alpha + \hskip.7pt\mathrm{s})\hskip.7pt d\tau\kern-0.8pt=0,\\ 2n\hskip.4pt\{\hskip-.7pt[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot ) +\hskip-.7pt\alpha\hskip.4pt\mathrm{r}\hskip.4pt(v,\cdot)\}\hskip-.7pt +\hskip-.7pt2[(n-1)\alpha\hskip.7pt dY\hskip-1.5pt-\hskip-.7pt Y\hskip-.7pt d\alpha]+\hskip-.7pt(n-2)\hskip.7pt d\hskip.4pt\mathrm{s}=0. \end{array} \end{align*} $$

If (1-3) holds as well, replacing $ 2\hskip .4pt\mathrm {r}\hskip .4pt(v,\cdot )$ here with $ -\hskip .4pt dY\hskip .7pt$ , we obtain $ n\hskip .4pt(\alpha \hskip .7pt dQ-\hskip .7pt dY) -2(Y\alpha +\hskip .7pt\mathrm {s})\hskip .7pt d\tau\kern-0.8pt=0$ and $ 2n\hskip .4pt[\nabla d\tau\kern-0.8pt](\nabla \hskip -1.5pt\alpha ,\cdot )+(n-2)(\alpha \hskip .7pt dY\hskip -1.5pt +d\hskip .4pt\mathrm {s})-\hskip -.7pt2Y\hskip -.7pt d\alpha =0$ . Thus, when (1-1)–(1-3) are all satisfied, (3-5) gives

(3-7) $$ \begin{align} \begin{array}{rl} \mathrm{(a)}& n\hskip.4pt(\alpha\hskip.7pt dQ-dY)-2(Y\alpha + \hskip.7pt\mathrm{s})\hskip.7pt d\tau\kern-0.8pt=0,\\ \mathrm{(b)}& n\hskip.4pt\alpha'\hskip-.7pt dQ+(n-2)(\alpha\hskip.7pt dY+ d\hskip.4pt\mathrm{s}) - 2Y\alpha'\hskip-.7pt d\tau\kern-0.8pt =0. \end{array} \end{align} $$

4 Ricci–Hessian equations on Kähler manifolds

The goal of this section is to prove Theorems B and D.

In any complex manifold, $ d\hskip .4pt\omega =0$ and $ \omega \hskip .4pt(J\cdot ,\cdot )$ is symmetric if $ \omega =i\hskip .4pt\partial \overline {\partial }\hskip .7pt\tau\kern-0.8pt$ , that is, if $ 2\hskip .7pt\omega =-\hskip .4pt d\hskip 1pt[\hskip -.4pt J^*\hskip -1.5pt d\tau\kern-0.8pt]$ for a real-valued function $ \tau\kern-0.8pt$ , with the $1$ -form $ J^*\hskip -1.5pt d\tau\kern-0.8pt=(d\tau\kern-0.8pt)\hskip -.4pt J$ , which sends any tangent vector field $ v$ to $ d\hskip -1.5pt_{J\hskip -.7ptv}\tau\kern-0.8pt$ . Our exterior-derivative and exterior-product conventions, for $1$ -forms $ \xi ,\xi '$ and vector fields $u,v$ , are

(4-1) $$ \begin{align} \begin{array}{l} [d\hskip.4pt\xi](u,v)=d_u[\hskip.4pt\xi(v)] - d_v[\hskip.4pt\xi(u)] - \xi([u,v]),\\ {}[\xi\wedge\xi'](u,v)=\xi(u)\xi'(v) - \xi(v)\xi'(u). \end{array} \end{align} $$

For a torsion-free connection $ \nabla \hskip -.7pt$ , (4-1) gives $ [d\hskip .4pt\xi ](u,v)=[\nabla \hskip -.7pt\!_u\xi ](v)-[\nabla \hskip -.7pt\!_v\xi ](u)$ , so that, if in addition $ \nabla \hskip -1.5pt J=0$ , on an almost-complex manifold,

(4-2) $$ \begin{align} 2i\hskip.4pt\partial\overline{\partial}\hskip.7pt\tau\kern-0.8pt =[\nabla d\tau\kern-0.8pt](J\cdot ,\cdot )-[\nabla d\tau\kern-0.8pt](\cdot ,J\cdot ). \end{align} $$

In the case of a Kähler metric g on a complex manifold $ M\hskip -.7pt$ , (1-1) implies that

(4-3) $$ \begin{align} i\hskip.4pt\alpha\hskip1pt\partial\overline{\partial}\hskip.7pt\tau\kern-0.8pt+\rho=\sigma\omega, \end{align} $$

$\omega ,\rho $ being the Kähler and Ricci forms, with both terms on the right-hand side of (4-2) equal, as the Hermitian symmetry of $ \nabla d\tau\kern-0.8pt$ follows from those of $ \rho $ and $ \omega $ .

We have the following result, due to Maschler [Reference Maschler17, Proposition 3.3].

Proof of Theorem D.

As $ dQ\wedge d\tau\kern-0.8pt\ne 0$ everywhere in $ \,U\hskip -.7pt$ , Theorem A implies (1-6) and, consequently, also the final clause about the five possible pairs.

Next, in (1-6), $4 d\theta =2d[\alpha \mathrm {s}+(2\alpha '+\alpha ^2)Y]=2[\alpha d\mathrm {s}+\mathrm {s}d\alpha +(2\alpha '+\alpha ^2)dY]$ , which, as $n=4$ , equals, in view of (3-5),

$$ \begin{align*} \alpha[n\hskip.4pt\alpha'\hskip-.7pt dQ+(n-2)(\alpha\hskip.7pt dY+ d\hskip.4pt\mathrm{s}) -2Y\alpha'\hskip-.7pt d\tau\kern-0.8pt]-\alpha'[n\hskip.4pt(\alpha\hskip.7pt dQ-dY)-2(Y\alpha+\hskip.7pt\mathrm{s})\hskip.7pt d\tau\kern-0.8pt], \end{align*} $$

and hence vanishes due to (3-7). However, the function $ \psi $ of $ \tau\kern-0.8pt$ defined in the theorem is an antiderivative of $ 1/\alpha ^2\hskip -.7pt$ , meaning that

(4-4) $$ \begin{align} \psi' = 1/\alpha^2. \end{align} $$

Namely, by (1-6), $ 4\varepsilon \hskip .4pt\psi ' = 1+2\alpha '\hskip -.7pt/\hskip -.7pt\alpha ^2 = (2\alpha '+\alpha ^2)/\hskip -.7pt\alpha ^2\hskip -.7pt =4\varepsilon /\hskip -.7pt\alpha ^2$ if $ \varepsilon \ne 0$ , and $ 3\psi ' = -6\alpha '\hskip -.7pt/\hskip -.7pt\alpha ^4 = 3/\hskip -.7pt\alpha ^2$ when $ \varepsilon =0$ , as $ 2\alpha ' = -\alpha ^2\hskip -.7pt$ .

Furthermore, $ d\hskip .7pt(\theta \psi +\alpha ^{-\hskip -1.5pt1}Y\hskip -.7pt-\hskip .7pt Q)=0$ . In fact, $ d\alpha =\alpha '\hskip -.7pt d\tau\kern-0.8pt$ in (3-5), and similarly for $ \psi $ , so that, from (4-4), $ d\hskip .7pt(\theta \psi )=\theta \hskip .7pt d\hskip .4pt\psi =\theta \psi '\hskip -.7pt d\tau\kern-0.8pt=\theta \alpha ^{-\hskip -.7pt2}d\tau\kern-0.8pt$ , and $ \alpha \hskip .7pt d\hskip .7pt(\hskip -.7pt\alpha ^{-\hskip -1.5pt1}Y\hskip -.4pt) =dY\hskip -.7pt-\alpha ^{-\hskip -1.5pt1}Y\alpha '\hskip -.7pt d\tau\kern-0.8pt$ . Also, $ 2(\theta -Y\alpha ')=(Y\alpha +\hskip .7pt\mathrm {s}\hskip -.4pt)\hskip .4pt\alpha $ from (1-6)–(1-7). These relations yield

$$ \begin{align*} -4\alpha d(\theta\psi+\alpha^{-1}Y- Q) &=4[(\alpha dQ-dY)-(\theta-Y\alpha')\alpha^{-1} d\tau\kern-0.8pt]\\ &=n(\alpha dQ- dY)-2(Y\alpha+\mathrm{s}) d\tau\kern-0.8pt, \end{align*} $$

with $ n=4$ , which equals $ 0$ by (3-7a).

