A parabolic approach to the Calabi–Yau problem in HKT geometry

Abstract

We consider the natural generalization of the parabolic Monge–Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex structure is locally flat and admits a compatible hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge–Ampère equation first introduced in Alekser and Verbitsky (Isr J Math 176:109–138, 2010). The result gives an alternative proof of a theorem of Alesker (Adv Math 241:192–219, 2013).

1 Introduction

Let M be a 4n-dimensional smooth manifold. A hypercomplex structure on M is a triple (I, J, K) of complex structures satisfying the quaternionic identities

$$\begin{aligned} IJ=-JI=K. \end{aligned}$$

A Riemannian metric g on (M, I, J, K) is called hyperhermitian if it is compatible with each of I, J, K. Every hyperhermitian metric induces the form

$$\begin{aligned} \Omega =\omega _J+i\omega _K\in \Lambda _I^{2,0}, \end{aligned}$$

where $\omega _J=g(J\cdot ,\cdot )$ and $\omega _K=g(K\cdot ,\cdot )$ are the fundamental forms of (g, J) and (g, K) respectively. The form $\Omega$ is nondegenerate, i.e. $\Omega ^n \ne 0$ everywhere, and determines g by the relation

$$\begin{aligned} \Omega (X,Y)=2g(JX,Y) \end{aligned}$$

(1)

for every X, Y vector fields of type (1, 0) with respect to I.

A hyperhermitian manifold (M, I, J, K, g) is called HKT (hyperkähler with torsion) if

$$\begin{aligned} \partial \Omega =0, \end{aligned}$$

where $\partial$ is with respect to I. HKT manifolds were first introduced by Howe and Papadopoulos in [15], but the characterization in terms of the form $\Omega$ is due to Grantcharov and Poon [13] who also showed the existence of HKT structures on some homogeneous hypercomplex manifolds. HKT structures have been studied intensively in the last years and many analogies to Kähler manifolds have been discovered (see e.g. [1,2,3,4,5, 9,10,11,12,13, 16, 21, 24, 26,27,28] and the references therein). In particular a pluripotential theory has been developed in HKT geometry, according to which every HKT form $\Omega$ can be locally written as

$$\begin{aligned} \Omega =\partial \partial _J v \end{aligned}$$

for some real smooth function v (see [3, 5]), where $\partial _J=J^{-1}{\bar{\partial }}J$ and the action of J on a k-form $\alpha$ is defined as

$$\begin{aligned} J\alpha (X_1,\dots ,X_k)=\alpha (JX_1,\dots ,JX_k). \end{aligned}$$

Furthermore, in analogy with the complex case, the space of quaternionic plurisubharmonic functions has been introduced

$$\begin{aligned} {\mathcal {H}}_\Omega =\{\varphi \in C^{\infty }(M): \Omega _\varphi :=\Omega +\partial \partial _J \varphi >0\}, \end{aligned}$$

where “$\Omega _\varphi > 0$”  means that $\Omega _\varphi$ induces a hyperhermitian metric via (1).

In [4] Alesker and Verbitsky introduced the following quaternionic version of the Calabi problem in analogy with the famous complex Calabi problem solved by Yau in [29]. Let (M, I, J, K, g) be a compact HKT manifold and let $F\in C^{\infty }(M)$ be a smooth real valued function. The quaternionic Monge–Ampère equation is

$$\begin{aligned} (\Omega +\partial \partial _J\varphi )^n=b\,\mathrm{e}^F\,\Omega ^n, \end{aligned}$$

(2)

where $F\in C^{\infty }(M)$ is the datum, while $(\varphi ,b) \in {\mathcal {H}}_\Omega \times \mathbb {R}_+$ is the unknown. Equation (2) is most naturally settled if the canonical bundle of (M, I) is holomorphically trivial. In this case there exists a holomorphic volume form $\Theta$ on (M, I) which satisfies the q-real condition $J\Theta ={\bar{\Theta }}$ and b is determined by F:

$$\begin{aligned} b=\frac{\int _M\Omega ^n\wedge {\bar{\Theta }}}{\int _M\mathrm{e}^F\Omega ^n\wedge {\bar{\Theta }}}. \end{aligned}$$

So far there are only partial results about the solvability of (2). In [4] it is proved the uniqueness and an a priori $C^0$ estimate for solutions to (2) when the canonical bundle of (M, I) is holomorphically trivial; in [2, 21] the $C^0$ estimate is established on any compact HKT manifold, without further assumptions and in [11] the problem is studied on some 8-dimensional nilmanifolds. In [1] it is proved that Eq. (2) can always be solved on HKT manifolds with locally flat hypercomplex structure which admit a compatible hyperkähler metric. Here we recall that a hypercomplex structure is called locally flat if it is locally isomorphic to $\mathbb {H}^n$. Manifolds of this kind were firstly considered by Sommese in [20] and simply called quaternionic manifolds. Recently in [9] the quaternionic Monge–Ampère equation has been solved on hyperkähler manifolds without the assumption of local flatness.

In the present paper we approach Eq. (2) via the following geometric flow

$$\begin{aligned} \varphi _t=\log \frac{(\Omega +\partial \partial _J\varphi )^n}{\Omega ^n}-F,\quad \varphi (x,0)=0, \end{aligned}$$

(3)

where the solution $\varphi$ is supposed to satisfy $\varphi (\cdot ,t)\in {\mathcal {H}}_\Omega$ for every t and the subscript t denotes the derivative of $\varphi$ with respect to the variable t. The same dynamic approach was pursued on Kähler manifolds [6], on Hermitian manifolds [14, 23] and on almost Hermitian manifolds [8].

Our main result is the following theorem which provides an alternative proof of Alesker’s Theorem [1].

