1 Introduction
Following Maschler [Reference Maschler17, page 367], one says that functions
$\tau\kern-0.8pt,\alpha ,\sigma $
on a Riemannian manifold
$(M,g)$
with the Ricci tensor
$\mathrm {r}$
satisfy a Ricci–Hessian equation if
$$ \begin{align} \alpha\nabla d\tau\kern-0.8pt + \mathrm{r} = \sigma g\quad\text{for some function } \sigma:M\to\mathrm{I\!R}{,} \end{align} $$
$\nabla $
being the Levi–Civita connection of g. We call (1-1) special when
$$ \begin{align} \alpha\nabla d\tau\kern-0.8pt\ne0 \mathrm{\ on\ a\ dense\ set,\ }\hskip.4pt\dim M=n>2\quad\mathrm{and}\quad \alpha \mathrm{\ is\ a\ }C^\infty \mathrm{\ function\ of\ } \tau\kern-0.8pt. \end{align} $$
Conditions (1-1)–(1-2) are satisfied in several situations that have been studied (see below), raising a natural question: Which functions
$\tau\kern-0.8pt\mapsto \alpha $
can be realized in this way? The present paper provides an answer in the Kähler case, outside of the classes that are already well understood; see Theorems D and E.
There are three well-known classes of examples leading to (1-1)–(1-2).
-
(I) Non-Einstein gradient Ricci almost-solitons [Reference Barros and Ribeiro1, Reference Pigola, Rigoli, Rimoldi and Setti20], including (non-Einstein) gradient Ricci solitons [Reference Hamilton and Isenberg15]. Here,
$\alpha $
is a nonzero constant. -
(II) Conformally Einstein metrics g, with
$\tau\kern-0.8pt>0$
and
$\alpha =(n-2)/\tau\kern-0.8pt$
, the Einstein metric being
${\hat {g\hskip 2pt}\hskip -1.3pt}=g/\tau\kern-0.8pt^2\hskip -.7pt$
. Compare with [Reference Derdzinski and Maschler11, (6.2)]. -
(III) Special Kähler–Ricci potentials
$\tau\kern-0.8pt$
on Kähler manifolds, at points where
$\mathrm {r}$
is not a multiple of g. See [Reference Derdzinski and Maschler11, Remark 7.4].
A special Kähler–Ricci potential [Reference Derdzinski and Maschler11, Section 7] on a Kähler manifold
$(M\hskip -.7pt,g)$
with the complex-structure tensor J is any nonconstant function
$ \tau\kern-0.8pt$
on
$ M$
having a real-holomorphic gradient
$ v=\nabla \hskip -.7pt\tau\kern-0.8pt$
for which, at points where
$ v\ne 0$
, all nonzero vectors orthogonal to
$ v$
and
$ J\hskip -.7pt v$
are eigenvectors of both
$ \nabla d\tau\kern-0.8pt$
and
$ \mathrm {r}$
. Such triples
$ (M\hskip -.7pt,g,\tau\kern-0.8pt)$
are completely understood, both locally [Reference Derdzinski and Maschler11] and in the compact case [Reference Derdzinski and Maschler12].
The classes (I)–(III) are far from disjoint: for instance [Reference Derdzinski and Maschler11, Corollary 9.3], in the Kähler category, if
$ n>4$
, class (II) is a special case of class (III).
We are interested in
$ M\hskip -.7pt,g,\tau\kern-0.8pt,\alpha ,\sigma $
satisfying (1-1)–(1-2) and such that
$$ \begin{align} 2\hskip.4pt\mathrm{r}\hskip.4pt(v,\cdot)=-\hskip.4pt dY \quad \mathrm{for} \quad v=\nabla\hskip-.7pt\tau\kern-0.8pt\quad\mathrm{and}\quad Y = \Delta\hskip-.4pt\tau\kern-0.8pt. \end{align} $$
Here, and throughout the paper, we use the notational conventions
$$ \begin{align} v=\nabla\hskip-.7pt\tau\kern-0.8pt,\quad Q=g(v,v),\quad Y =\Delta\hskip-.4pt\tau\kern-0.8pt,\quad n=\dim M \end{align} $$
whenever
$ (M\hskip -.7pt,g)$
is a Riemannian manifold and
$ \tau\kern-0.8pt:M\to \mathrm {I\!R}$
. As we point out near the end of Section 2, with J denoting the complex-structure tensor,
$$ \begin{align} \begin{array}{l} \text{for K}\ddot{\text{a}}\text{hler metrics } g, \text{ conditions }({1\mbox{-}1})\text{--}({1\mbox{-}2}) \text{ imply } ({1\mbox{-}3}),\\ \text{and the gradient } v = \nabla\tau\kern-0.8pt\text{ is a real-holomorphic vector field}\\ \text{or, equivalently}, Jv \text{ is a real-holomorphic } g\text{-Killing field.} \end{array} \end{align} $$
Assuming (1-1)–(1-2), we may treat the derivatives
$\alpha ' = \hskip .7pt d\alpha /d\tau\kern-0.8pt$
and
$\alpha "$
both as functions of the real variable
$\tau\kern-0.8pt$
and as functions
$M\to \mathrm {I\!R}$
. In Sections 3 and 4, we prove the following two results, as well as Theorem D, stated below.
Theorem A. Under the hypotheses (1-1)–(1-3), at points where
$\alpha "+\alpha \alpha '\hskip -.7pt\ne 0$
and
$d\tau\kern-0.8pt\ne 0$
, both
$Q=g(\nabla \hskip -.7pt\tau\kern-0.8pt,\hskip -1.5pt\nabla \hskip -.7pt\tau\kern-0.8pt)$
and
$Y = \Delta \hskip -.4pt\tau\kern-0.8pt$
are, locally, functions of
$\tau\kern-0.8pt$
.
Theorem B. Let functions
$\tau\kern-0.8pt,\alpha ,\sigma $
satisfy a special Ricci–Hessian equation (1-1), with (1-2), on a Kähler manifold
$(M\hskip -.7pt,g)$
of real dimension
$n\ge 4$
. If
$\alpha \,d\alpha $
and
$d\tau\kern-0.8pt$
are nonzero at all points of an open submanifold U of M, and:
-
(i)
$n>4$
; or -
(ii)
$n=4$
and
$d\sigma \wedge d\tau\kern-0.8pt=0$
identically in U; or finally, -
(iii)
$dQ\wedge d\tau\kern-0.8pt=0$
everywhere in
$U\hskip -.7pt$
, where
$Q=g(\nabla \hskip -.7pt\tau\kern-0.8pt,\hskip -1.5pt\nabla \hskip -.7pt\tau\kern-0.8pt)$
,
then
$\tau\kern-0.8pt:U\to \mathrm {I\!R}$
is a special Kähler–Ricci potential on the Kähler manifold
$(U\hskip -.7pt,g)$
.