Finally, (1-8) and the second relation in (1-7) easily give (1-9a). Thus, by (1-4) and (3-5), $ (Q\alpha '+F')Q=(Q\alpha '+F')\hskip .7pt d_v\tau\kern-0.8pt=Q d_v\alpha +d_v F$ , which, due to (1-9a), equals $ d_v Y\hskip -1.5pt-\alpha \hskip .7pt d_vQ$ . At the same time, $ -\imath _v$ applied to (3-3i) yields $ d_v Y\hskip -1.5pt-\alpha \hskip .7pt d_vQ=-\hskip -.7pt2Q\sigma $ . We thus get (1-9b). To obtain (1-9c), note that, from (1-4), $ \Delta \alpha =Q\alpha "+\hskip .7pt Y\alpha '$ , which, by (1-6) and (1-9a), equals $ (Y\hskip -1.5pt-Q\alpha )\alpha ' = F\hskip -.7pt\alpha ' = -\hskip -.7pt F"\hskip -.7pt$ , where the last equality trivially follows from (1-8).

Theorem D has a partial converse: if a nonconstant function $ \tau\kern-0.8pt$ with real-holomorphic gradient $ v=\nabla \hskip -.7pt\tau\kern-0.8pt$ on a Kähler surface $ (M\hskip -.7pt,g)$ and a function $\alpha $ of the real variable $ \tau\kern-0.8pt$ satisfy (1-6) and (1-7), then they must also satisfy the Ricci–Hessian equation (1-1) with $ \sigma $ given by (3-1) for $ n=4$ .

In fact, $ b(v,\cdot )=0$ , where $ b$ denotes the traceless Hermitian symmetric $ 2$ -tensor field $ \alpha \nabla d\tau\kern-0.8pt+\hskip .7pt\mathrm {r}\hskip .7pt-\sigma g$ . Namely, (1-3)–(1-5) and (2-3b) yield $ 4b(v,\cdot )=2\alpha \,dQ-2\hskip .7pt dY\hskip -1.5pt-4\sigma \hskip .7pt d\tau\kern-0.8pt$ which, due to (3-1) and (3-5), equals $ 2\alpha \,dQ-2\hskip .7pt dY\hskip -1.5pt-(Y\alpha +\mathrm {s})\hskip .7pt d\tau\kern-0.8pt$ , and so $ -4\alpha \hskip .7pt b(v,\cdot )= 2\hskip .7pt\alpha ^2\hskip -.7pt d\hskip .7pt(\theta \psi +\alpha ^{-\hskip -1.5pt1}Y\hskip -1.5pt-Q)+ (\alpha \hskip .7pt\mathrm {s}+4\varepsilon Y\hskip -1.5pt-2\hskip .4pt\theta )\hskip .7pt d\tau\kern-0.8pt$ . (Note that, by (1-6) and (4-4), $ 4\varepsilon =2\alpha '\hskip -1.5pt+\hskip .7pt\alpha ^2$ and $ 2\hskip .7pt\alpha ^2\hskip -.7pt d\hskip .7pt(\theta \psi )=2\hskip .4pt\theta \hskip .7pt d\tau\kern-0.8pt$ .) Thus, $ b=0$ , since $ b$ corresponds, via g, to a complex-linear bundle morphism $ {T\hskip -.3ptM}\to {T\hskip -.3ptM}\hskip -.7pt$ .

5 The local Kähler potentials

This and the following seven sections are devoted to proving Theorem E.

In an open set $\hskip .7pt M\subseteq \mathrm {I\!R}\hskip -.7pt^4\hskip -.7pt$ with the Cartesian coordinates $ x,x'\hskip -.7pt,u,u'$ arranged into the complex coordinates $ (x+ix'\hskip -.7pt,u+iu')$ for the complex plane carrying the standard complex structure $ J$ , one has $ J^*\hskip -1.5pt dx=-\hskip .4pt dx'$ and $ J^*\hskip -1.5pt du=-\hskip .4pt du'\hskip -.7pt$ , so that, if a $ C^\infty \hskip -1.5pt$ function $ f$ on M only depends on $ x$ and $ u$ , the relation $ 2\hskip .4pt i\hskip .4pt\partial \hskip 1.7pt\overline {\hskip -2pt\partial }\hskip -.7pt f =-\hskip .4pt d\hskip 1pt[\hskip -.4pt J^*\hskip -1.5pt d\hskip -.9ptf]$ yields, with subscripts denoting partial differentiations,

$$ \begin{align*} 2\hskip .4pt i\hskip .4pt\partial \hskip 1.7pt\overline {\hskip -2pt\partial }\hskip -.7pt f= f\hskip -.7pt\hskip -1.5pt_{x\hskip -.4pt x}\hskip .7pt dx\wedge dx'\hskip -.7pt +f\hskip -.7pt\hskip -1.5pt_{x\hskip -.4pt u}(dx\wedge du'+\hskip .7pt du\wedge dx') +f\hskip -.7pt\hskip -1.5pt_{u\hskip -.4pt u}\hskip .7pt du\wedge du'\hskip -.7pt, \end{align*} $$

since $ d\hskip -.9ptf=f\hskip -.7pt\hskip -1.5pt_x\hskip .7pt dx+f\hskip -.7pt\hskip -1.5pt_u\hskip .7pt du$ . Furthermore, we set

(5-1) $$ \begin{align} v=\partial\hskip-.4pt_x\quad\text{and}\quad w=\partial_u\quad\mathrm{(the\ real\ coordinate\ vector\ fields).} \end{align} $$

For the Kähler metric g on M having the Kähler form $ \omega =2\hskip .4pt i\hskip .4pt\partial \hskip 1.7pt\overline {\hskip -2pt\partial }\hskip .4pt\phi $ , where the function $ \phi :M\to \mathrm {I\!R}$ is assumed to depend on $ x$ and $ u$ only, $ 2\phi $ is a Kähler potential [Reference Besse2, page 85] of g, and the above formula for $ 2\hskip .4pt i\hskip .4pt\partial \hskip 1.7pt\overline {\hskip -2pt\partial }\hskip -.7pt f\hskip -.7pt$ , with $ f=\phi $ , becomes

(5-2) $$ \begin{align} \omega=\phi_{x\hskip-.4pt x}\hskip.7pt dx\wedge dx'\hskip-.7pt +\phi_{x\hskip-.4pt u}(dx\wedge du'+\hskip.7pt du\wedge dx') +\phi_{u\hskip-.4pt u}\hskip.7pt du\wedge du'. \end{align} $$

Generally, for a skew-Hermitian $ 2$ -form $ \zeta =Q\hskip .7pt dx\wedge dx'\hskip -.7pt +S\hskip .7pt(dx\wedge du'+\hskip .7pt du\wedge dx') +B\hskip .7pt du\wedge du'$ and the Hermitian symmetric $ 2$ -tensor field $ a$ with $ \zeta =a(J\hskip .7pt\cdot \,,\cdot )$ , one has

$$ \begin{align*} a&=Q(dx\otimes dx+\hskip .7pt dx'\otimes dx')+S\hskip .7pt(dx\otimes du+\hskip .7pt du\otimes dx +dx'\otimes du'+\hskip .7pt du'\otimes dx')\\& \quad +B(du\, \otimes du+du'\otimes du'), \end{align*} $$

due to (4-1), and so the components of a relative to the coordinates $ (x,x'\hskip -.7pt,u,u')$ form the matrix

(5-3) $$ \begin{align} \left[\begin{matrix} Q&0&S&0\\ 0&Q&0&S\\ S&0&B&0\\ 0&S&0&B\end{matrix}\right]\hskip-1.5pt\hskip-.7pt\quad\mathrm{with\ the\ determinant\ }(QB - S^2)^2. \end{align} $$

When $ a=g$ , (5-2) with $ \zeta =\omega $ gives $ (Q,S,B) =(\phi _{x\hskip -.4pt x},\hskip .7pt\phi _{x\hskip -.4pt u},\hskip .7pt\phi _{u\hskip -.4pt u})$ . Thus,

(5-4) $$ \begin{align} \phi_{x\hskip-.4pt x}>0\quad\text{and}\quad\varPi>0 \quad \mathrm{for} \ \varPi = \phi_{x\hskip-.4pt x}\phi_{u\hskip-.4pt u}\hskip-.7pt-\phi^{\hskip-.4pt2}_{\hskip-.7pt x\hskip-.4pt u}, \end{align} $$

which amounts to Sylvester’s criterion for positive definiteness of g, namely, positivity of the upper left subdeterminants of (5-3). From now on, we set