Theorem 1

Let (M, I, J, K, g) be a compact HKT manifold with (I, J, K) locally flat and assume that there exists a hyperkähler metric ${\hat{g}}$ on (M, I, J, K). Then there exists a long-time solution $\varphi \in C^{\infty }(M\times \mathbb {R}_+)$ to the parabolic quaternionic Monge–Ampère equation (3) such that

$$\begin{aligned} \tilde{\varphi }=\varphi -\int _M\varphi \, \Omega ^n\wedge \bar{\Omega }_{{\hat{g}}}^n \end{aligned}$$

converges in $C^{\infty }$-topology to a smooth function $\tilde{\varphi }_{\infty }\in C^{\infty }(M)$. Moreover if

$$\begin{aligned} b:=\frac{\int _M\Omega ^n\wedge \bar{\Omega }_{{\hat{g}}}^n}{\int _M\mathrm{e}^F\Omega ^n\wedge \bar{\Omega }_{{\hat{g}}}^n}, \end{aligned}$$

then $(\tilde{\varphi }_\infty ,b)$ solves the quaternionic Monge–Ampère equation (2).

Recently we have been made aware of the paper [30] where the parabolic quaternionic Monge–Ampère equation is studied and its long-time behaviour is described with techniques different from ours.

Now we describe the layout of the proof. Since (3) is strongly parabolic, it admits a unique maximal solution $\varphi \in C^{\infty }(M\times [0,T))$.

• Step 1. From the equation we directly deduce a uniform $C^0$ bound on $\varphi _t$ (Lemma 5).

• Step 2. The $C^0$ estimate for solutions of the quaternionic Calabi–Yau equation (2) then implies a uniform bound on $\mathrm{osc}\,\varphi$ (Lemma 6).

• Step 3. We use the existence of the hyperkähler metric and the local flatness of the hypercomplex structure in order to establish a uniform upper bound on $\Delta _{{\hat{g}}}\varphi$ (Lemma 7).

• Step 4. A general result in [7] implies a uniform Hölder estimate on the second derivatives of $\varphi$, thus a classical bootstrapping argument using Schauder estimates implies $T=\infty$ and a uniform bound on $|\nabla ^k\varphi |$ for $k\ge 1$ (Lemmas 8 and 9).

• Step 5. We prove the convergence of $\tilde{\varphi }$ using an argument due to Phong–Sturm [18] based on an adapted Mabuchi-type functional (Lemma 10).

We point out that the local flatness of the hypercomplex structure plays a role in steps 3 and 4, while the existence of a background hyperkähler metric is only used in step 3.

Remark

Flow (3) can be regarded as a geometric flow in Hermitian Geometry. Here we assume that the canonical bundle of (M, I) is trivial and we fix a q-real complex volume form $\Theta$ on (M, I). As shown in [4] one has

$$\begin{aligned} (\Omega +\partial \partial _J\varphi )^n\wedge \bar{\Theta }=i^n(\omega -i\partial \bar{\partial }\varphi )^n\wedge \Phi , \quad \Omega ^n\wedge \bar{\Theta }=i^n\omega ^n\wedge \Phi \end{aligned}$$

where $\omega$ is the fundamental form of (g, I) and $\Phi$ is a real (n, n)-form which is positive in a weak sense. By setting $u=-\varphi$ we can then rewrite (3) as

$$\begin{aligned} u_t=-\log \frac{(\omega +i\partial \bar{\partial }u)^n\wedge \Phi }{\omega ^n\wedge \Phi }+F,\quad u(0)=0. \end{aligned}$$

(4)

Equation (4) reminds the parabolic k-Hessian flow

$$\begin{aligned} u_t=\log \frac{(\chi +i\partial \bar{\partial }u)^k\wedge \alpha ^{n-k}}{\alpha ^n}+F,\quad u(0)=0 \end{aligned}$$

(5)

studied by Phong and Tô on a complex n-dimensional Hermitian manifold $(M,\alpha )$ in [19], where $1\le k\le n$ and $\chi$ is real k-positive (1, 1)-form. According to [19] (5) has always a long-time solution whose normalization converges in $C^\infty$-topology to a solution of the k-Hessian equation. Equation (4) differs from the parabolic n-Hessian flow since the role of $\alpha ^n$ is replaced by the form $\Phi$ which is positive in a weak sense and the theorem of Phong and Tô cannot be directly applied.

Moreover the authors would like to thank the anonymous referee who carefully read the first version of the present paper and made many very useful remarks which allowed them to considerably improve the presentation of the results.

2 Preliminaries

Let (M, I, J, K, g) be a compact HKT manifold with HKT form $\Omega$. Let $\partial$ be the $\partial$-operator with respect to I and $\partial _J:=J^{-1}\bar{\partial }J:\Lambda ^{r,0}_I\rightarrow \Lambda ^{r+1,0}_I$. Then

$$\begin{aligned} \partial \partial _J=-\partial _J\partial , \end{aligned}$$

see [26]. Moreover we assume that the canonical bundle of (M, I) is holomorphically trivial and we let $\Theta$ be a q-real holomophic volume form on (M, I). Note that, since $\Omega$ is easily seen to be q-real, $\Omega ^n\wedge \bar{\Theta }$ is a real volume form; indeed, J acts trivially on top forms and thus

$$\begin{aligned} \overline{\Omega ^n\wedge \bar{\Theta }}=J\Omega ^n\wedge J\bar{\Theta }=\Omega ^n\wedge \bar{\Theta }. \end{aligned}$$

The HKT metric induces the quaternionic Laplacian operator

$$\begin{aligned} \Delta _g\varphi :=\frac{\partial \partial _J\varphi \wedge \Omega ^{n-1}}{\Omega ^n} \end{aligned}$$

for $\varphi \in C^{\infty }(M)$. It is well-known that $\Delta _g$ is elliptic and it is straightforward to show that for $\eta ,\psi \in C^{\infty }(M)$ we have

$$\begin{aligned} \int _M(\Delta _g\eta )\psi \,\Omega ^n\wedge \bar{\Theta }= \int _M\eta (\Delta _g\psi )\,\Omega ^n\wedge \bar{\Theta }. \end{aligned}$$