With
$v,Q,Y\hskip .7pt$
as in (1-4), a function
$\tau\kern-0.8pt$
on a Riemannian manifold
$(M\hskip -.7pt,g)$
has
$dQ\wedge d\tau\kern-0.8pt=0$
if and only if Q is locally, at points where
$d\tau\kern-0.8pt\ne 0$
, a function of
$\tau\kern-0.8pt$
. This amounts to requiring the integral curves of v to be reparameterized geodesics (since, due to (2-2) below, the latter condition means that
$\nabla \hskip -.7pt\!_vv$
is a functional multiple of
$ v$
). Such functions
$ \tau\kern-0.8pt$
, called transnormal, have been studied extensively [Reference Bolton3, Reference Miyaoka18, Reference Wang21], and are referred to as isoparametric when, in addition,
$ dY\hskip -1.5pt\wedge d\tau\kern-0.8pt=0$
.
Theorem B renders the transnormal case
$ dQ\wedge d\tau\kern-0.8pt=0$
, as well as real dimensions
$ n>4$
, rather uninteresting in the context of special Ricci–Hessian equations (1-1)–(1-2) on Kähler manifolds, since at
$ d\alpha $
-generic points (see the end of Section 2), one then ends up with examples (I) or (III) above, compare with Remark 4.3, of which the former is the subject of a large existing literature, and the latter, as mentioned earlier, has been completely described. This is why our next two results focus exclusively on the real dimension four and functions
$ \tau\kern-0.8pt$
with
$ dQ\wedge d\tau\kern-0.8pt$
not identically zero.
Remark C. Equation (1-1), with (1-2), remains satisfied after
$\tau\kern-0.8pt$
and the function
$ \tau\kern-0.8pt\mapsto \alpha =\alpha (\tau\kern-0.8pt)$
have been subjected to an affine modification in the sense of being replaced with
$\hat \tau\kern-0.8pt$
and
$\hat \tau\kern-0.8pt\mapsto \hat \alpha (\hat \tau\kern-0.8pt)$
given by
$\hat \tau\kern-0.8pt=p+\tau\kern-0.8pt/c$
and
$\hat \alpha (\hat \tau\kern-0.8pt) =c\hskip .4pt\alpha (c\hskip .4pt\hat \tau\kern-0.8pt-cp)$
for real constants
$ c\ne 0$
and
$ p$
.
Theorem D. If the special Ricci–Hessian equation (1-1) and (1-2) both hold for functions
$\tau\kern-0.8pt,\alpha ,\sigma $
on a Kähler manifold
$(M\hskip -.7pt,g)$
of real dimension four, while
$dQ\wedge d\tau\kern-0.8pt\ne 0$
everywhere in an open connected set
$U\subseteq M\hskip -.7pt$
, then the function
$\alpha $
of the variable
$\tau\kern-0.8pt$
and its derivative
$\alpha ' = \hskip .7pt d\alpha /d\tau\kern-0.8pt$
satisfy, on U, the equation
$$ \begin{align} \alpha"\hskip.4pt+\,\alpha\alpha'\hskip.4pt=\,0,\text{\ that\ is\ }2\alpha'\hskip-1.5pt+\hskip.7pt\alpha^2=\,4\varepsilon\,\text{\ with\ a\ constant\ }\varepsilon\in\mathrm{I\!R}. \end{align} $$
In addition, for Q and Y as in (1-4), and the scalar curvature
$\mathrm{s}$
, the functions
$$ \begin{align} 2\hskip.4pt\theta=\alpha\hskip.7pt\mathrm{s}+4\varepsilon Y\quad\text{and}\quad \kappa=\theta\psi + \alpha^{-\hskip-1.5pt1}Y-Q \text{ are both constant,} \end{align} $$
$\psi $
being given by
$ 4\varepsilon \hskip .4pt\psi =\tau\kern-0.8pt-2/\hskip -.7pt\alpha $
, if
$ \varepsilon \ne 0$
, or
$ 3\psi =2/\hskip -.7pt\alpha ^3\hskip -.7pt$
, when
$ \varepsilon =0$
. Furthermore,
$ \sigma $
in (1-1) and the function F of the variable
$ \tau\kern-0.8pt$
characterized by
$$ \begin{align} 4\varepsilon F=\theta(2-\tau\kern-0.8pt\alpha)+4\varepsilon\kappa\alpha \ \text{for} \ \varepsilon\ne0\quad \text{and}\quad F= \kappa\alpha-2\hskip.4pt\theta/(3\alpha^2) \ \text{if} \ \varepsilon=0, \end{align} $$
and thus depending on the real constants
$\theta ,\kappa $
, satisfy the conditions
$$ \begin{align} \mathrm{(a)}\hskip6ptY\hskip-1.5pt-Q\alpha=F,\quad \mathrm{(b)}\hskip6pt2\sigma=-(Q\alpha'+F'),\quad \mathrm{(c)}\hskip6pt\Delta\alpha=F\hskip-.7pt\alpha' = -\hskip-.7pt F". \end{align} $$
Finally, up to affine modifications (see Remark C), the pair
$(\alpha (\tau\kern-0.8pt),\varepsilon )$
is one of the following five:
$ (2,1),\,\,(2/\hskip -.7pt\tau\kern-0.8pt,0),\,\,(2\hskip -.4pt\tanh \tau\kern-0.8pt,1),\,\,(2\hskip -.4pt\coth \tau\kern-0.8pt,1),\,\,(2\hskip -.4pt\cot \tau\kern-0.8pt,-\hskip -1.5pt1)$
.
Theorem E. Each of the five options listed in Theorem D, namely,
$$ \begin{align*} (2,1),\quad(2/\hskip-.7pt\tau\kern-0.8pt,0),\quad(2\hskip-.4pt\tanh\tau\kern-0.8pt,1),\quad(2\hskip-.4pt\coth\tau\kern-0.8pt,1),\quad(2\hskip-.4pt\cot\tau\kern-0.8pt,-\hskip-1.5pt1), \end{align*} $$
is realized by a special Ricci–Hessian equation (1-1)–(1-2) on a real-analytic Kähler manifold
$ (M\hskip -.7pt,g)$
of real dimension four such that, with
$v=\nabla \hskip -.7pt\tau\kern-0.8pt$
and
$ Q=g(v,v)$
, one has
$dQ\wedge d\tau\kern-0.8pt\ne 0$
somewhere in M and
$Jv$
lies in a two-dimensional Abelian Lie algebra of Killing fields.
For
$(2,1)$
and
$(2/\hskip -.7pt\tau\kern-0.8pt,0)$
, one can choose
$(M\hskip -.7pt,g)$
to be compact and biholomorphic to the two-point blow-up of 
, where g is the Wang–Zhu toric Kähler– Ricci soliton [Reference Wang and Zhu22, Theorem 1.1] or respectively the Chen–LeBrun–Weber conformally-Einstein Kähler metric [Reference Chen, LeBrun and Weber6, Theorem A].
In contrast with the final clause of Theorem E, we do not know whether the remaining three options,
$ (2\hskip -.4pt\tanh \tau\kern-0.8pt,1)$
,
$ (2\hskip -.4pt\coth \tau\kern-0.8pt,1)$
, and
$ (2\hskip -.4pt\cot \tau\kern-0.8pt,-\hskip -1.5pt1)$
, may be realized on a compact Kähler surface. An analytic-continuation phenomenon described below (Section 13) may hint at the plausibility of trying to obtain such compact examples via small deformations of the Wang–Zhu or Chen–LeBrun–Weber metric, combined with suitable affine modifications.