(5-5) $$ \begin{align} \begin{array}{l} (\tau\kern-0.8pt,\lambda,Q,S,B)=\,(\phi_x,\phi_u,\phi_{x\hskip-.4pt x}, \hskip.7pt\phi_{x\hskip-.4pt u},\hskip.7pt\phi_{u\hskip-.4pt u}),\ \text{so that}\\Q>0 \text{ and } \varPi=QB - S^2 > 0 \text{ due to }({5\mbox{-}4}). \end{array} \end{align} $$

With $ \mathrm {div},\hskip .7pt\Delta $ denoting the g-divergence and g-Laplacian, for $ \tau\kern-0.8pt,\lambda ,Q$ in (5-5):

1. (a) the functions $ \tau\kern-0.8pt,\hskip .7pt\lambda $ have the holomorphic g-gradients $ v=\partial \hskip -.4pt_x$ and $ w=\partial \hskip -.4pt_u$ ;

2. (b) the other coordinate fields $Jv$ and $Jw$ are holomorphic g-Killing fields;

3. (c) $Q=g(v,v)$ and $\Delta \hskip -.4pt\tau\kern-0.8pt=\mathrm {div}\,v =[\hskip .7pt\log \varPi ]_x$ , while $\Delta \lambda =\mathrm {div}\,w=[\hskip .7pt\log \varPi ]_u$ .

Namely, (5-1) and (5-3) yield item (a). Also, item (b) follows since $ \phi $ only depends on $ x$ and $ u$ . Finally, (5-3) has the determinant $ \varPi ^2\hskip -.7pt$ , and so $ \varPi \hskip .7pt dx\wedge dx'\hskip -.7pt\wedge du\wedge du'$ is the volume form of g, on which $ \unicode{xA3} _v,\unicode{xA3} _w$ act (see (5-1)) via partial differentiations $ \partial \hskip -.4pt_x,\,\partial \hskip -.4pt_u$ of the $ \varPi \hskip .7pt$ factor. Thus, $ \mathrm {div}\,v=[\hskip .7pt\log \varPi ]_x$ and $ \mathrm {div}\,w=[\hskip .7pt\log \varPi ]_u$ , compare with [Reference Kobayashi and Nomizu16, page 281]. Finally, by item (a), (5-1) and (5-5), $ g(v,v)=d_v\tau\kern-0.8pt=\partial \hskip -.4pt_x\tau\kern-0.8pt =\partial \hskip -.4pt_x\phi _x=\phi _{x\hskip -.4pt x}=Q$ .

For our $ (\tau\kern-0.8pt,\lambda )=(\phi _x,\phi _u)$ , the mapping $ (x,u)\mapsto (\tau\kern-0.8pt,\lambda )$ is locally diffeomorphic due to (5-4), which makes $ (Q,S,B)=(\phi _{x\hskip -.4pt x},\phi _{x\hskip -.4pt u},\phi _{u\hskip -.4pt u})$ , locally, a triple of real-valued functions of the new variables $ \tau\kern-0.8pt,\lambda $ . Thus, from the chain rule, in matrix form,

(5-6) $$ \begin{align} \begin{array}{ll} \mathrm{(i)}&\left[\begin{matrix}\partial_x\hskip-1.5pt &\hskip-1.5pt\partial_u\cr\end{matrix}\right] =\left[\begin{matrix}\partial\hskip-1pt_{\tau\kern-0.8pt}\hskip-1.5pt &\hskip-1.5pt\partial\hskip-1.5pt_\lambda\cr \end{matrix}\right] \left[\begin{matrix} Q&S\\ S&B\\\end{matrix}\right]\hskip-1.5pt\hskip-.7pt,\quad\mathrm{(ii)}\quad \left[\begin{matrix}d\tau\kern-0.8pt\cr d\lambda\\\end{matrix}\right] =\left[\begin{matrix} Q&S\\ S&B\\\end{matrix}\right] {\left[\begin{matrix}dx\\ du\\\end{matrix}\right]},\hskip7pt\mathrm{and\ so}\\[12pt] \mathrm{(iii)}&(QB - S^2) \left[\begin{matrix}\partial\hskip-1pt_{\tau\kern-0.8pt}\hskip-1.5pt &\hskip-1.5pt\partial\hskip-1.5pt_\lambda\cr\end{matrix}\right] =\left[\begin{matrix}\partial_x\hskip-1.5pt&\hskip-1.5pt\partial_u\cr\end{matrix} \right] {\left[\begin{matrix} B\hskip-3pt&-\hskip-1.5ptS\\ -\hskip-1.5ptS\hskip-3pt&Q\\\end{matrix}\right]},\\[12pt] \mathrm{(iv)}&(QB - S^2)\left[\begin{matrix}dx\\ du\\\end{matrix}\right] =\left[\begin{matrix} B\hskip-3pt&-\hskip-1.5ptS\\ -\hskip-1.5ptS\hskip-3pt&Q\\\end{matrix}\right]\left[\begin{matrix}d\tau\kern-0.8pt\cr d\lambda\\\end{matrix}\right]. \end{array} \end{align} $$

With subscripts still denoting partial differentiations, the obvious integrability conditions $ Q_u\hskip -.7pt-S_x=S_u\hskip -.7pt-B_x=0$ and (5-5) give, due to (5-6i),

(5-7) $$ \begin{align} SQ_{\tau\kern-0.8pt}+BQ_\lambda=QS_{\tau\kern-0.8pt} +SS_\lambda,\quad SS_{\tau\kern-0.8pt} + BS_\lambda=QB_{\tau\kern-0.8pt} +SB_{\lambda},\quad Q>0,\quad QB>S^2. \end{align} $$

Conversely, if functions $ Q,S,B$ of the variables $ \tau\kern-0.8pt,\lambda $ satisfy (5-7), then, locally:

1. (d) $(Q,S,B) =(\phi _{x\hskip -.4pt x},\phi _{x\hskip -.4pt u},\phi _{u\hskip -.4pt u})$ for a function $ \phi $ , with (5-4), of the variables $ x,u$ related to $ \tau\kern-0.8pt,\lambda $ via $ (\tau\kern-0.8pt,\lambda )=(\phi _x,\phi _u)$ , and $ Q,S,B$ determine each of $\phi ,x,u$ uniquely up to additive constants.

In fact, the equalities in (5-7) state precisely that the vector fields $ Q\partial\hskip -1pt _{\tau\kern-0.8pt} + S\hskip -.4pt\partial \hskip -1.5pt_\lambda $ and $ S\hskip -.4pt\partial \hskip -1pt_{\tau\kern-0.8pt} + B\hskip -.7pt\partial \hskip -1.5pt_\lambda $ commute or, equivalently, the $ 1$ -forms

$$ \begin{align*} (QB-S^2)^{-\hskip -1.5pt1}\hskip -.7pt(B\hskip .7pt d\tau\kern-0.8pt-S\hskip .7pt d\lambda ) \text{ and } (QB-S^2)^{-\hskip -1.5pt1}\hskip -.7pt(Q\,d\lambda -S\hskip .7pt d\tau\kern-0.8pt), \end{align*} $$

dual to them, are closed, and we may declare these vector fields (or $ 1$ -forms) to be $ \partial _x,\,\partial _u$ or respectively $ dx,\,du$ . Now that, locally, $ x,u$ are defined, up to additive constants, we obtain $ \phi $ by solving the system $ (\phi _x,\phi _u)=(\tau\kern-0.8pt,\lambda )$ , where $\tau\kern-0.8pt,\lambda $ are treated as functions of $ x,u$ via the resulting locally diffeomorphic coordinate change $(\tau\kern-0.8pt,\lambda )\mapsto (x,u)$ . Closedness of $\tau\kern-0.8pt\,dx+\lambda \,du$ and the equality $(Q,S,B) =(\phi _{x\hskip -.4pt x},\phi _{x\hskip -.4pt u},\phi _{u\hskip -.4pt u})$ are obvious: our choice of $ dx$ and $ \,du$ gives (5-6iv), and hence (5-6ii), so that $ d\tau\kern-0.8pt\wedge dx+d\lambda \wedge du=0$ .

The g-Laplacians of $\tau\kern-0.8pt$ and $\lambda $ can also be expressed as:

1. (e) $\Delta \hskip -.4pt\tau\kern-0.8pt=Q_{\tau\kern-0.8pt} + S_\lambda $ and $\Delta \lambda =S_{\tau\kern-0.8pt} + B_\lambda $ ; while

2. (f) $\varPi \hskip -1.5pt_x =(Q_{\tau\kern-0.8pt} + S_\lambda )\varPi $ and $ \varPi \hskip -1.5pt_u =(S_{\tau\kern-0.8pt} + B_\lambda )\varPi $ for $ \varPi = QB-S^2$ .