Moreover the following formula will be useful: for every $\alpha , \beta \in \Lambda ^{1,0}_I$

$$\begin{aligned} \frac{\alpha \wedge J\bar{\beta }\wedge \Omega ^{n-1}}{\Omega ^n}= - \frac{1}{2n} g(\alpha ,\bar{\beta }). \end{aligned}$$

(6)

The basic example of hyperhermitian manifold is given by an open set A of ${\mathbb {R}}^{4n}$ with the standard hyperhermitian structure

$$\begin{aligned} I_0=\begin{pmatrix} 0 &{} -\mathbbm {1}_n &{} 0 &{} 0\\ \mathbbm {1}_n &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{}-\mathbbm {1}_n\\ 0 &{} 0 &{} \mathbbm {1}_n &{} 0 \end{pmatrix},\quad J_0=\begin{pmatrix} 0 &{} 0 &{} -\mathbbm {1}_n &{} 0\\ 0 &{} 0 &{} 0 &{} \mathbbm {1}_n\\ \mathbbm {1}_n &{} 0 &{} 0 &{} 0\\ 0 &{} -\mathbbm {1}_n &{} 0 &{} 0 \end{pmatrix},\quad K_0=\begin{pmatrix} 0 &{} 0 &{} 0 &{} -\mathbbm {1}_n\\ 0 &{} 0 &{} -\mathbbm {1}_n &{} 0\\ 0 &{} \mathbbm {1}_n &{} 0 &{} 0\\ \mathbbm {1}_n &{} 0 &{} 0 &{} 0 \end{pmatrix}, \end{aligned}$$

where $\mathbbm {1}_n$ is the $n\times n$ identity matrix. In this case for an ${\mathbb {H}}$-valued function $u:\mathbb {R}^{4n}\rightarrow {\mathbb {H}}$ the following derivatives are defined

$$\begin{aligned} \partial _{q^r}u:=\partial _{x_0^r}u e_0-\sum _{i=1}^3 \partial _{x_i^r}u e_i ,\quad \partial _{ {\bar{q}}^r}u:=\sum _{i=0}^3e_{i} \partial _{x_i^r}u , \end{aligned}$$

where to shorten the notation we denote the quaternions 1, i, j, k with $e_0,e_1,e_2,e_3$ and the coordinates on ${\mathbb {R}}^{4n}$ are taken as $(x^1_{0},\dots ,x^n_0,x^{1}_{1},\dots ,x^n_1,x^1_2,\dots ,x^n_2,x^1_3,\dots ,x^{n}_{3})$ in order to identify $\mathbb {R}^{4n}$ with ${\mathbb {H}}^n$. We denote by

$$\begin{aligned} \mathrm{Hyp}(n,{\mathbb {H}})=\{U\in {\mathbb {H}}^{n,n}:{\bar{U}}=\,^tU\} \end{aligned}$$

the space of hyperhermitian matrices. Any $U\in \mathrm{Hyp}(n,{\mathbb {H}})$ has real eigenvalues and we can consider the subset $\mathrm{Hyp}^+(n,{\mathbb {H}})$ of positive-definite hyperhermitian matrices.

Any hyperhermitian Riemannian metric g on $(A,I_0,J_0,K_0)$ defines a smooth map $G:A\rightarrow \mathrm{Hyp}^{+}(n,{\mathbb {H}})$,

$$\begin{aligned} G_{rs}:=g(\partial _{q^r},\partial _{q^s}), \end{aligned}$$

where g is extended ${\mathbb {H}}$-semilinearly in its components, i.e.

$$\begin{aligned} g(X,Y\lambda )=g(X,Y)\lambda ,\quad g(X\lambda ,Y)=\bar{\lambda }g(X,Y) \end{aligned}$$

for every $\lambda \in {\mathbb {H}}$, $X,Y\in \Gamma (TA)$.

In some occasions we will make use of the following real representation of quaternionic matrices $\iota :{\mathbb {H}}^{n,n}\rightarrow \{U\in \mathbb {R}^{4n,4n}:I_0UI_0=J_0UJ_0=K_0UK_0=-U\}$,

$$\begin{aligned} \iota (A+iB+jC+kD):= \begin{pmatrix} A &{} B &{} C &{} D\\ -B &{} A &{} -D &{}C\\ -C &{} D &{} A &{}-B\\ -D &{} -C &{} B&{} A \end{pmatrix}. \end{aligned}$$

The map $\iota$ is an isomorphism of real algebras and

$$\begin{aligned} \iota (\mathrm{Hyp}(n, \mathbb {H})) = \mathrm{Hyp}(n, \mathbb {R}), \end{aligned}$$

where

$$\begin{aligned} \mathrm{Hyp}(n, \mathbb {R})= \{U\in \mathrm {Sym}(4n,\mathbb {R}): I_0UI_0=J_0UJ_0=K_0UK_0=-U\}. \end{aligned}$$

For any smooth function $u:A\rightarrow {\mathbb {R}}$ it is defined the quaternionic Hessian matrix

$$\begin{aligned} (\mathrm{Hess}_{{\mathbb {H}}}u)_{rs}:= u_{{\bar{r}} s}, \end{aligned}$$

where we set $u_{{\bar{r}} s} = \partial _{{\bar{q}}^r}\partial _{ q^s} u$. $\mathrm{Hess}_{{\mathbb {H}}}u$ is a hyperhermitian quaternionic matrix, in particular the entries $(\mathrm{Hess}_{{\mathbb {H}}}u)_{rr}$ are real.

The following lemma will be useful. We refer to [3, Proposition 4.1] for a proof (see also [22]).