For the pairs
$ (2,1)$
and
$ (2/\hskip -.7pt\tau\kern-0.8pt,0)$
in Theorem D, the constancy conclusions of (1-7) are well known: [Reference Chow, LeBrun and Wang7, page 201], [Reference Derdziński9, page 417, Proposition 3(i) and page 419, (40)].
The paper is organized as follows. Section 2 contains the preliminaries. Consequences of special Ricci–Hessian equations, leading to proofs of Theorems A, B and D, are presented in the next two sections. Sections 5–12 are devoted to proving Theorem E: we rephrase it as solvability of the system (6-1) of quasi-linear first-order partial differential equations, subject to the additional conditions (6-2), which allows us to derive our claim from the Cartan–Kähler theorem for exterior differential systems.
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
$$ \begin{align} \delta[\nabla d\tau\kern-0.8pt]=\mathrm{r}(v,\cdot) +dY\quad\text{with } v=\nabla\tau\kern-0.8pt\quad\text{and}\quad Y=\Delta\tau\kern-0.8pt. \end{align} $$
Also, given a function
$ \tau\kern-0.8pt$
on a Riemannian manifold,
$$ \begin{align} 2[\nabla d\tau\kern-0.8pt](v,\cdot)=dQ,\quad\text{where } v=\nabla\hskip-.7pt\tau\kern-0.8pt\quad\mathrm{and}\quad Q=g(v,v), \end{align} $$
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
$$ \begin{align} \mathrm{(a)}\quad \delta\hskip.4pt[\nabla d\tau\kern-0.8pt]=\imath_v\mathrm{r}\hskip.7pt\, +\hskip.7pt\,dY,\quad\mathrm{(b)}\quad 2\imath_v[\nabla d\tau\kern-0.8pt]=dQ\quad\text{with ({1\mbox{-}4}).} \end{align} $$
Finally, for the Ricci tensor
$ \mathrm {r}$
and scalar curvature
$ \mathrm {s}$
of any Riemannian metric,
$$ \begin{align} 2\hskip.7pt\delta\mathrm{r}=d\hskip.4pt\mathrm{s}, \end{align} $$
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
$$ \begin{align} g(\Delta v,\cdot )=-\mathrm{r}\hskip.4pt(v,\cdot), \end{align} $$
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}$
,
$$ \begin{align} \text{the harmonic-flow condition ({2\mbox{-}5}) amounts to ({1\mbox{-}3}).} \end{align} $$
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)$
,
$$ \begin{align} \begin{array}{l} \text{conditions\ ({1\mbox{-}1})--({1\mbox{-}2})\ imply\ real}\text{-}\mathrm{holomorphicity\ of\ the}\\ \mathrm{gradient\ } v \hskip-.4pt = \hskip-1.5pt \ \nabla\tau\kern-0.8pt\mathrm{, \ while \ } J \hskip-.4ptv\mathrm{\ is \ then \ a \ holomorphic \ Killing}\\ \text{field, due to the resulting Hermitian symmetry of } \nabla d\tau\kern-0.8pt. \end{array} \end{align} $$
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
$$ \begin{align} ({1\mbox{-}3}), \text{with } ({1\mbox{-}4}) \text{ for all real-holomorphic gradients } v=\nabla\hskip-.7pt\tau\kern-0.8pt\hskip.4pt. \end{align} $$
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),
$$ \begin{align} n\hskip.4pt\sigma=Y\alpha + \hskip.7pt\mathrm{s},\quad\text{where } n=\dim M. \end{align} $$
Applying
$ 2\hskip .4pt\imath _v$
or
$ 2\hskip .4pt\delta $
to (1-1), we obtain, from (2-3)–(2-4) and (1-4),
$$ \begin{align} \begin{array}{rl} \mathrm{(i)}& \alpha\hskip.7pt dQ + 2\hskip.4pt\mathrm{r}\hskip.4pt(v,\cdot)=2\sigma\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, +\,2\alpha\hskip.7pt[\mathrm{r}\hskip.4pt(v,\cdot)\,+\hskip.7pt\,dY]\,+\hskip.7pt\,d\hskip.7pt\mathrm{s}\, =\,2\hskip.7pt d\sigma. \end{array} \end{align} $$
In the case where (1-1)–(1-2) hold along with (1-3), one may rewrite (3-2) as
$$ \begin{align} \begin{array}{rl} \mathrm{(i)}& \alpha\hskip.7pt dQ\,\hskip.4pt-\,dY = 2\sigma\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, +\,\alpha\hskip.7pt dY\hskip.7pt+\hskip.7pt\,d\hskip.7pt\mathrm{s}=2\hskip.7pt d\sigma, \end{array} \end{align} $$
which, in view of (2-2) and (3-1), amounts to nothing other than
$$ \begin{align} \begin{array}{rl} \mathrm{(i)}& d(Q\alpha - Y)=(Q\alpha'\hskip.7pt+\,2\sigma)\hskip.7pt d\tau\kern-0.8pt,\\ \mathrm{(ii)}& d\hskip.4pt[Q\alpha'\hskip.7pt+\,(n-2)\hskip.4pt\sigma]=(Q\alpha"\hskip.7pt+\,Y\alpha')\hskip.7pt d\tau\kern-0.8pt, \end{array} \end{align} $$
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
$$ \begin{align} d\alpha=\alpha'\hskip-.7pt d\tau\kern-0.8pt,\quad\nabla\hskip-1.5pt\alpha\, =\,\alpha'\hskip-.7ptv,\quad 2[\nabla d\tau\kern-0.8pt](\nabla\hskip-1.5pt\alpha,\cdot )\, =\alpha'\hskip-.4pt dQ,\quad\mathrm{where\ }\alpha' = \hskip.7pt d\alpha/d\tau\kern-0.8pt, \end{align} $$
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$
,
$$ \begin{align} \begin{array}{l} Q\alpha\hskip.7pt-\hskip.7pt Y\hskip.7pt\mathrm{\ and\ }\hskip.7pt\,Q\alpha' \ + \hskip.7pt(n-2)\hskip.4pt\,\sigma \ \mathrm{\ are\ functions\ of\ } \ \tau\kern-0.8pt\mathrm{,\ with}\\ \mathrm{the\ respective\ } \tau\kern-0.8pt \text{-derivatives } Q\alpha'+\hskip.7pt2\sigma\,\mathrm{\ and\ }\,Q\alpha"+\,Y\alpha',\\ \mathrm{which,\hskip-1.5pt\ consequently\hskip-1.5pt,\hskip-1.5pt\ must\hskip-.4pt\ themselves\hskip-.4pt\ be\hskip-.4pt\ functions\hskip-.4pt\ of\hskip-.4pt\ }\hskip.7pt\tau\kern-0.8pt\hskip-.7pt. \end{array} \end{align} $$
Proof of Theorem A.
At the points in question, using (3-6) to equate both
$ Q\alpha -Y\hskip .7pt$
and
$ Q\alpha "+\,Y\alpha '$
to some specific functions of
$ \tau\kern-0.8pt$
, we obtain a system of two linear equations with the nonzero determinant
$ \alpha "+\alpha \alpha '\hskip -.7pt$
, imposed on the unknowns
$ Q,\hskip -.7pt Y\hskip -.7pt$
, and our assertion follows since
$ \alpha "+\alpha \alpha '$
is also a function of
$ \tau\kern-0.8pt$
.