To see this, first note that, by (5-6i), $ (QB-S^2)_x=Q(QB-S^2)_{\tau\kern-0.8pt} + S(QB -S^2)_\lambda = Q(QB_{\tau\kern-0.8pt} + SB_{\lambda} -SS_{\tau\kern-0.8pt}) -S(Q\hskip -.4ptS_{\tau\kern-0.8pt} + SS_\lambda -BQ_\lambda )-S^2\hskip -.7ptS_\lambda +QBQ_{\tau\kern-0.8pt}$ . Using (5-7) to replace the two three-term sums in parentheses by $ B\hskip -.7ptS_\lambda $ and $ SQ_{\tau\kern-0.8pt}$ , we thus obtain the first part of item (f). For the second one, we similarly rewrite $ (QB-S^2)_u =S(QB-S^2)_{\tau\kern-0.8pt} + B(QB-S^2)_\lambda $ as $ S(QB_{\tau\kern-0.8pt}-BS_\lambda -SS_{\tau\kern-0.8pt}) -S^2\hskip -.7ptS_{\tau\kern-0.8pt} + B(BQ_\lambda + SQ_{\tau\kern-0.8pt} -SS_\lambda )+QBB_\lambda $ , and use analogous three-term replacements based on (5-7). Now item (e) follows from item (c), (5-5), and item (f).

Proof. By (1-9b), Equation (1-1) is, in case (i), equivalent to

(5-8) $$ \begin{align} 2\alpha\nabla d\tau\kern-0.8pt + 2\hskip.7pt\mathrm{r}\hskip.7pt=-(Q\alpha'+F')\hskip.4pt g, \end{align} $$

where all the terms are Hermitian symmetric $ 2$ -tensor fields, and hence correspond, via g, to complex-linear bundle morphisms $ {T\hskip -.3ptM}\to {T\hskip -.3ptM}\hskip -.7pt$ . Thus, (5-8) amounts to:

1. (f) equalities of the images of both sides in (5-8) under $ \imath _v$ and $ \imath _w$ .

The equality of the $ \imath _v$ -images is, by (2-3b) and (1-3), the result of applying $ d$ , via (3-5), to the relation (1-9a) in item (i): $ Y\hskip -1.5pt-Q\alpha =F\hskip -.7pt$ . This last relation and item (e), with $ Y = \Delta \hskip -.4pt\tau\kern-0.8pt$ due to (1-4), show that condition (i) implies the equality $ Q_{\tau\kern-0.8pt}\hskip -.4pt+S\hskip -1.5pt_\lambda \hskip -.7pt =Q\alpha +F$ in item (ii). Defining $ G$ to be $ S_{\tau\kern-0.8pt} + B_\lambda -S\hskip -.7pt\alpha $ , we get $ S_{\tau\kern-0.8pt} + B_\lambda =S\hskip -.7pt\alpha +G$ . Next, the equality of the $ \imath _w$ -images in (5-8) reads

(5-9) $$ \begin{align} \alpha\,dS - d\hskip.7pt\Delta\lambda=-(Q\alpha'+F')\,d\lambda. \end{align} $$

In fact, the first term equals $ \alpha \,dS$ since, for the two commuting gradients $ v=\nabla \hskip -.7pt\tau\kern-0.8pt$ and $ w=\nabla \hskip -.7pt\lambda $ , one has $ 2\nabla \!_w d\tau\kern-0.8pt=\hskip .7pt d\hskip .4pt[g(v,w)]$ or, in local coordinates, $ 2w\hskip .4pt^kv_{,\hskip .4pt jk}=w\hskip .4pt^kv_{,\hskip .4pt jk}+v\hskip .4pt^kw_{,\hskip .4pt jk} =(v\hskip .4pt^kw_k)\hskip -.4pt_{,\hskip .4pt j}$ and $ S=\phi _{x\hskip -.4pt u} =g(v,w)$ by (5-3) and (5-1). The second term is $ -d\hskip .7pt\Delta \lambda $ due to item (a) and (2-8). By item (e), $ G=S_{\tau\kern-0.8pt} + B_\lambda -S\hskip -.7pt\alpha =\Delta \lambda -S\hskip -.7pt\alpha $ , and so (5-9) becomes $ \alpha \,dS-d\hskip .4pt(G+S\hskip -.7pt\alpha )=-(Q\alpha '+F')\,d\lambda $ , that is, according to (3-5),

$$ \begin{align*} dG=(Q\alpha'+F')\,d\lambda - S\hskip.7pt d\alpha\, =\,(Q\alpha'+F')\,d\lambda - S\hskip-.7pt\alpha'\hskip-.7pt d\tau\kern-0.8pt \end{align*} $$

or, in other words, $ G_{\tau\kern-0.8pt}=-S\hskip -.7pt\alpha '$ and $ G\hskip -1.5pt_\lambda =Q\alpha '+F'\hskip -.7pt$ . Consequently, condition (i) implies condition (ii), since (5-7) arises as the integrability conditions $ Q_u\hskip -.7pt-S_x=S_u\hskip -.7pt-B_x=0$ combined with (5-4), and the equality $ dQ=Q_{\tau\kern-0.8pt}\hskip .7pt d\tau\kern-0.8pt+Q_\lambda \hskip .7pt d\lambda $ yields $ dQ\wedge d\tau\kern-0.8pt= -Q_\lambda \hskip .7pt d\tau\kern-0.8pt\wedge d\lambda $ .

Conversely, assuming condition (ii), we get condition (i) from item (f). Namely, as we saw above, the equality of the $ \imath _v$ -images in (5-8) arises by applying $ d$ to $ Y\hskip -1.5pt-Q\alpha =F\hskip -.7pt$ , that is—compare with item (e)—to $ Q_{\tau\kern-0.8pt}\hskip -.4pt+S\hskip -1.5pt_\lambda = Q\alpha +F\hskip -.7pt$ . Also, (5-9) follows from condition (ii) and item (e):

$$ \begin{align*} \alpha\,dS-d\Delta\lambda & =\alpha\,dS-d(S_{\tau\kern-0.8pt} + B_\lambda) =\alpha\,dS-d(S\hskip-.7pt\alpha+G) =-dG-S d\alpha =-dG-S\alpha'\, d\tau\kern-0.8pt\\ &=G_{\tau\kern-0.8pt} \, d\tau\kern-0.8pt-dG = - G_\lambda\, d\lambda =-(Q\alpha'+F')\,d\lambda. \end{align*} $$

This completes the proof.

Note that items (e), (f), Theorem 5.1(ii) and (5-1) give

$$ \begin{align*} \Delta\hskip-.4pt\tau\kern-0.8pt=Q\alpha+F\hskip-.7pt,\quad\Delta\lambda=S\hskip-.7pt\alpha+G,\quad d_v\varPi = \varPi\Delta\hskip-.4pt\tau\kern-0.8pt,\quad d_w\varPi = \varPi\Delta\lambda. \end{align*} $$

6 Some linear algebra

We now proceed to discuss the first-order system equivalent, as we saw in the last section (Theorem 5.1), to the Kähler-potential problem, the solution of which amounts to proving Theorem E. The main result established here, Theorem 6.3, will lead—in Section 9—to a unique-extension property of integral lines, which results in applicability of the Cartan–Kähler theorem to our situation.