Lemma 2

Let g be a HKT metric on $(A,I_0,J_0,K_0)$. Then the matrix associated to g is

$$\begin{aligned} G=\kappa \mathrm{Hess}_{{\mathbb {H}}}u, \end{aligned}$$

where $u\in C^{\infty }(A,\mathbb {R})$ is such that $\Omega =\partial \partial _J u$ is the HKT form associated to g and $\kappa >0$ is a universal constant.

Next we recall the formulation of the quaternionic Monge–Ampère equation (2) on open sets of ${\mathbb {H}}^n$; for a hyperhermitian $U \in {\mathbb {H}}^{n,n}$ we will denote by $\det U$ its Moore determinant (see [17]).

Lemma 3

Let g be a HKT metric on $(A,I_0,J_0,K_0)$, $\varphi :A\rightarrow \mathbb {R}$ a smooth function and $\Omega$ the HKT form of g, then

$$\begin{aligned} (\Omega +\partial \partial _J\varphi )^n=\frac{\det (G+\kappa \,\mathrm{Hess}_{{\mathbb {H}}}\varphi )}{\det G}\,\Omega ^n,\quad n\frac{\partial \partial _J\varphi \wedge \Omega ^{n-1}}{\Omega ^n}= \kappa \,\mathrm {Re}\left( \mathrm{tr}(G^{-1}\mathrm{Hess}_{{\mathbb {H}}}\varphi )\right) . \end{aligned}$$

Proof

The first formula is [3, Corollary 4.6] and the second is simply obtained by linearizing the first one at the origin and using $\det \iota (U)=(\det U)^4$ for any hyperhermitian matrix U (see [3, Theorem 2.4]),

$$\begin{aligned} \frac{d}{ds}\bigg \vert _{s=0} \log \frac{(\Omega + \partial \partial _J (s\varphi ))^n}{\Omega ^n} =\frac{d}{ds}\bigg \vert _{s=0}\log \frac{\det (G+\kappa \mathrm {Hess}_\mathbb {H}(s\varphi ))}{\det G} \end{aligned}$$

which gives

$$\begin{aligned} \begin{aligned} n\frac{\partial \partial _J \varphi \wedge \Omega ^{n-1}}{\Omega ^n}&=\frac{d}{ds}\bigg \vert _{s=0} \log \frac{\det \iota \left( G+\kappa \mathrm {Hess}_\mathbb {H}(s\varphi )\right) ^{1/4}}{\det \iota (G)^{1/4}}\\&=\frac{1}{4}\frac{d}{ds}\bigg \vert _{s=0}\log \frac{\det \left( \iota (G+\kappa \mathrm {Hess}_\mathbb {H}(s\varphi ))\right) }{\det \iota (G)}\\&= \frac{\kappa }{4} \mathrm {tr}\left( \iota (G)^{-1}\iota (\mathrm {Hess}_\mathbb {H}\varphi ) \right) = \kappa \mathrm {Re}\left( \mathrm {tr}\left( G^{-1} \mathrm {Hess}_\mathbb {H}\varphi \right) \right) \end{aligned} \end{aligned}$$

as claimed. $\square$

Finally, we provide a lemma which will be helpful in the proof of the main theorem.

Lemma 4

Let $U:A\rightarrow \mathrm{Hyp}^+(n,{\mathbb {H}})$ be a smooth map and assume that there exists $p\in A$ such that U(p) is diagonal. Let ${\hat{g}}$ be a hyperhermitian metric on A such that the induced matrix ${\hat{G}}$ is the identity. Then

$$\begin{aligned} \Delta _{{\hat{g}}}\log (\det U)=-\frac{\kappa }{n}\sum _{r,s,t=1}^n\sum _{i=0}^3 \frac{1}{U_{ss}} \frac{1}{U_{tt}} |U_{st,x^r_i} |^2 +\sum _{s=1}^n\frac{1}{U_{ss}} \Delta _{{\hat{g}}} U_{ss} \end{aligned}$$

at p, where the subindex “$x^r_{i}$”denotes the derivative with respect to the corresponding real coordinate.

Proof

Since $\det \iota (U)=(\det U)^4$ we directly compute

$$\begin{aligned} \begin{aligned} \partial _{{\bar{q}}^r}\partial _{q^r} \log (\det U)&=\sum _{i=0}^3\partial _{x^r_i}^2 \log (\det U) =\frac{1}{4}\sum _{i=0}^3\partial _{x^r_i}^2 \log (\det \iota (U))\\&=\frac{1}{4}\sum _{i=0}^3\partial _{x^r_i}\mathrm {tr}\left( \iota (U)^{-1} \iota (U)_{,x^r_i} \right) \\&=\frac{1}{4}\sum _{i=0}^3 \mathrm {tr}\left( -\iota (U)^{-1}\iota (U)_{,x^r_i}\iota (U)^{-1} \iota (U)_{,x^r_i} + \iota (U)^{-1}\iota (U)_{,x^r_ix^r_i} \right) \\&=\frac{1}{4}\sum _{i=0}^3 \mathrm {tr}\left( \iota \left( -U^{-1}U_{,x^r_i}U^{-1} U_{,x^r_i} + U^{-1}U_{,x^r_ix^r_i}\right) \right) \\&=\sum _{i=0}^3 \mathrm {Re}\left( \mathrm {tr}\left( -U^{-1}U_{,x^r_i}U^{-1} U_{,x^r_i} + U^{-1}U_{,x^r_ix^r_i} \right) \right) \end{aligned} \end{aligned}$$