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
$$ \begin{align} \begin{array}{rl} \mathrm{(a)}& n\hskip.4pt(\alpha\hskip.7pt dQ-dY)-2(Y\alpha + \hskip.7pt\mathrm{s})\hskip.7pt d\tau\kern-0.8pt=0,\\ \mathrm{(b)}& n\hskip.4pt\alpha'\hskip-.7pt dQ+(n-2)(\alpha\hskip.7pt dY+ d\hskip.4pt\mathrm{s}) - 2Y\alpha'\hskip-.7pt d\tau\kern-0.8pt =0. \end{array} \end{align} $$
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
$$ \begin{align} \begin{array}{l} [d\hskip.4pt\xi](u,v)=d_u[\hskip.4pt\xi(v)] - d_v[\hskip.4pt\xi(u)] - \xi([u,v]),\\ {}[\xi\wedge\xi'](u,v)=\xi(u)\xi'(v) - \xi(v)\xi'(u). \end{array} \end{align} $$
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,
$$ \begin{align} 2i\hskip.4pt\partial\overline{\partial}\hskip.7pt\tau\kern-0.8pt =[\nabla d\tau\kern-0.8pt](J\cdot ,\cdot )-[\nabla d\tau\kern-0.8pt](\cdot ,J\cdot ). \end{align} $$
In the case of a Kähler metric g on a complex manifold
$ M\hskip -.7pt$
, (1-1) implies that
$$ \begin{align} i\hskip.4pt\alpha\hskip1pt\partial\overline{\partial}\hskip.7pt\tau\kern-0.8pt+\rho=\sigma\omega, \end{align} $$
$\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 $
.
Remark 4.1. As an obvious consequence of the last line in (2-7), if g is a Kähler metric, conditions (1-1)–(1-2) are equivalent to (4-3) along with (1-2) and real-holomorphicity of the gradient
$ v=\nabla \hskip -.7pt\tau\kern-0.8pt$
.
Remark 4.2. For the Kähler form
$\omega $
of a Kähler manifold
$ (M\hskip -.7pt,g)$
of real dimension
$ n\ge 4$
, the operator
$ \zeta \mapsto \zeta \wedge \hskip 1pt\omega $
acting on differential q-forms is injective if
$ q=2$
and
$ n>4$
, or
$ q=1$
. Namely, the contraction of
$ \zeta \wedge \hskip 1pt\omega $
against
$ \omega $
yields a nonzero constant times
$ (n-4)\hskip .7pt\zeta +2\langle \omega ,\zeta \rangle \hskip .4pt\omega $
(if
$ q=2$
), or times
$ (n-2)\hskip .7pt\zeta $
(if
$ q=1$
). In the former case,
$ \zeta $
with
$ \zeta \wedge \hskip 1pt\omega =0$
is thus a multiple of
$ \omega $
, and hence is
$ 0$
.
Remark 4.3. Whenever (4-3) with a constant
$\alpha $
holds on a Kähler manifold of real dimension
$ n\ge 4$
, constancy of
$ \sigma $
follows (from Remark 4.2, as
$ d\sigma \wedge \hskip 1pt\omega =0$
).
We have the following result, due to Maschler [Reference Maschler17, Proposition 3.3].
Lemma 4.4. Condition (1-1) on a Kähler manifold
$ (M\hskip -.7pt,g)$
of real dimension
$ n>4$
implies that
$ d\sigma \wedge d\alpha =0$
. Equivalently, wherever
$ d\alpha $
is nonzero,
$ \sigma $
must, locally, be a function of
$ \alpha $
.
Proof. By (4-3),
$ 0=d\rho =d\sigma \wedge \hskip 1pt\omega -d\alpha \wedge i\hskip .4pt\hskip 1pt\partial \overline {\partial }\hskip .7pt\tau\kern-0.8pt$
, so that
$ d\alpha \wedge d\sigma \wedge \hskip 1pt\omega =0$
, and our assertion is immediate from Remark 4.2.
Proof of Theorem B.
In all three cases (i)–(iii),
$ d\sigma \wedge d\tau\kern-0.8pt=0$
. For case (i), this follows from Lemma 4.4 while, when
$ dQ\wedge d\tau\kern-0.8pt=0$
on
$ \,U\hskip -.7pt$
, we see that, in view of the equality
$ \alpha '\hskip -.7pt dQ+\alpha \hskip .7pt dY = d(2\sigma -\mathrm {s})$
arising from (3-3ii) and (3-5),
$ Y$
and
$ 2\sigma -\mathrm {s}$
are, locally, functions of
$ \tau\kern-0.8pt$
, and hence so is
$ \sigma $
, as a consequence of (3-1) with
$ n\ge 4$
. Now, [Reference Derdzinski and Maschler11, Corollary 9.2] yields our claim.
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
$$ \begin{align} \psi' = 1/\alpha^2. \end{align} $$
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
$$ \begin{align} v=\partial\hskip-.4pt_x\quad\text{and}\quad w=\partial_u\quad\mathrm{(the\ real\ coordinate\ vector\ fields).} \end{align} $$
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
$$ \begin{align} \omega=\phi_{x\hskip-.4pt x}\hskip.7pt dx\wedge dx'\hskip-.7pt +\phi_{x\hskip-.4pt u}(dx\wedge du'+\hskip.7pt du\wedge dx') +\phi_{u\hskip-.4pt u}\hskip.7pt du\wedge du'. \end{align} $$
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
$$ \begin{align} \left[\begin{matrix} Q&0&S&0\\ 0&Q&0&S\\ S&0&B&0\\ 0&S&0&B\end{matrix}\right]\hskip-1.5pt\hskip-.7pt\quad\mathrm{with\ the\ determinant\ }(QB - S^2)^2. \end{align} $$
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,
$$ \begin{align} \phi_{x\hskip-.4pt x}>0\quad\text{and}\quad\varPi>0 \quad \mathrm{for} \ \varPi = \phi_{x\hskip-.4pt x}\phi_{u\hskip-.4pt u}\hskip-.7pt-\phi^{\hskip-.4pt2}_{\hskip-.7pt x\hskip-.4pt u}, \end{align} $$
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
$$ \begin{align} \begin{array}{l} (\tau\kern-0.8pt,\lambda,Q,S,B)=\,(\phi_x,\phi_u,\phi_{x\hskip-.4pt x}, \hskip.7pt\phi_{x\hskip-.4pt u},\hskip.7pt\phi_{u\hskip-.4pt u}),\ \text{so that}\\Q>0 \text{ and } \varPi=QB - S^2 > 0 \text{ due to }({5\mbox{-}4}). \end{array} \end{align} $$
With
$ \mathrm {div},\hskip .7pt\Delta $
denoting the g-divergence and g-Laplacian, for
$ \tau\kern-0.8pt,\lambda ,Q$
in (5-5):
-
(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$
; -
(b) the other coordinate fields
$Jv$
and
$Jw$
are holomorphic g-Killing fields; -
(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,