Theorem 5.1 reduces constructing local examples of special Ricci–Hessian equations (1-1)–(1-2) with $ dQ\wedge d\tau\kern-0.8pt\ne 0$ , on Kähler surfaces, which is a fourth-order problem in the Kähler potential $ 2\phi $ , to solving the following system of quasi-linear first-order partial differential equations, with subscripts representing partial derivatives:

(6-1) $$ \begin{align} \begin{array}{rl} \mathrm{(a)}&Q_{\tau\kern-0.8pt} + S\hskip-1.5pt_\lambda-Q\alpha-F=0,\\\mathrm{(b)}&S_{\tau\kern-0.8pt} + B_\lambda-S\hskip-.7pt\alpha-G=0,\\\mathrm{(c)}&QB_{\tau\kern-0.8pt} + SB_\lambda -SS_{\tau\kern-0.8pt}-B\hskip-.7ptS_\lambda=0,\\\mathrm{(d)}&SQ_{\tau\kern-0.8pt} + BQ_\lambda-Q\hskip-.4ptS_{\tau\kern-0.8pt} -SS_\lambda=0,\\\mathrm{(e)}&G_{\tau\kern-0.8pt} + S\hskip-.7pt\alpha' = 0,\\\mathrm{(f)}&G\hskip-1.5pt_\lambda-Q\alpha'\hskip-.7pt-F' = 0, \end{array} \end{align} $$

on which one imposes the additional conditions

(6-2) $$ \begin{align} QB>S^2,\quad Q\,>\,0,\quad Q_\lambda\hskip.7pt\ne\,0\,\mathrm{\ \ \ everywhere.} \end{align} $$

Generally, if $ Q,S,B$ are real-valued functions of the real variables $ \tau\kern-0.8pt,\lambda $ and subscripts denote partial differentiations, writing $ d[(QB\hskip -1.5pt-\hskip -1.5ptS^2)^{-\hskip -1.5pt1}\hskip -.7pt(B\hskip .7pt d\tau\kern-0.8pt\hskip -.7pt -\hskip -.7ptS\hskip .7pt d\lambda )]=\varPhi \,d\tau\kern-0.8pt\wedge d\lambda $ and $ d[(QB\hskip -1.5pt-\hskip -1.5ptS^2)^{-\hskip -1.5pt1}\hskip -.7pt(S\hskip .7pt d\tau\kern-0.8pt\hskip -.7pt -\hskip -.7ptQ\,d\lambda )]=\varPsi \,d\tau\kern-0.8pt\wedge d\lambda $ at points where $ QB\ne S^2\hskip -.7pt$ , one easily verifies that

(6-3) $$ \begin{align} \left[\begin{matrix} \varPhi\\ \varPsi\end{matrix}\right]= \left[\begin{matrix} S&B\\ Q&S\end{matrix}\right] \left[\begin{matrix} QB_{\tau\kern-0.8pt} + SB_\lambda -SS_{\tau\kern-0.8pt}-B\hskip-.7ptS_\lambda\cr SQ_{\tau\kern-0.8pt} + BQ_\lambda-Q\hskip-.4ptS_{\tau\kern-0.8pt} -SS\hskip-1.5pt_\lambda\end{matrix}\right]\hskip-1.5pt. \end{align} $$

For the remainder of this section, we treat the letter symbols in (6-1)–(6-2) as real variables, so as to derive some consequences of conditions (6-1)–(6-2) just by using linear algebra.

Lemma 6.1. If $ (Q,S,B\hskip -.7pt,G,Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt}, G_{\tau\kern-0.8pt},Q_\lambda ,S_\lambda , B_\lambda ,G_\lambda )\in \mathrm {I\!R}\hskip -.7pt^{12}$ satisfies the conditions (6-1a)–(6-1d), with some $ (\alpha ,F)\in \mathrm {I\!R}\hskip -.7pt^2\hskip -.7pt$ , then

(6-4) $$ \begin{align} \begin{array}{rl} \mathrm{(i)}&QB_{\tau\kern-0.8pt} + BQ_{\tau\kern-0.8pt} -2SS_{\tau\kern-0.8pt}-(QB-S^2)\alpha-B F+SG=0,\\ \mathrm{(ii)}&QB_\lambda + BQ_\lambda -2SS\hskip-1.5pt_\lambda-QG+S F=0. \end{array} \end{align} $$

Proof. The left-hand side of (6-4i) is, obviously,

$$ \begin{align*}(QB_{\tau\kern-0.8pt} + SB_\lambda -SS_{\tau\kern-0.8pt}-B\hskip-.7ptS_\lambda) +(Q_{\tau\kern-0.8pt} + S\hskip-1.5pt_\lambda-Q\alpha-F)B -(S_{\tau\kern-0.8pt} + B_\lambda-S\alpha-G)S, \end{align*} $$

and each sum in the parentheses vanishes due to (6-1). Similarly, (6-4ii) has the left-hand side

$$ \begin{align*}(SQ_{\tau\kern-0.8pt} + BQ_\lambda-Q\hskip-.4ptS_{\tau\kern-0.8pt} -SS\hskip-1.5pt_\lambda) +(S_{\tau\kern-0.8pt} + B_\lambda-S\hskip-.7pt\alpha-G)Q -(Q_{\tau\kern-0.8pt} + S\hskip-1.5pt_\lambda-Q\alpha-F)S=0+0+0. \end{align*} $$

With subscripts denoting partial differentiations, (6-1e–f) and (6-4) read

(6-5) $$ \begin{align} \begin{array}{rl} \mathrm{(i)}&dG=-S\hskip-.7pt\alpha'd\tau\kern-0.8pt\, +\,(Q\alpha'+\hskip.7pt F')\,d\lambda,\\ \mathrm{(ii)}&d\varPi=\hskip.7pt(\varPi\hskip-.7pt\alpha +B F-SG)\,d\tau\kern-0.8pt\, +\,\hskip-.7pt(QG-S F)\,d\lambda\,\mathrm{\ for\ }\,\varPi=QB-S^2\hskip-.7pt. \end{array} \end{align} $$

Since $ d\varPi \hskip -.7pt =\varPi \hskip .4pt[(Q_{\tau\kern-0.8pt} + S_\lambda )\,dx +(S_{\tau\kern-0.8pt} + B_\lambda )\,du]$ due to item (f), one can also obtain (6-5ii) from (5-6iv), with $ \varPi =QB-S^2\hskip -.7pt$ , and (6-1a–b).

Proof. Assertion (a) follows as $ \varPsi $ is positive or negative definite in $ (\dot \tau\kern-0.8pt,\dot \lambda )$ . Next, one easily verifies that, for $ \varPsi $ as in assertion (a), and the rows of the above matrix,

Depending on whether $ \dot \lambda \ne 0$ (or $ \dot \lambda =0$ and hence $ \dot \tau\kern-0.8pt\ne 0$ ), the $ 6\times 6$ matrix with the rows followed by (or respectively by ), and then by $ (0,0,0,0,0,\varPsi )$ displayed above, is upper triangular, with the diagonal entries $ 1,1,Q,B,\dot \lambda ,\varPsi $ or respectively $ 1,1,Q,B,\hskip -1.5pt-\hskip -.7pt\dot \tau\kern-0.8pt,\varPsi $ , all nonzero by assertion (a). These six new rows thus form a basis of $ \mathrm {I\!R}\hskip -.7pt^6$ (so that the matrix has rank $ 6$ ), while the first four are linear combinations of the first four original rows, which proves both assertions (b) and (c).

Theorem 6.3. Given vectors $ (\dot \tau\kern-0.8pt,\dot \lambda ,Q,S,B\hskip -.7pt,G)\in \mathrm {I\!R}\hskip -.7pt^6$ and $ (\alpha ,F\hskip -.7pt,\alpha '\hskip -.7pt,F')\in \mathrm {I\!R}\hskip -.7pt^4$ with $ (\dot \tau\kern-0.8pt,\dot \lambda )\ne (0,0)$ and $ QB>S^2\hskip -.7pt$ , the set of all

(6-6) $$ \begin{align} (Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt},G_{\tau\kern-0.8pt}, Q_\lambda,S_\lambda,B_\lambda,G_\lambda) \,\in\,\mathrm{I\!R}\hskip-.7pt^8 \end{align} $$

satisfying (6-1) forms a two-dimensional affine subspace $ \mathcal {A}$ of $ \mathrm {I\!R}\hskip -.7pt^8\hskip -.7pt$ . Furthermore, the restriction to $ \mathcal {A}$ of the linear operator $ \varPhi :\mathrm {I\!R}\hskip -.7pt^8\hskip -.7pt\to \mathrm {I\!R}\hskip -.7pt^4$ sending (6-6) to

(6-7) $$ \begin{align} (\dot Q,\dot S,\dot B\hskip-.7pt,\dot G) =(Q_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+Q_\lambda\dot\lambda,\, S_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+S_\lambda\dot\lambda,\, B_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+B_\lambda\dot\lambda,\, G_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+G_\lambda\dot\lambda) \end{align} $$

is an affine isomorphism $ \varPhi :\mathcal {A}\to \mathcal {L}$ onto the two-dimensional affine subspace $ \mathcal {L}$ of $ \mathrm {I\!R}\hskip -.7pt^4$ consisting of all $ (\dot Q,\dot S,\dot B\hskip -.7pt,\dot G)$ such that

(6-8) $$ \begin{align} \begin{array}{l} \dot G=-S\hskip-.7pt\alpha'\hskip-.7pt\dot\tau\kern-0.8pt + (Q\alpha'+F')\dot\lambda,\\ Q\hskip-.7pt\dot B+B\dot Q-2S\hskip-.7pt\dot S =[(QB-S^2)\alpha+B F-SG]\dot\tau\kern-0.8pt+(QG-S F)\dot\lambda. \end{array} \end{align} $$

7 The existence of solutions

After the preceding foray into linear algebra, we now return to treating (6-1) as a system of quasi-linear first-order partial differential equations with four unknown real-valued functions $ Q,S,B\hskip -.7pt,G$ of the real variables $ \tau\kern-0.8pt,\lambda $ , subject to the additional conditions (6-2).