and at the point p where U takes a diagonal form

$$\begin{aligned} \begin{aligned} \Delta _{{\hat{g}}} \log (\det U)&=\frac{\kappa }{n}\sum _{r,s,t=1}^n \sum _{i=0}^3 \mathrm {Re}\left( -U^{ss}U_{st,x^r_i}U^{tt} U_{ts,x^r_i}+ U^{ss}U_{ss,x^r_ix^r_i} \right) \\&=-\frac{\kappa }{n}\sum _{r,s,t=1}^n \sum _{i=0}^3 \frac{1}{U_{ss}}\frac{1}{U_{tt}}|U_{st,x^r_i}|^2 + \sum _{s=1}^n\frac{1}{U_{ss}}\Delta _{{\hat{g}}}U_{ss} \end{aligned} \end{aligned}$$

and the claim follows. $\square$

3 Proof of the result

Let (M, I, J, K, g) be a HKT manifold with HKT form $\Omega$. Every $\varphi \in {\mathcal {H}}_\Omega$ induces a HKT metric $g_\varphi$ and a quaternionic Laplacian $\Delta _\varphi :=\Delta _{g_\varphi }$. Consider the operator

$$\begin{aligned} P:{\mathcal {H}}_\Omega \rightarrow C^{\infty }(M),\quad P(\varphi )=\log \frac{(\Omega +\partial \partial _J\varphi )^n}{\Omega ^n}-F . \end{aligned}$$

The first variation of P is

$$\begin{aligned} P_{*|\varphi }(\psi )=n\frac{\partial \partial _J\psi \wedge (\Omega +\partial \partial _J\varphi )^{n-1}}{(\Omega +\partial \partial _J\varphi )^{n}}=n\Delta _\varphi \psi . \end{aligned}$$

Since $\Delta _{\varphi }$ is a strongly elliptic operator, Eq. (3) is always well-posed and it admits a unique maximal solution $\varphi \in C^{\infty }(M\times [0,T))$. Assume further that the canonical bundle of (M, I) is holomorphically trivial and let $\Theta \in \Lambda ^{2n,0}_I$ be a q-real holomorphic volume form. We then denote

$$\begin{aligned} \tilde{\varphi }= \varphi -\int _M\varphi \,\Omega ^n\wedge \bar{\Theta }. \end{aligned}$$

We start by proving $C^0$ bounds for the time derivatives $\varphi _t$ and $\tilde{\varphi }_t$ and then use these to prove the $C^0$ estimate for $\tilde{\varphi }$. In what follows we denote by C all the uniform constants (which may be different from line to line).

Lemma 5

There exists a uniform constant C such that

$$\begin{aligned} |\varphi _t(x,t)|\le C,\quad |\tilde{\varphi }_t(x,t)|\le C\, \end{aligned}$$

for every $(x,t)\in M\times [0,T)$.

Proof

Since

$$\begin{aligned} \frac{\partial }{\partial t} \log \frac{(\Omega +\partial \partial _J\varphi )^n}{\Omega ^n}=n \frac{\partial \partial _J\varphi _t\wedge \Omega ^{n-1}_\varphi }{\Omega ^{n}_\varphi }=n\Delta _\varphi \varphi _t, \end{aligned}$$

we have

$$\begin{aligned} \varphi _{tt}=n\Delta _\varphi \varphi _t \end{aligned}$$

and the parabolic maximum principle implies the a priori $C^{0}$ estimate for $\varphi _t$. The estimate on $\tilde{\varphi }_t$ immediately follows. $\square$

Lemma 6

We have

$$\begin{aligned} \max _M \varphi -\min _M \varphi \le C \end{aligned}$$

and

$$\begin{aligned} |\tilde{\varphi }|\le C, \end{aligned}$$

for a uniform constant C.

Proof

Since $\left|\varphi _t\right|$ is bounded and

$$\begin{aligned} (\Omega +\partial \partial _J\varphi )^n=\mathrm{e}^{F+\varphi _t}\,\Omega ^n, \end{aligned}$$

for every fixed t, $\varphi (\cdot ,t)$ solves the quaternionic Monge–Ampère equation (2) with datum $F+\varphi _t$. In view of the $C^0$ estimate for solutions to the quaternionic Monge–Ampère equation [2, 4, 21], $\varphi$ satisfies the bound

$$\begin{aligned} \max _M \varphi -\min _M \varphi \le C, \end{aligned}$$

(7)

where C depends only on (M, I, J, K, g) and an upper bound of $\max |F+\varphi _t|$. Therefore Lemma 5 implies that the constant C in (7) may be chosen so that it only depends on (M, I, J, K, g) and an upper bound of $\max |F|.$ Now, let $(x,t)\in M\times [0,T)$, since $\tilde{\varphi }$ is normalized, there exist (y, t) such that $\tilde{\varphi }(y,t)=0$, and thus we obtain $|\tilde{\varphi }(x,t)|=|\tilde{\varphi }(x,t)-\tilde{\varphi }(y,t)|=|\varphi (x,t)-\varphi (y,t)|\le C$ and the claim follows. $\square$

Lemma 7

Assume that (I, J, K) is locally flat and that there exists a hyperkähler metric ${\hat{g}}$ on (M, I, J, K). Then

$$\begin{aligned} \Delta _{{\hat{g}}}\varphi \le C, \end{aligned}$$

for a uniform constant C.