$$ \begin{align} \begin{array}{ll} \mathrm{(i)}&\left[\begin{matrix}\partial_x\hskip-1.5pt &\hskip-1.5pt\partial_u\cr\end{matrix}\right] =\left[\begin{matrix}\partial\hskip-1pt_{\tau\kern-0.8pt}\hskip-1.5pt &\hskip-1.5pt\partial\hskip-1.5pt_\lambda\cr \end{matrix}\right] \left[\begin{matrix} Q&S\\ S&B\\\end{matrix}\right]\hskip-1.5pt\hskip-.7pt,\quad\mathrm{(ii)}\quad \left[\begin{matrix}d\tau\kern-0.8pt\cr d\lambda\\\end{matrix}\right] =\left[\begin{matrix} Q&S\\ S&B\\\end{matrix}\right] {\left[\begin{matrix}dx\\ du\\\end{matrix}\right]},\hskip7pt\mathrm{and\ so}\\[12pt] \mathrm{(iii)}&(QB - S^2) \left[\begin{matrix}\partial\hskip-1pt_{\tau\kern-0.8pt}\hskip-1.5pt &\hskip-1.5pt\partial\hskip-1.5pt_\lambda\cr\end{matrix}\right] =\left[\begin{matrix}\partial_x\hskip-1.5pt&\hskip-1.5pt\partial_u\cr\end{matrix} \right] {\left[\begin{matrix} B\hskip-3pt&-\hskip-1.5ptS\\ -\hskip-1.5ptS\hskip-3pt&Q\\\end{matrix}\right]},\\[12pt] \mathrm{(iv)}&(QB - S^2)\left[\begin{matrix}dx\\ du\\\end{matrix}\right] =\left[\begin{matrix} B\hskip-3pt&-\hskip-1.5ptS\\ -\hskip-1.5ptS\hskip-3pt&Q\\\end{matrix}\right]\left[\begin{matrix}d\tau\kern-0.8pt\cr d\lambda\\\end{matrix}\right]. \end{array} \end{align} $$
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),
$$ \begin{align} SQ_{\tau\kern-0.8pt}+BQ_\lambda=QS_{\tau\kern-0.8pt} +SS_\lambda,\quad SS_{\tau\kern-0.8pt} + BS_\lambda=QB_{\tau\kern-0.8pt} +SB_{\lambda},\quad Q>0,\quad QB>S^2. \end{align} $$
Conversely, if functions
$ Q,S,B$
of the variables
$ \tau\kern-0.8pt,\lambda $
satisfy (5-7), then, locally:
-
(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:
-
(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 -
(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).
Theorem 5.1. In 
with the complex coordinates
$ (x+ix'\hskip -.7pt,u+iu')$
, given an open subset M and a function
$ \phi :M\to \mathrm {I\!R}$
of the real variables
$ x,u$
, having the property (5-4), let g be the Kähler metric on M with the Kähler potential
$ 2\phi $
. The following two conditions are equivalent.
-
(i) The special Ricci–Hessian equation (1-1)–(1-2) holds on M for
$ \tau\kern-0.8pt=\phi _x$
, and
$ dQ\wedge d\tau\kern-0.8pt\ne 0$
everywhere in
$ M\hskip -.7pt$
, with
$ Q=g(\nabla \hskip -.7pt\tau\kern-0.8pt,\hskip -1.5pt\nabla \hskip -.7pt\tau\kern-0.8pt)$
. Thus, by Theorem D, one has (1-6) and (1-9a), where
$ Y = \Delta \hskip -.4pt\tau\kern-0.8pt$
and F is the function of
$ \tau\kern-0.8pt$
characterized by (1-8) for the constants
$ \theta ,\kappa $
in (1-7). -
(ii) The triple
$ (Q,S,B)=(\phi _{x\hskip -.4pt x},\phi _{x\hskip -.4pt u},\phi _{u\hskip -.4pt u})$
of functions of the new variables
$ (\tau\kern-0.8pt,\lambda ) =(\phi _x,\phi _u)$
satisfies (5-7) with
$ Q_\lambda \ne 0$
everywhere, as well as the equations
$ S_{\tau\kern-0.8pt} + B_\lambda = S\hskip -.7pt\alpha +G$
,
$ \,Q_{\tau\kern-0.8pt} + S\hskip -1.5pt_\lambda = Q\alpha +F\hskip -.7pt$
,
$ \,G_{\tau\kern-0.8pt} = -S\hskip -.7pt\alpha '\hskip -.7pt$
,
$ G\hskip -1.5pt_\lambda =Q\alpha '+F'$
for some function
$ G$
and
$ (\hskip 2.5pt)' = d/d\tau\kern-0.8pt$
.
Proof. By (1-9b), Equation (1-1) is, in case (i), equivalent to
$$ \begin{align} 2\alpha\nabla d\tau\kern-0.8pt + 2\hskip.7pt\mathrm{r}\hskip.7pt=-(Q\alpha'+F')\hskip.4pt g, \end{align} $$
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:
-
(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
$$ \begin{align} \alpha\,dS - d\hskip.7pt\Delta\lambda=-(Q\alpha'+F')\,d\lambda. \end{align} $$
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:
$$ \begin{align} \begin{array}{rl} \mathrm{(a)}&Q_{\tau\kern-0.8pt} + S\hskip-1.5pt_\lambda-Q\alpha-F=0,\\\mathrm{(b)}&S_{\tau\kern-0.8pt} + B_\lambda-S\hskip-.7pt\alpha-G=0,\\\mathrm{(c)}&QB_{\tau\kern-0.8pt} + SB_\lambda -SS_{\tau\kern-0.8pt}-B\hskip-.7ptS_\lambda=0,\\\mathrm{(d)}&SQ_{\tau\kern-0.8pt} + BQ_\lambda-Q\hskip-.4ptS_{\tau\kern-0.8pt} -SS_\lambda=0,\\\mathrm{(e)}&G_{\tau\kern-0.8pt} + S\hskip-.7pt\alpha' = 0,\\\mathrm{(f)}&G\hskip-1.5pt_\lambda-Q\alpha'\hskip-.7pt-F' = 0, \end{array} \end{align} $$
on which one imposes the additional conditions
$$ \begin{align} QB>S^2,\quad Q\,>\,0,\quad Q_\lambda\hskip.7pt\ne\,0\,\mathrm{\ \ \ everywhere.} \end{align} $$
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
$$ \begin{align} \left[\begin{matrix} \varPhi\\ \varPsi\end{matrix}\right]= \left[\begin{matrix} S&B\\ Q&S\end{matrix}\right] \left[\begin{matrix} QB_{\tau\kern-0.8pt} + SB_\lambda -SS_{\tau\kern-0.8pt}-B\hskip-.7ptS_\lambda\cr SQ_{\tau\kern-0.8pt} + BQ_\lambda-Q\hskip-.4ptS_{\tau\kern-0.8pt} -SS\hskip-1.5pt_\lambda\end{matrix}\right]\hskip-1.5pt. \end{align} $$
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
$$ \begin{align} \begin{array}{rl} \mathrm{(i)}&QB_{\tau\kern-0.8pt} + BQ_{\tau\kern-0.8pt} -2SS_{\tau\kern-0.8pt}-(QB-S^2)\alpha-B F+SG=0,\\ \mathrm{(ii)}&QB_\lambda + BQ_\lambda -2SS\hskip-1.5pt_\lambda-QG+S F=0. \end{array} \end{align} $$
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
$$ \begin{align} \begin{array}{rl} \mathrm{(i)}&dG=-S\hskip-.7pt\alpha'd\tau\kern-0.8pt\, +\,(Q\alpha'+\hskip.7pt F')\,d\lambda,\\ \mathrm{(ii)}&d\varPi=\hskip.7pt(\varPi\hskip-.7pt\alpha +B F-SG)\,d\tau\kern-0.8pt\, +\,\hskip-.7pt(QG-S F)\,d\lambda\,\mathrm{\ for\ }\,\varPi=QB-S^2\hskip-.7pt. \end{array} \end{align} $$
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).