Subscripts again denote partial differentiations, while $\alpha $ and F are functions of the variable $ \tau\kern-0.8pt$ , also depending on three fixed real constants $ \varepsilon ,\theta ,\kappa $ , so that

(7-1) $$ \begin{align} \begin{array}{l} 2\alpha'+\hskip.7pt\alpha^2=4\varepsilon\mathrm{,\hskip-1.5pt\ where\ }\,(\,\,)' = d/d\tau\kern-0.8pt,\\ 4\varepsilon F = \hskip.7pt\theta(2\hskip-.4pt-\hskip-.4pt\tau\kern-0.8pt\alpha)+4\varepsilon\kappa\alpha\,\mathrm{\ if\ }\hskip.7pt\varepsilon\ne0,\\ F=\,\kappa\alpha\hskip.4pt-\hskip.4pt2\hskip.4pt\theta/(3\alpha^2)\,\mathrm{\ when\ }\,\varepsilon\hskip.7pt=\hskip.7pt0. \end{array} \end{align} $$

In addition, it is natural to assume here that

(7-2) $$ \begin{align} \tau\kern-0.8pt\,\mathrm{\ ranges\ over\ the\ domain\ of\ }\,\alpha\,\mathrm{\ (which\ is\ also\ the\ domain\ of\ }\,F). \end{align} $$

Consequently, $\alpha $ and F satisfy the ordinary differential equations

(7-3) $$ \begin{align} \alpha"\hskip.4pt+\,\alpha\alpha'\hskip.4pt=\,0,\quad F"\hskip.4pt=\,-F\hskip-.7pt\alpha'. \end{align} $$

We prove Theorem 7.1 at the end of Section 10. As we point out in Remark 10.2, there is an infinite-dimensional freedom of choosing the data described in the second paragraph of Theorem 7.1.

8 The associated exterior differential system

By an exterior differential system on a manifold M, one means an ideal $\mathcal {I}$ in the graded algebra $ \varOmega \hskip .4pt^*\hskip -1.5pt M\hskip -.7pt$ , closed under exterior differentiation; its integral manifolds (or integral elements) are those submanifolds of M (or subspaces of its tangent spaces) on which every form in $\mathcal {I}$ vanishes [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 16 and 65]. When such objects have dimension $ 1$ or $ 2$ , we call them integral curves/surfaces or lines/planes.

If $ E\subseteq {T_zM}$ is a $ p\hskip .7pt$ -dimensional integral element of $ \mathcal {I}\hskip -.7pt$ , one sets [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 67–68]:

(8-1) $$ \begin{align} \begin{array}{l} H(E)=\{v\in{T_zM}:\zeta(v,e_1,\ldots,e_p)=0\,\mathrm{\ for\ all\ }\hskip.7pt\zeta\in\mathcal{I}\cap\varOmega\hskip.4pt^{p\hskip.7pt+1}\hskip-1.5pt M\}\\ \mathrm{and\ }\,r(E)\hskip.7pt=\hskip.7pt\dim H(E)-(p+1)\mathrm{,\ for\ any\ basis\ }\,e_1,\ldots,e_p\mathrm{\ of\ }E, \end{array} \end{align} $$

so that $ H(E)$ is a vector subspace of $ {T_zM}\hskip -.7pt$ , not depending on $ e_1,\ldots ,e_p$ since

(8-2) $$ \begin{align} H(E)=\{v\in{T_zM}:\mathrm{span}(v,E)\,\mathrm{\ is\ an\ integral\ element\ of\ }\mathcal{I}\}. \end{align} $$

For fixed real constants $ \varepsilon ,\theta ,\kappa $ , let the open subset $\mathcal {Y}$ of $\mathrm {I\!R}\hskip -.7pt^6$ consist of all points $ (\tau\kern-0.8pt,\lambda ,Q,S,B\hskip -.7pt,G)\in \mathrm {I\!R}\hskip -.7pt^6$ such that Q and $QB-S^2$ are both positive, while $\tau\kern-0.8pt$ lies in the domains of $\alpha $ and F, chosen so as to satisfy (7-1). Consider now

(8-3) $$ \begin{align} \begin{array}{l} \mathrm{the\ exterior\ differential\ system\ }\,\,\mathcal{I}\,\hskip.4pt\hskip.7pt\mathrm{ \ on\ this\ }\,\hskip.4pt\hskip.7pt\mathcal{Y}\,\mathrm{\ generated}\\ \mathrm{by\ \, the\ two\ } \ 1\text{-}\mathrm{forms\ }\ \ dG + S\hskip-.7pt\alpha'd\tau\kern-0.8pt\, -\,(Q\alpha'+\hskip.7pt F')\,d\lambda\,\,\mathrm{\ and}\\ d\hskip.4pt(QB\hskip-1.5pt-\hskip-1.5ptS^2)\ \hskip-1.5pt-\hskip-1.5pt \ [(QB\hskip-1.5pt \ - \ \hskip-1.5ptS^2)\ \alpha\hskip-1.5pt +\hskip-1.5ptB F\hskip-1.5pt-\hskip-1.5ptSG]\hskip.7pt d\tau\kern-0.8pt\hskip-1.5pt -\hskip-1.5pt(QG\hskip-1.5pt-\hskip-1.5ptS F)\hskip.7pt d\lambda,\\ \mathrm{the\ }\,\hskip.7pt2\text{-}\mathrm{form\ \ }\,dQ\wedge d\lambda\,\, +\,\,d\tau\kern-0.8pt\wedge\hskip.4pt dS\, - \,(Q\alpha\hskip.7pt+\,F)\,d\tau\kern-0.8pt\wedge d\lambda,\\ \mathrm{the\ \ }\,\hskip.7pt2\text{-}\mathrm{form\ \ \ }\hskip.4pt\,dS\wedge d\lambda\hskip.7pt\, +\hskip.7pt\,d\tau\kern-0.8pt\hskip.7pt\wedge dB\hskip.7pt\, -\,(S\hskip-.7pt\alpha\hskip.7pt+\hskip.7ptG)\,d\tau\kern-0.8pt\wedge d\lambda,\\ \mathrm{their\hskip-.4pt\ exterior\hskip-.4pt\ derivatives,\hskip-.7pt\ and\hskip-.4pt\ the\hskip-.4pt\ exterior\hskip-.4pt\ \ derivatives\hskip-.4pt\ of}\\ (QB\hskip-1.5pt-\hskip-1.5ptS^2)^{-\hskip-1.5pt1}\hskip-.7pt \ (B\hskip.7pt d\tau\kern-0.8pt\hskip-.7pt-\hskip-.7ptS\hskip.7pt d\lambda) \ \mathrm{\ and\ } \ (QB\hskip-1.5pt-\hskip-1.5ptS^2)^{-\hskip-1.5pt1} \ \hskip-.7pt \ (S\hskip.7pt d\tau\kern-0.8pt\hskip-.7pt \ - \ \hskip-.7ptQ\,d\lambda)\hskip-.4pt. \end{array} \end{align} $$

The choice of $\mathcal {I}$ is justified as follows. We want $\mathcal {I}$ to consist of differential forms that are expected to vanish on all graphs of solutions to (6-1) with (6-2). Since such a graph is horizontal (in the sense that $ d\tau\kern-0.8pt\wedge d\lambda \ne 0$ on it), forms generating $\mathcal {I}$ will result from multiplying by $ d\tau\kern-0.8pt\wedge d\lambda $ the left-hand sides in (6-1), along with those in its consequence (6-4), as including the latter changes nothing except for enriching the differential system. Each of the terms $ \varXi Z_{\tau\kern-0.8pt}$ , or $ \varXi Z\hskip -2.2pt_\lambda $ , or $ \varTheta $ , where $ Z\in \{Q,S,B\hskip -.7pt,G\}$ (with various $ \varXi $ depending on $ Q,S,B\hskip -.7pt,G$ , and $ \varTheta $ which may further depend also on $ \alpha ,F\hskip -.7pt,\alpha '\hskip -.7pt,F'$ ) then becomes $ \varXi \,dZ\wedge d\lambda $ , or $ \varXi \,d\tau\kern-0.8pt\wedge dZ$ , or simply $ \varTheta \,d\tau\kern-0.8pt\wedge d\lambda $ .