Proof

Let

$$\begin{aligned} Q=2\sqrt{\frac{1}{n}\mathrm {tr}_{ {\hat{g}}}g_{\varphi }}-\varphi . \end{aligned}$$

Fix $T'<T$ and let $(x_0,t_0)$ be a point where Q achieves its maximum in $M\times [0,T']$. We may assume without loss of generality that $t_0>0$. Since (I, J, K) is locally flat, then in a neighborhood of $x_0$ we can locally identify M with an open set A of ${\mathbb {H}}^n$. Let G and ${\hat{G}}$ be the hyperhermitian matrices in A induced by g and ${\hat{g}}$ respectively. We may further assume that $G=\mathrm{Hess}_{{\mathbb {H}}}v$ in A, that ${\hat{G}}$ is the identity in A and that $U=\mathrm{Hess}_{{\mathbb {H}}}(v+\kappa \varphi )$ is diagonal at $x_0$. Let $u=v+\kappa \varphi$. Then in A we have

$$\begin{aligned} Q=2\sqrt{\Delta _{{\hat{g}}} u}-\varphi . \end{aligned}$$

Computing at $(x_0,t_0)$, we have

$$\begin{aligned} \Delta _{\varphi } Q=\frac{\kappa }{n} \frac{1}{\sqrt{\Delta _{{\hat{g}}} u}} \sum _{r=1}^n \frac{1}{u_{r{\bar{r}}}} \left( -\frac{1}{2} \frac{1}{\Delta _{{\hat{g}}} u}|\Delta _{{\hat{g}}}\,u_r|^2+\Delta _{{\hat{g}}} u_{r {\bar{r}}}\right) -\Delta _{\varphi }\varphi \end{aligned}$$

and, applying Lemmas 3 and 4 we infer

$$\begin{aligned} \begin{aligned} \partial _tQ=&\,\frac{1}{\sqrt{\Delta _{{\hat{g}}} u}}\Delta _{{\hat{g}}} \varphi _t-\varphi _t= \frac{1}{\sqrt{\Delta _{{\hat{g}}} u}}\Delta _{{\hat{g}}} (\log \det (U)-\log \det (G)-F)-\varphi _t\\ =&\, \frac{1}{\sqrt{\Delta _{{\hat{g}}} u}} \left( -\frac{\kappa }{n}\sum _{r,s,t=1}^n\sum _{i=0}^3 \frac{1}{u_{s {\bar{s}}}} \frac{1}{u_{t {\bar{t}}}} |u_{s {\bar{t}},x^r_i} |^2 +\sum _{r=1}^n\frac{1}{u_{r {\bar{r}}}} \Delta _{{\hat{g}}} u_{r {\bar{r}}}-\Delta _{{\hat{g}}}\log \det (G) -\Delta _{{\hat{g}}} F\right) - \varphi _t \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned}&\partial _tQ-\frac{n}{\kappa }\Delta _{\varphi } Q\\&\quad =\frac{1}{\sqrt{\Delta _{{\hat{g}}} u}} \left( \frac{1}{2\Delta _{{\hat{g}}} u} \sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}}|\Delta _{{\hat{g}}}\,u_r|^2\right. \\&\qquad \left. -\frac{\kappa }{n}\sum _{r,s,t=1}^n\sum _{i=0}^3 \frac{1}{u_{s {\bar{s}}}} \frac{1}{u_{t {\bar{t}}}} |u_{s {\bar{t}},x^r_i} |^2 -\Delta _{{\hat{g}}} (F+\log \det (G))\right) +\frac{n}{\kappa }\Delta _{\varphi }\varphi -\varphi _t. \end{aligned}$$

Using the Cauchy–Schwarz inequality and [1, Proposition 3.1] we obtain

$$\begin{aligned} \begin{aligned} \sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}}|\Delta _{{\hat{g}}} u_{r}|^2&=\sum _{r=1}^n\sum _{i=0}^3\frac{1}{u_{r{\bar{r}}}}(\Delta _{{\hat{g}}} u_{x^r_i})^2=\frac{\kappa ^2}{n^2}\sum _{r=1}^n\sum _{i=0}^3\frac{1}{u_{r{\bar{r}}}}\left( \sum _{s=1}^n\frac{\sqrt{u_{s\bar{s}}}}{\sqrt{u_{s\bar{s}}}}u_{s\bar{s},x^r_i}\right) ^2\\&\le \frac{\kappa ^2}{n^2}\sum _{r,s,t=1}^nu_{t\bar{t}}\sum _{i=0}^3\frac{1}{u_{r{\bar{r}}}} \frac{1}{u_{s\bar{s}}}(u_{s\bar{s},x^r_i})^2=\frac{\kappa }{n}\Delta _{\hat{g}} u\sum _{r,s=1}^n\sum _{i=0}^3\frac{1}{u_{r{\bar{r}}}} \frac{1}{u_{s\bar{s}}}(u_{s\bar{s},x^r_i})^2\\&\le 2\frac{\kappa }{n} \Delta _{{\hat{g}}} u \sum _{r,s,t=1}^n \sum _{i=0}^3 \frac{1}{u_{ s{\bar{s}}}} \frac{1}{u_{ t{\bar{t}}}} | u_{s{\bar{t}}, x^r_i} |^2 \end{aligned} \end{aligned}$$

i.e.

$$\begin{aligned} \frac{1}{2\Delta _{{\hat{g}}} u}\sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}}|\Delta _{{\hat{g}}} u_{r}|^2\le \frac{\kappa }{n}\sum _{r,s,t=1}^n\sum _{i=0}^3 \frac{1}{u_{s {\bar{s}}}} \frac{1}{u_{t {\bar{t}}}} |u_{ s {\bar{t}},x^r_i} |^2 , \end{aligned}$$

from which it follows

$$\begin{aligned} 0&\le \partial _tQ-\frac{n}{\kappa }\Delta _{\varphi } Q\le \frac{n}{\kappa }\Delta _{\varphi }\varphi -\frac{\Delta _{{\hat{g}}} (F+\log \det (G))}{\sqrt{\Delta _{{\hat{g}}} u}}-\varphi _t\\&\le \frac{n}{\kappa }-\frac{n}{\kappa ^2}\Delta _{\varphi }v-\frac{\Delta _{{\hat{g}}} (F+\log \det (G))}{\sqrt{\Delta _{{\hat{g}}} u}}-\varphi _t \end{aligned}$$

at $(x_0,t_0)$, where we have used that it is a maximum point as well as the relation

$$\begin{aligned} \Delta _{\varphi }\varphi =1-\frac{1}{\kappa }\Delta _{\varphi }v. \end{aligned}$$