Lemma 6.2. Let
$ (\dot \tau\kern-0.8pt,\dot \lambda ,Q,S,B\hskip -.7pt,G)\in \mathrm {I\!R}\hskip -.7pt^6$
have
$ (\dot \tau\kern-0.8pt,\dot \lambda )\ne (0,0)$
and
$ QB>S^2\hskip -.7pt$
.
-
(a)
$QB>0$
and
$ \varPsi \ne 0$
, where
$ \varPsi =B\dot \tau\kern-0.8pt^2\hskip -.7pt-2S\dot \tau\kern-0.8pt\dot \lambda +Q\dot \lambda ^2\hskip -.7pt$
. -
(b)
$(0,0,0,0,0,0)$
is the only vector
$ (Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt}, Q_\lambda ,S_\lambda ,B_\lambda ) \in \mathrm {I\!R}\hskip -.7pt^6$
with
$$ \begin{align*} \left[\begin{matrix} 1&0&0&0&1&0\\ 0&1&0&0&0&1\\ 0&-\hskip-.7ptS&Q&0&-\hskip-.7ptB&S\\ S&-Q&0&B&-\hskip-.7ptS&0\\ \dot\tau\kern-0.8pt&0&0&\dot\lambda&0&0\\ 0&\dot\tau\kern-0.8pt&0&0&\dot\lambda&0\\ 0&0&\dot\tau\kern-0.8pt&0&0&\dot\lambda\end{matrix}\right] \left[\begin{matrix} Q_{\tau\kern-0.8pt}\cr S_{\tau\kern-0.8pt}\cr B_{\tau\kern-0.8pt}\cr Q_\lambda\cr S_\lambda\cr B_\lambda\end{matrix}\right] =\left[\begin{matrix} 0\cr0\cr0\cr0\cr0\cr0\end{matrix}\right]\hskip-1.5pt. \end{align*} $$
-
(c) The first four rows of the above
$ 7\times 6$
matrix are linearly independent.
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
$$ \begin{align} (Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt},G_{\tau\kern-0.8pt}, Q_\lambda,S_\lambda,B_\lambda,G_\lambda) \,\in\,\mathrm{I\!R}\hskip-.7pt^8 \end{align} $$
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
$$ \begin{align} (\dot Q,\dot S,\dot B\hskip-.7pt,\dot G) =(Q_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+Q_\lambda\dot\lambda,\, S_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+S_\lambda\dot\lambda,\, B_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+B_\lambda\dot\lambda,\, G_{\tau\kern-0.8pt}\dot\tau\kern-0.8pt+G_\lambda\dot\lambda) \end{align} $$
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
$$ \begin{align} \begin{array}{l} \dot G=-S\hskip-.7pt\alpha'\hskip-.7pt\dot\tau\kern-0.8pt + (Q\alpha'+F')\dot\lambda,\\ Q\hskip-.7pt\dot B+B\dot Q-2S\hskip-.7pt\dot S =[(QB-S^2)\alpha+B F-SG]\dot\tau\kern-0.8pt+(QG-S F)\dot\lambda. \end{array} \end{align} $$
Proof. That
$ \mathcal {A}\subseteq \mathrm {I\!R}\hskip -.7pt^8\hskip -.7pt$
, or
$ \mathcal {L}\subseteq \mathrm {I\!R}\hskip -.7pt^4\hskip -.7pt$
, if nonempty, is an affine subspace, clearly follows since (6-1), or (6-8), is a system of nonhomogeneous linear equations imposed on (6-6) or respectively
$ (\dot Q,\dot S,\dot B\hskip -.7pt,\dot G)$
. Also,
$ \mathcal {A}$
is nonempty, and two-dimensional, being the pre-image of
$ (Q\alpha +F\hskip -.7pt,S\hskip -.7pt\alpha +G,0,0,-S\hskip -.7pt\alpha '\hskip -.7pt,Q\alpha '+F')$
under the obvious linear operator
$ \mathrm {I\!R}\hskip -.7pt^8\hskip -.7pt\to \mathrm {I\!R}\hskip -.7pt^6\hskip -.7pt$
. Namely, this operator is surjective: it equals the direct sum of the identity operator
$ \mathrm {I\!R}\hskip -.7pt^2\hskip -.7pt\to \mathrm {I\!R}\hskip -.7pt^2\hskip -.7pt$
, acting on
$ (G_{\tau\kern-0.8pt},G_\lambda )$
, and an operator
$ \mathrm {I\!R}\hskip -.7pt^6\hskip -.7pt\to \mathrm {I\!R}\hskip -.7pt^4$
, which has rank
$ 4$
due to Lemma 6.2(c). Next,
$ \varPhi $
maps
$ \mathcal {A}$
into
$ \mathcal {L}$
, which one sees adding (6-1e), or (6-4i), times
$ \dot \tau\kern-0.8pt$
to (6-1f), or (6-4ii), times
$ \dot \lambda $
, and then using (6-7). Thus,
$ \mathcal {L}$
is nonempty, and
$ \dim \mathcal {L}=2$
, the matrix of the homogeneous system associated with (6-8) having rank
$ 2$
since Lemma 6.2(a) gives
$ Q\ne 0$
. Let
$ \mathcal {A}\hskip -.4pt'\hskip -.7pt\subseteq \mathrm {I\!R}\hskip -.7pt^8$
and
$ \mathcal {L}\hskip -.4pt'\hskip -.7pt\subseteq \mathrm {I\!R}\hskip -.7pt^4$
now be the vector subspaces parallel to
$ \mathcal {A}\, $
and
$ \mathcal {L}$
. Our assertion will thus follow once we establish injectivity of
$ \varPhi :\mathcal {A}\to \mathcal {L}$
, that is, injectivity of its linear part
$ \varPhi :\mathcal {A}\hskip -.4pt'\hskip -.7pt\to \mathcal {L}\hskip -.4pt'\hskip -.7pt$
. Equivalently, we need to show that zero is the only vector (6-6) lying in
$ \mathcal {A}\hskip -.4pt'$
and having the
$ \varPhi $
-image
$ (0,0,0,0)$
or, in other words, the only solution to the matrix equation in Lemma 6.2, with
$ (\dot G,G_{\tau\kern-0.8pt},G_\lambda )=(0,0,0)$
. This, however, is precisely what Lemma 6.2(b) states.