The $ 2$ -forms in the fourth and fifth lines of (8-3) arise as above from (6-1a) and (6-1b), the exterior derivatives of the $ 1$ -forms in the last line—from (6-1c) and (6-1d) via (6-3). The rest of (8-3) is based on an additional principle: if our exterior differential system ends up containing $ \xi \wedge d\tau\kern-0.8pt$ and $ \xi \wedge d\lambda $ , for some $ 1$ -form $ \xi $ , we are free to include $ \xi $ among the system’s generators, as the horizontal integral surfaces/planes then obviously remain unaffected. The $ 1$ -form $ \xi $ in the second, or third, line of (8-3) corresponds in this way to (6-1e–f) or respectively (6-4).

Instead of invoking the general ‘additional principle’, one can also justify the second and third lines in (8-3) directly from the fact that (6-5) is a consequence of (6-1), which causes the two $ 1$ -forms to vanish on all graphs of solutions to (6-1).

9 The unique-extension theorem

In $ \mathrm {I\!R}\hskip -.7pt^6$ with the coordinates $ \tau\kern-0.8pt,\lambda ,Q,S,B\hskip -.7pt,G$ , given a subspace $ E\subseteq \mathrm {I\!R}\hskip -.7pt^6\hskip -.7pt$ ,

(9-1) $$ \begin{align} \mathrm{we\ call\ }\,E\,\mathrm{\ horizontal\ when\ }\,(d\tau\kern-0.8pt,d\lambda):\hskip-1.5pt E\hskip.7pt\to\mathrm{I\!R}\hskip-.7pt^2\mathrm{\ is\ injective.} \end{align} $$

Remark 9.2. The only $ 1$ -forms in $\mathcal {I}$ are, obviously, the functional combinations of those in the second and third lines of (8-3). Due to their linear independence at every point, the simultaneous kernel of these two $ 1$ -forms is a codimension-two distribution $ \mathcal {D}$ on $ \mathcal {Y}\hskip -.7pt$ , that is, a vector subbundle of $ T\hskip -.4pt\mathcal {Y}\hskip -.7pt$ , and its fiber at any $ (\tau\kern-0.8pt,\lambda ,Q,S,B\hskip -.7pt,G)\in \mathcal {Y}$ consists of all $ (\dot \tau\kern-0.8pt,\dot \lambda ,\dot Q,\dot S,\dot B\hskip -.7pt,\dot G)\in \mathrm {I\!R}\hskip -.7pt^6$ with (6-8). Hence,

(9-2) $$ \begin{align} \begin{array}{l} \mathrm{vectors\ \,}\hskip.7pt(\dot\tau\kern-0.8pt, \ \dot\lambda,\dot Q,\dot S,\dot B\hskip-.7pt,\dot G)\hskip.7pt \ \mathrm{\ at\ } \hskip.7pt(\tau\kern-0.8pt, \ \lambda,Q,S,B\hskip-.7pt,G)\hskip.7pt \ \mathrm{\ spanning\ horizontal}\\ \text{integral lines of } \mathcal{I} \text{ are characterized by (6-8) and } (\dot\tau\kern-0.8pt,\dot\lambda)\hskip.7pt\ne(0,0), \end{array} \end{align} $$

where $ \alpha ,F\hskip -.7pt,\alpha '\hskip -.7pt,F'$ satisfy (7-1)–(7-2).

Proof. Any horizontal plane in $ \mathrm {I\!R}\hskip -.7pt^6$ has, by (9-1), a unique basis of the form

(9-3) $$ \begin{align} (1,0,Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt},G_{\tau\kern-0.8pt}),\quad (0,1,Q_\lambda,S_\lambda,B_\lambda, G_\lambda). \end{align} $$

The span of (9-3) is an integral plane of $\mathcal {I}$ if and only if all the $ 1$ -forms (and $ 2$ -forms) listed in (8-3) yield the value $ 0$ when evaluated on both vectors in (9-3) or respectively on the pair (9-3). Due to the two final paragraphs of Section 8, and (6-3), this is equivalent to (6-1) and (6-4), and hence (Lemma 6.1) just to (6-1).

Every vector in (9-2) is a linear combination of a unique pair (9-3) satisfying (6-1): as the coefficients of the combination must be $ \dot \tau\kern-0.8pt$ and $ \dot \lambda $ , this is immediate from Theorem 6.3. In other words, every horizontal integral line of $\mathcal {I}$ lies within a unique horizontal integral plane. Remark 9.1 now allows us to drop the last occurrence of the word ‘horizontal’, completing the proof.

10 Existence of integral surfaces

The next fact—used below to derive our Theorem 7.1—is a special case of the celebrated Cartan–Kähler theorem [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 81–82]. Since our phrasing differs from that of [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4], we devote the next section to clarifying how our version amounts to adapting the one in [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4] to our particular case.

The symbols $ \mathcal {Y}\hskip -.7pt,\mathcal {I}\hskip .7pt$ and $ \mathcal {D}$ stand here for more general objects than those in Sections 8–9. The definition (9-1) of horizontality, for integral elements, is used more generally, as well as extended, in an obvious fashion, to integral manifolds.

Theorem 10.1. Let real-analytic functions $ \tau\kern-0.8pt,\lambda $ and $1$ -forms $ \xi _1,\ldots ,\xi _q$ on a manifold $\mathcal {Y}$ , where $ 0<\hskip .7pt q<\dim \mathcal {Y}\hskip -1.5pt$ , have the property that $ d\tau\kern-0.8pt,d\lambda ,\,\xi _1,\ldots ,\xi _q$ are linearly independent at every point. Denoting by $ \mathcal {D}\hskip .7pt$ and $ \mathcal {I}$ the distribution on $ \mathcal {Y}\hskip -.7pt$ arising as the simultaneous kernel of the $1$ -forms $ \xi _1,\ldots ,\xi _q$ and respectively the exterior differential system on $ \mathcal {Y}\hskip -.7pt$ generated by $ \xi _1,\ldots ,\xi _q$ and, possibly, some higher-degree forms, along with their exterior derivatives, let us suppose that

(10-1) $$ \begin{align} \begin{array}{l} \text{every\ \,horizontal\ \,integral\ \,line\ \,of\ \,}\mathcal{I}\hskip-.7pt\text{,\ at\ \,any\ point}\\ \text{of\ }\mathcal{Y}\hskip-1.5pt\text{,\ is\ contained\ in\ a\ unique\ integral\ plane\ of\ }\,\mathcal{I}\hskip-.7pt. \end{array} \end{align} $$

Then, every horizontal real-analytic integral curve of $\mathcal {I}$ is contained, locally, in a locally-unique horizontal real-analytic integral surface. Examples of such curves are provided by unparameterized integral curves of any real-analytic vector field without zeros forming a horizontal local section of the vector bundle $ \mathcal {D}\hskip .7pt$ over $\mathcal {Y}$ . Also,

(10-2) $$ \begin{align} \text{integral lines of } \mathcal{I} \text{ are the same as lines tangent to } \mathcal{D}. \end{align} $$

Under the assumptions of Theorem 10.1, let $ k=\dim \mathcal {Y}\hskip -.7pt$ . For all $ p\hskip .7pt$ -dimensional horizontal integral elements $ E =E_p$ of $ \mathcal {I}\hskip -.7pt$ , with $ p\in \{0,1\}$ , and for $ r(E)=\dim H(E)-(p+1)$ in (8-1):

(10-3) $$ \begin{align} \begin{array}{rl} \mathrm{(a)}&\mathrm{the\ integer\ }\,r(E)\,\mathrm{\ has\ a\ fixed\ nonnegative\ value,\ namely,}\\ \mathrm{(b)}& \dim H(E_0)=k- q\mathrm{\ and\ } r(E_0) =k- q-1\mathrm{\ when\ }p=0,\\ \mathrm{(c)}&\dim H\hskip-.7pt(E_1)=2\mathrm{\ and\ } r(E_1) =0\mathrm{\ in\ the\ case\ where\ }p=1. \end{array} \end{align} $$

This is obvious from (10-2) or respectively (10-1).