Hence

$$\begin{aligned} \frac{n}{\kappa ^2}\Delta _{\varphi }v \le \frac{n}{\kappa }- \frac{\Delta _{{\hat{g}}} (F+\log \det (G))}{\sqrt{\Delta _{{\hat{g}}} u}}-\varphi _t \end{aligned}$$

at $(x_0,t_0)$. Since $|\varphi _t|$ is uniformly bounded we obtain

$$\begin{aligned} \Delta _{\varphi }v(x_0,t_0) \le C + \frac{C}{\sqrt{\sum _{r=1}^n u_{r{\bar{r}}}(x_0,t_0)}} \end{aligned}$$

(8)

for a uniform constant C. In terms of u and G Eq. (3) writes as

$$\begin{aligned} \frac{1}{\kappa }u_t=\log \det \left( U \right) -\log \det (G)-F \end{aligned}$$

and then

$$\begin{aligned} \frac{1}{\kappa }u_t(x_0,t_0)=\log \prod _{r=1}^{n} u_{r{\bar{r}}}(x_0,t_0)-\log \det (G(x_0))-F(x_0). \end{aligned}$$

Lemma 5 implies that $|u_t|$ is uniformly bounded and we deduce that

$$\begin{aligned} \frac{1}{C} \le \prod _{r=1}^{n}u_{r{\bar{r}}}(x_0,t_0) \le C. \end{aligned}$$

Thus in particular by the geometric-arithmetic mean inequality we have $\sum _{r=1}^{n}u_{r{\bar{r}}}(x_0,t_0) \ge C$. Since

$$\begin{aligned} \Delta _{\varphi }v(x_0,t_0)=\frac{\kappa }{n}\sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}(x_0,t_0)}v_{r{\bar{r}} }(x_0), \end{aligned}$$

by (8) we finally deduce

$$\begin{aligned} \sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}(x_0,t_0)}\le C. \end{aligned}$$

Therefore

$$\begin{aligned} \Delta _{{\hat{g}}}u(x_0,t_0)=\frac{\kappa }{n}\sum _{r=1}^n u_{r{\bar{r}}}(x_0,t_0)\le \frac{\kappa }{n} \frac{1}{(n-1)!} \left( \sum _{r=1}^n\frac{1}{u_{r{\bar{r}}}(x_0,t_0)} \right) ^{n-1}\prod _{r=1}^{n}u_{r{\bar{r}}}(x_0,t_0)\le C. \end{aligned}$$

It follows

$$\begin{aligned} 2\sqrt{\Delta _{{\hat{g}}} u(x,t)}\le C+\varphi (x,t)-\varphi (x_0,t_0)\le C+\mathrm {osc}\varphi \qquad \text{ in } M\times [0,T'], \end{aligned}$$

from which, using Lemma 6, we get

$$\begin{aligned} \Delta _{{\hat{g}}} u\le C \end{aligned}$$

for a uniform C and the claim is proved. $\square$

Lemma 8

Assume that (I, J, K) is locally flat and that there exists a hyperhermitian metric ${\hat{g}}$ on (M, I, J, K) such that

$$\begin{aligned} \Delta _{{\hat{g}}}\varphi \le C \end{aligned}$$

for a uniform constant C. Then for $0<\alpha <1$ we have

$$\begin{aligned} \Vert \nabla ^2\varphi \Vert _{C^{\alpha }}\le C \end{aligned}$$

for a uniform constant C.

Proof

We prove the result by applying [7, Theorem 5.1]. First we state some algebraic preliminaries (which are analogous to the complex case [25, Section 2]). Note that, in the notation introduced in Sect. 2, the real representation $\iota :{\mathbb {H}}^{n,n}\rightarrow \{H\in \mathbb {R}^{4n,4n}:I_0HI_0=J_0HJ_0=K_0HK_0=-H\}$ of quaternionic matrices is monotonic in the sense that when $H_1,H_2$ are hyperhermitian one has

$$\begin{aligned} H_1\le H_2\Rightarrow \iota (H_1)\le \iota (H_2), \end{aligned}$$

where $H_1 \le H_2$ means that all the eigenvalues of $H_2-H_1$ are non-negative.

Let $\mathrm{p}:\mathbb {R}^{4n,4n}\rightarrow \{H\in \mathbb {R}^{4n,4n}:I_0HI_0=J_0HJ_0=K_0HK_0=-H\}$ be the projection defined as

$$\begin{aligned} \mathrm{p}(N):=\frac{1}{4}(N-I_0NI_0-J_0NJ_0-K_0NK_0). \end{aligned}$$

Then for any real valued smooth function f and any hyperhermitian matrix H we have

$$\begin{aligned} \iota (\mathrm{Hess}_{\mathbb {H}}f)=4 \mathrm{p}(\mathrm{Hess}_{\mathbb {R}}f),\quad \det (\iota (H))=(\det H)^4. \end{aligned}$$

Thus, once local quaternionic coordinates are fixed, working as in the proof of Lemma 7, we can rewrite Eq. (3) as

$$\begin{aligned} \frac{1}{\kappa }u_t=\frac{1}{4} \log \det \left( 4\mathrm{p}(\mathrm{Hess}_{\mathbb {R}}u)\right) -\log \det (G)-F, \end{aligned}$$

where $u=\kappa v+\kappa \varphi$ and v is a HKT potential of $\Omega$. We rewrite the last equation as

$$\begin{aligned} \frac{1}{\kappa }u_t=\Phi (\mathrm{p}(\mathrm{Hess}_{\mathbb {R}}u))-\log \det (G)-F \end{aligned}$$