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
$$ \begin{align} \begin{array}{l} 2\alpha'+\hskip.7pt\alpha^2=4\varepsilon\mathrm{,\hskip-1.5pt\ where\ }\,(\,\,)' = d/d\tau\kern-0.8pt,\\ 4\varepsilon F = \hskip.7pt\theta(2\hskip-.4pt-\hskip-.4pt\tau\kern-0.8pt\alpha)+4\varepsilon\kappa\alpha\,\mathrm{\ if\ }\hskip.7pt\varepsilon\ne0,\\ F=\,\kappa\alpha\hskip.4pt-\hskip.4pt2\hskip.4pt\theta/(3\alpha^2)\,\mathrm{\ when\ }\,\varepsilon\hskip.7pt=\hskip.7pt0. \end{array} \end{align} $$
In addition, it is natural to assume here that
$$ \begin{align} \tau\kern-0.8pt\,\mathrm{\ ranges\ over\ the\ domain\ of\ }\,\alpha\,\mathrm{\ (which\ is\ also\ the\ domain\ of\ }\,F). \end{align} $$
Consequently,
$\alpha $
and F satisfy the ordinary differential equations
$$ \begin{align} \alpha"\hskip.4pt+\,\alpha\alpha'\hskip.4pt=\,0,\quad F"\hskip.4pt=\,-F\hskip-.7pt\alpha'. \end{align} $$
Theorem 7.1. For any fixed
$ \alpha ,F\hskip .7pt$
as in (7-1)–(7-2), real-analytic solutions 
to (6-1)–(6-2) exist, locally, on a neighborhood of any
$ (\tau\kern-0.8pt,\lambda )\in \mathrm {I\!R}\hskip -.7pt^2$
with the property (7-2).
More precisely, one obtains a locally unique such solution 
by prescribing 
and the partial derivatives 
real-analytically along an arbitrary real-analytic embedded curve
$ t\mapsto (\tau\kern-0.8pt,\lambda )\in \mathrm {I\!R}\hskip -.7pt^2\hskip -.7pt$
, so as to satisfy (6-1), (6-2), (7-2), and the condition 
, where
$ (\,\,)\dot {\,}=d/dt$
.
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]:
$$ \begin{align} \begin{array}{l} H(E)=\{v\in{T_zM}:\zeta(v,e_1,\ldots,e_p)=0\,\mathrm{\ for\ all\ }\hskip.7pt\zeta\in\mathcal{I}\cap\varOmega\hskip.4pt^{p\hskip.7pt+1}\hskip-1.5pt M\}\\ \mathrm{and\ }\,r(E)\hskip.7pt=\hskip.7pt\dim H(E)-(p+1)\mathrm{,\ for\ any\ basis\ }\,e_1,\ldots,e_p\mathrm{\ of\ }E, \end{array} \end{align} $$
so that
$ H(E)$
is a vector subspace of
$ {T_zM}\hskip -.7pt$
, not depending on
$ e_1,\ldots ,e_p$
since
$$ \begin{align} H(E)=\{v\in{T_zM}:\mathrm{span}(v,E)\,\mathrm{\ is\ an\ integral\ element\ of\ }\mathcal{I}\}. \end{align} $$
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
$$ \begin{align} \begin{array}{l} \mathrm{the\ exterior\ differential\ system\ }\,\,\mathcal{I}\,\hskip.4pt\hskip.7pt\mathrm{ \ on\ this\ }\,\hskip.4pt\hskip.7pt\mathcal{Y}\,\mathrm{\ generated}\\ \mathrm{by\ \, the\ two\ } \ 1\text{-}\mathrm{forms\ }\ \ dG + S\hskip-.7pt\alpha'd\tau\kern-0.8pt\, -\,(Q\alpha'+\hskip.7pt F')\,d\lambda\,\,\mathrm{\ and}\\ d\hskip.4pt(QB\hskip-1.5pt-\hskip-1.5ptS^2)\ \hskip-1.5pt-\hskip-1.5pt \ [(QB\hskip-1.5pt \ - \ \hskip-1.5ptS^2)\ \alpha\hskip-1.5pt +\hskip-1.5ptB F\hskip-1.5pt-\hskip-1.5ptSG]\hskip.7pt d\tau\kern-0.8pt\hskip-1.5pt -\hskip-1.5pt(QG\hskip-1.5pt-\hskip-1.5ptS F)\hskip.7pt d\lambda,\\ \mathrm{the\ }\,\hskip.7pt2\text{-}\mathrm{form\ \ }\,dQ\wedge d\lambda\,\, +\,\,d\tau\kern-0.8pt\wedge\hskip.4pt dS\, - \,(Q\alpha\hskip.7pt+\,F)\,d\tau\kern-0.8pt\wedge d\lambda,\\ \mathrm{the\ \ }\,\hskip.7pt2\text{-}\mathrm{form\ \ \ }\hskip.4pt\,dS\wedge d\lambda\hskip.7pt\, +\hskip.7pt\,d\tau\kern-0.8pt\hskip.7pt\wedge dB\hskip.7pt\, -\,(S\hskip-.7pt\alpha\hskip.7pt+\hskip.7ptG)\,d\tau\kern-0.8pt\wedge d\lambda,\\ \mathrm{their\hskip-.4pt\ exterior\hskip-.4pt\ derivatives,\hskip-.7pt\ and\hskip-.4pt\ the\hskip-.4pt\ exterior\hskip-.4pt\ \ derivatives\hskip-.4pt\ of}\\ (QB\hskip-1.5pt-\hskip-1.5ptS^2)^{-\hskip-1.5pt1}\hskip-.7pt \ (B\hskip.7pt d\tau\kern-0.8pt\hskip-.7pt-\hskip-.7ptS\hskip.7pt d\lambda) \ \mathrm{\ and\ } \ (QB\hskip-1.5pt-\hskip-1.5ptS^2)^{-\hskip-1.5pt1} \ \hskip-.7pt \ (S\hskip.7pt d\tau\kern-0.8pt\hskip-.7pt \ - \ \hskip-.7ptQ\,d\lambda)\hskip-.4pt. \end{array} \end{align} $$
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$
,
$$ \begin{align} \mathrm{we\ call\ }\,E\,\mathrm{\ horizontal\ when\ }\,(d\tau\kern-0.8pt,d\lambda):\hskip-1.5pt E\hskip.7pt\to\mathrm{I\!R}\hskip-.7pt^2\mathrm{\ is\ injective.} \end{align} $$
Remark 9.1. If
$ E_1$
is an integral line of
$\mathcal {I}$
in (8-3) and
$ E_1\subseteq E_2$
for a unique horizontal integral plane
$ E_2$
, then
$ E_2$
is the only integral plane containing
$ E_1$
. Namely, another such plane
$ E_2'$
, being nonhorizontal, would intersect the kernel of
$ (d\tau\kern-0.8pt,d\lambda )$
along a line. As
$ E_3\subseteq H\hskip -.7pt(E_1)$
for the vector subspace
$ H(E)$
in (8-2) and the three-dimensional span
$ E_3$
of
$ E_2$
and
$ E_2'$
, all planes in
$ E_3$
containing the line
$ E_1$
, other than
$ E_2'$
, would be horizontal integral planes, making
$ E_2$
nonunique.