11 Where Theorem 10.1 comes from

Here is the Cartan–Kähler theorem, cited verbatim from [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 81–82]:

Let $ \mathcal {I}\subset \hskip -.4pt\varOmega \hskip .4pt^*\hskip -1.5pt(M)$ be a real analytic differential ideal. Let $ P\subset \hskip -.7pt M$ be a connected, $ p\hskip .7pt$ -dimensional, real analytic, Kähler-regular integral manifold of $ \mathcal {I}\hskip -.7pt$ .

Suppose that $ r=r(P)$ is a nonnegative integer. Let $ R\subset \hskip -.7pt M$ be a real analytic submanifold of M which is of codimension $ r$ , which contains $ P\hskip -.7pt$ , and which satisfies the condition that $ {T\hskip -2.9pt_x\hskip -.4pt R}$ and $ H({T\hskip -2.9pt_x\hskip -.4pt R})$ are transverse in $ {T\hskip -2.9pt_x\hskip -.4pt M}$ for all $x\in P$ .

Then, there exists a real analytic integral manifold of $\mathcal {I}$ , X, which is connected and $(p+1)$ -dimensional and which satisfies $ P\subset X\subset R$ . This manifold is unique in the sense that any other real analytic integral manifold of $\mathcal {I}$ with these properties agrees with X on an open neighborhood of P.

As we verify in the following paragraphs, the hypotheses of our Theorem 10.1 imply those listed above, for $ (p,r)=(1,0)$ , the manifolds $ M\hskip -.7pt,R$ above which are both replaced by our $\mathcal {Y}$ , and the same ideal $\mathcal {I}$ as ours. By our $\mathcal {Y}$ and $\mathcal {I}$ , we mean the ‘general’ ones (see the three lines preceding Theorem 10.1), rather than the very special choices of $\mathcal {Y}$ and $\mathcal {I}$ made in Section 8.

Furthermore, P mentioned above is our (arbitrary) horizontal real-analytic integral curve of $\mathcal {I}$ . The resulting manifold X corresponds to the horizontal real-analytic integral surface of $\mathcal {I}$ claimed to exist in Theorem 10.1.

We now proceed to explain why our horizontal integral curve must automatically be Kähler-regular [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, page 81], meaning that its tangent lines are all Kähler-regular in the sense of [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, page 68, Definition 1.7]. To verify this last claim, we first apply Cartan’s test [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, page 74, Theorem 1.11]. Namely, in the notation of [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, page 74, Theorem 1.11], $ n=1$ (as we are dealing with tangent lines). Due to the relation $ \dim H\hskip -.7pt(E_0)=k-q$ in (10-3b), and (10-2), $ H\hskip -.7pt(E_0)$ is of codimension $ q$ in the tangent space of $\mathcal {Y}$ containing it, which is the same as the codimension, in the $ (2k-1)$ -dimensional Grassmann manifold $ \mathrm {Gr}_1\hskip .7pt\mathcal {Y}$ of lines tangent to $ \mathcal {Y}\hskip -.7pt$ , of the $ (2k-q-1)$ -dimensional submanifold $ V\hskip -3pt_1(\mathcal {I})$ formed by all integral lines of $ \mathcal {I}\hskip -.7pt$ . Cartan’s test thus shows that every line $ E_1$ tangent to our horizontal integral curve is ordinary [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, page 73, Definition 1.9]. The Kähler-regularity of $ E_1$ now trivially follows, as r in [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 67–68] has the constant value $\hskip .7pt0\hskip .7pt$ according to (10-3c). This is also the value $r=r(P)$ in the italicized statement cited above from [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4] (compare with [Reference Bryant, Chern, Gardner, Goldschmidt and Griffiths4, pages 81–82, the lines preceding Theorem 2.2]).

12 Proof of Theorem E

Let $\nabla \hskip .7pt$ (or g) be a connection (or a pseudo-Riemannian metric) on a $C^\infty $ manifold $ M\hskip -.7pt$ . We call $\nabla \hskip .7pt$ or g real-analytic if, in a suitable coordinate system around every point of $M\hskip -.7pt$ , its components $\varGamma _{\hskip -2.1ptjk}^{\hskip .7pt l}$ (or $g_{jk}$ ) are real-analytic functions of the coordinates. The $C^\infty $ structure of M then contains a unique real-analytic structure (maximal atlas) making $\nabla \hskip .7pt$ or g real-analytic. (The atlas consists of all coordinate systems just mentioned; their mutual transition mappings are real-analytic due to real-analyticity of affine mappings, or isometries, between manifolds with real-analytic connections/metrics, which follows since such mappings appear linear in geodesic coordinates.) Real-analyticity of a metric g obviously implies that of its Levi–Civita connection $ \nabla \hskip .7pt$ (and vice versa, since $ \nabla \hskip -.7pt g=0$ ).

For a real-analytic (Riemannian) Kähler metric g on a complex manifold $ M\hskip -.7pt$ , the unique real-analytic structure described above coincides with that induced by the complex structure of $ M\hskip -.7pt$ . In fact, local holomorphic coordinate functions, being g-harmonic, must be real-analytic relative to the former structure, as a consequence of the standard regularity theory of elliptic partial differential equations applied to the g-Laplacian $\Delta $ .

13 The analytic-continuation phenomenon

We elaborate here on the plausibility of small deformations mentioned in the lines following Theorem E, beginning with the coth-cot analytic continuation. The real-analytic function $\mathrm {I\!R}\ni y\mapsto y^{-\hskip -1.5pt1}\hskip -1.5pt\tanh y$ , with the value $1$ at $y=0$ , being even, has the form $\varSigma (y^2)$ for some real-analytic function $\varSigma $ . Now, $ (\varepsilon ,\tau\kern-0.8pt)\mapsto \beta \hskip -.4pt_\varepsilon \hskip -.4pt(\tau\kern-0.8pt)=\tau\kern-0.8pt \varSigma (\varepsilon \tau\kern-0.8pt^2)$ is a real-analytic function on an open subset of $ \mathrm {I\!R}\hskip -.7pt^2$ and $ \beta _\varepsilon (\tau\kern-0.8pt)$ equals $ \varepsilon ^{-\hskip -1.5pt1/2}\tanh (\varepsilon ^{1/2}\tau\kern-0.8pt)$ , or $ \tau\kern-0.8pt$ , or $ |\varepsilon |^{-1/2}\tan (|\varepsilon |^{1/2}\tau\kern-0.8pt)$ , depending on whether $ \varepsilon>0$ , or $ \varepsilon =0$ , or $\varepsilon <0$ . For $\alpha _\varepsilon (\tau\kern-0.8pt)=2/\beta _\varepsilon (\tau\kern-0.8pt)$ , the analogous expressions are

$$ \begin{align*} 2\varepsilon^{1/2}\coth(\varepsilon^{1/2}\tau\kern-0.8pt)\ (\mathrm{if\ }\varepsilon>0),\quad 2/\tau\kern-0.8pt\ (\mathrm{if\ }\varepsilon=0),\quad 2|\varepsilon|^{1/2}\cot(|\varepsilon|^{1/2}\tau\kern-0.8pt)\ (\mathrm{if\ }\varepsilon<0). \end{align*} $$

All $ \alpha \hskip -.4pt_\varepsilon $ with $ \varepsilon>0$ , as well as those with $ \varepsilon <0$ , are thus affine (in fact, linear) modifications (see Remark C) of $ \alpha _1$ or respectively $ \alpha _{-\hskip -1.5pt1}$ , and $ \alpha _0(\tau\kern-0.8pt)=2/\hskip -.7pt\tau\kern-0.8pt$ .

For a tanh-coth analytic-continuation argument, we define $ (t,\tau\kern-0.8pt)\mapsto \alpha _t(\tau\kern-0.8pt)$ by $ \alpha _t(\tau\kern-0.8pt) =2(e^{\tau\kern-0.8pt}- te^{-{\tau\kern-0.8pt}})/(e^{\tau\kern-0.8pt}+te^{-{\tau\kern-0.8pt}})$ . Thus, with $ q$ such that $ 2q=\log |t|$ , if $ t>0$ (or $ t<0$ ), $ \alpha _t(\tau\kern-0.8pt)=2\hskip -.4pt\tanh (\tau\kern-0.8pt-q)$ or respectively $ \alpha _t(\tau\kern-0.8pt)=2\hskip -.4pt\coth (\tau\kern-0.8pt-q)$ . Again, all $ \alpha _t$ for $ t>0$ , or those with $ t<0$ , are affine (this time, translational) modifications of $ \alpha _1$ , or of $ \alpha _{-1}$ , while $ \alpha _0(\tau\kern-0.8pt)=2$ .