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,
$$ \begin{align} \begin{array}{l} \mathrm{vectors\ \,}\hskip.7pt(\dot\tau\kern-0.8pt, \ \dot\lambda,\dot Q,\dot S,\dot B\hskip-.7pt,\dot G)\hskip.7pt \ \mathrm{\ at\ } \hskip.7pt(\tau\kern-0.8pt, \ \lambda,Q,S,B\hskip-.7pt,G)\hskip.7pt \ \mathrm{\ spanning\ horizontal}\\ \text{integral lines of } \mathcal{I} \text{ are characterized by (6-8) and } (\dot\tau\kern-0.8pt,\dot\lambda)\hskip.7pt\ne(0,0), \end{array} \end{align} $$
where
$ \alpha ,F\hskip -.7pt,\alpha '\hskip -.7pt,F'$
satisfy (7-1)–(7-2).
Theorem 9.3. Every horizontal integral line of the system
$\mathcal {I}$
defined by (8-3) is contained in a unique integral plane of
$ \mathcal {I}\hskip -.7pt$
, and this unique plane is also horizontal.
Proof. Any horizontal plane in
$ \mathrm {I\!R}\hskip -.7pt^6$
has, by (9-1), a unique basis of the form
$$ \begin{align} (1,0,Q_{\tau\kern-0.8pt},S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt},G_{\tau\kern-0.8pt}),\quad (0,1,Q_\lambda,S_\lambda,B_\lambda, G_\lambda). \end{align} $$
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
$$ \begin{align} \begin{array}{l} \text{every\ \,horizontal\ \,integral\ \,line\ \,of\ \,}\mathcal{I}\hskip-.7pt\text{,\ at\ \,any\ point}\\ \text{of\ }\mathcal{Y}\hskip-1.5pt\text{,\ is\ contained\ in\ a\ unique\ integral\ plane\ of\ }\,\mathcal{I}\hskip-.7pt. \end{array} \end{align} $$
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,
$$ \begin{align} \text{integral lines of } \mathcal{I} \text{ are the same as lines tangent to } \mathcal{D}. \end{align} $$
Remark 10.2. Due to Theorem 9.3, our
$ \mathcal {Y}$
and
$ \mathcal {I}\hskip -.7pt$
, introduced in Section 8, satisfy the hypotheses of Theorem 10.1, with
$ q=2$
, the coordinate functions
$ \tau\kern-0.8pt,\lambda $
, and the two
$ 1$
-forms in the second and third lines of (8-3). Therefore, our
$ \mathcal {D}$
(see Remark 9.2) then corresponds to
$ \mathcal {D}$
in Theorem 10.1, and hence satisfies (10-2). Horizontal integral curves of our
$\mathcal {I}$
thus are, by (9-2), precisely those curves that have parameterizations 
with (6-8), where
$ (\,\,)\dot {\,}=\,d/dt$
, and
$ (\dot \tau\kern-0.8pt,\dot \lambda )\hskip .7pt\ne (0,0)$
for all
$ t$
. Choosing such a curve as in the sentence preceding (10-2), we obtain the additional data 
required in Theorem 7.1 by applying to 
the inverse of the affine isomorphism
$ \varPhi :\mathcal {A}\to \mathcal {L}$
of Theorem 6.3. This clearly results in the infinite-dimensional freedom of choices mentioned at the end of Section 6.
Proof of Theorem 7.1.
The image of the mapping 
in the second paragraph of Theorem 7.1 is a horizontal real-analytic integral curve of
$ \mathcal {I}\hskip -.7pt$
. In fact, horizontality follows since
$ t\mapsto (\tau\kern-0.8pt,\lambda )$
is an embedding, while, as 
, the resulting tangent directions are integral lines of
$\mathcal {I}$
as an immediate consequence of Theorem 9.3 combined with (9-2).
The integral surface of
$\mathcal {I}$
arising in Theorem 10.1, being horizontal (Theorem 9.3), forms, locally, the graph of a function 
, which is a solution to (6-1) according to the description of
$\mathcal {I}$
in (8-3) and the paragraph following (8-3). To realize the condition
$ Q_\lambda \hskip -.7pt\ne 0$
required in (6-2), we solve (6-1), at a given point
$ (\tau\kern-0.8pt,\lambda ,Q,S,B\hskip -.7pt,G)\in \mathcal {Y}\hskip -.7pt$
, by setting
$ (Q_{\tau\kern-0.8pt},Q_\lambda )=(0,1)$
, which uniquely determines
$ S_{\tau\kern-0.8pt},B_{\tau\kern-0.8pt},G_{\tau\kern-0.8pt}, S_\lambda ,B_\lambda ,G_\lambda $
, and then choosing the quadruple (6-7), with fixed
$ (\dot \tau\kern-0.8pt,\dot \lambda )\hskip .7pt\ne (0,0)$
, associated with the resulting octuple (6-6).
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):
$$ \begin{align} \begin{array}{rl} \mathrm{(a)}&\mathrm{the\ integer\ }\,r(E)\,\mathrm{\ has\ a\ fixed\ nonnegative\ value,\ namely,}\\ \mathrm{(b)}& \dim H(E_0)=k- q\mathrm{\ and\ } r(E_0) =k- q-1\mathrm{\ when\ }p=0,\\ \mathrm{(c)}&\dim H\hskip-.7pt(E_1)=2\mathrm{\ and\ } r(E_1) =0\mathrm{\ in\ the\ case\ where\ }p=1. \end{array} \end{align} $$
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 $
.
Proof of Theorem E.
Combining Theorems 7.1 and 5.1, we obtain the first assertion of Theorem E.
For the second one, we invoke the existence results of [Reference Chen, LeBrun and Weber6, Reference Wang and Zhu22]. In both cases,
$dQ\wedge d\tau\kern-0.8pt\ne 0$
somewhere, and the metric is real-analytic. The former claim follows, for instance, since a compact Kähler surface with a nontrivial holomorphic gradient
$\nabla \hskip -.7pt\tau\kern-0.8pt$
having
$dQ\wedge d\tau\kern-0.8pt=0$
identically for
$ Q=g(\nabla \hskip -.7pt\tau\kern-0.8pt,\hskip -1.5pt\nabla \hskip -.7pt\tau\kern-0.8pt)$
must necessarily [Reference Derdzinski10, Section 1] be biholomorphic to 
or a 
bundle over 
(rather than the two-point blow-up of 
). The latter, in the case of [Reference Wang and Zhu22], is due to a general reason: all Ricci solitons are real-analytic [Reference Dancer and Wang8, Lemma 3.2]. So are, however, all Riemannian Einstein metrics [Reference DeTurck and Kazdan13, Theorem 5.2], and the Chen–LeBrun–Weber metric of [Reference Chen, LeBrun and Weber6] is conformal to an Einstein metric
$ {\hat {g\hskip 2pt}\hskip -1.3pt}$
, while again, for a general reason [Reference Derdziński9, page 417, Proposition 3(ii)], the conformal change leading from
${\hat {g\hskip 2pt}\hskip -1.3pt}$
to g has a canonical form (up to a constant factor, it is the multiplication by the cubic root of the norm-squared of the self-dual Weyl tensor). This causes g to be real-analytic as well.
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$
.
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