(9)

where for $N\in \mathrm{Sym}(4n,\mathbb {R})$ such that $\det N>0$ we set

$$\begin{aligned} \Phi (N)=\frac{1}{4} \log \det \left( 4 N\right) . \end{aligned}$$

Fix positive constants $C_1<C_2$ and let

$$\begin{aligned} {\mathcal {E}}:=\left\{ N\in \mathrm{Sym}(4n,\mathbb {R}):C_1 {\mathbbm {1}}_{4n}\le N\le C_2{\mathbbm {1}}_{4n}\right\} . \end{aligned}$$

Then ${\mathcal {E}}$ is a compact convex subset of $\mathrm{Sym}(4n,\mathbb {R})$. We observe that $\Phi$ and $\mathrm{p}$ satisfy the assumptions in [7, Theorem 5.1]. Indeed

• $\Phi$ is uniformly elliptic in ${\mathcal {E}}$;

• $\Phi$ is concave in ${\mathcal {E}}$;

• $\mathrm{p}$ is linear;

• if $N\ge 0$, then $\mathrm{p}(N)\ge 0$ and $C^{-1} \Vert N\Vert \le \Vert \mathrm{p}(N)\Vert \le C\Vert N\Vert$ for C uniform.

Therefore, if we show that $\mathrm{p}(\mathrm{Hess}_{\mathbb {R}}u)\in {\mathcal {E}}$ for a suitable choice of $C_1$ and $C_2$ Eq. (9) belongs to the class of equations considered in [7, Theorem 5.1].

Without loss of generality we can fix $x_0\in M$ and assume that ${\hat{G}}$ is the identity at $x_0$. Our assumption $\Delta _{{\hat{g}}} \varphi \le C$ implies

$$\begin{aligned} \sum _{r=1}^n u_{r{\bar{r}}}\le C \end{aligned}$$

(10)

at $x_0$ for a uniform $C>0$ and thus

$$\begin{aligned} \mathrm{Hess}_{{\mathbb {H}}}u\le C {\mathbbm {1}}_n. \end{aligned}$$

On the other hand, Eq. (3) writes as

$$\begin{aligned} \frac{1}{\kappa }u_t=\log \det (\mathrm{Hess}_{{\mathbb {H}}}u)-\log \det (G)-F. \end{aligned}$$

Thus by Lemma 5

$$\begin{aligned} \prod _{i=1}^n \lambda _i=\det (\mathrm{Hess}_{{\mathbb {H}}}u)\ge \det (G)\mathrm {e}^{-\frac{1}{\kappa } |u_t|+F} \ge C, \end{aligned}$$

where $\lambda _1,\dots ,\lambda _n$ are the eigenvalues of $\mathrm{Hess}_{{\mathbb {H}}}u$ and $C>0$ is a uniform constant. From (10) we also infer $\sum _{i=1}^n \lambda _i \le C$ at $x_0$ which then implies a uniform lower bound for each $\lambda _i$ at the point $x_0$, but such bound does not depend on $x_0$.

Therefore

$$\begin{aligned} C_1{\mathbbm {1}}_n\le \mathrm{Hess}_{{\mathbb {H}}}u\le C_2{\mathbbm {1}}_n. \end{aligned}$$

By applying $\iota$ we get

$$\begin{aligned} C_1{\mathbbm {1} }_{4n}\le 4\mathrm{p}(\mathrm{Hess}_{{\mathbb {R}}}u)\le C_2{\mathbbm {1}}_{4n}. \end{aligned}$$

Then we can work as in the proof of [8, Lemma 6.1].

We assume that the domain of u is $B \times [0,T)$ with B diffeomorphic to the unit ball in $\mathbb {R}^{4n}$. If $T<1$, then Lemma 5 implies

$$\begin{aligned} |u|\le CT+C\le C \end{aligned}$$

for a uniform C and [7, Theorem 5.1] implies the result. If $T\ge 1$ we define, for any $a\in (0,T-1)$

$$\begin{aligned} {\hat{u}}(x,t): =u(x,t+a)-\inf _{B\times [a,a+1)} u(x,t) \end{aligned}$$

for all $t\in [0,1)$. We immediately deduce

$$\begin{aligned} \frac{1}{\kappa }{\hat{u}}_t=\log \det (\mathrm{Hess}_\mathbb {H}{\hat{u}})- \log \det (G) -F, \qquad \sup _{B\times [0,1)} | {\hat{u}}(x,t)| \le C. \end{aligned}$$

Invoking again [7, Theorem 5.1], chosen $\varepsilon \in (0,\frac{1}{2})$ and $\alpha \in (0,1)$ we have

$$\begin{aligned} \Vert \nabla ^2u\Vert _{C^{\alpha }(B\times [a+\varepsilon ,a+1))}= \Vert \nabla ^2{\hat{u}}\Vert _{C^{\alpha }(B\times [\varepsilon ,1))}\le C \end{aligned}$$

where the constant C depends on $\varepsilon$ and $\alpha$. As a was chosen arbitrarily in $(0,T-1)$ we have

$$\begin{aligned} \Vert \nabla ^2 u\Vert _{C^{\alpha }(B\times [\varepsilon ,T))}\le C, \end{aligned}$$

and the lemma follows.$\square$

Lemma 9

Assume that there exists $0<\alpha <1$ such that

$$\begin{aligned} \Vert \nabla ^2\varphi \Vert _{C^{\alpha }}\le C \end{aligned}$$

for a uniform constant C. Then $T=\infty$ and for every $k\ge 1$

$$\begin{aligned} \Vert \nabla ^k\varphi \Vert _{C^{0}}\le C \end{aligned}$$

for a uniform constant C.

Proof

Our assumptions imply that the spatial derivatives of $\varphi$ satisfy a uniformly parabolic equation and uniform bounds on $\Vert \nabla ^k\varphi \Vert _{C^{0}}$ with $k\ge 1$ follow by Schauder theory and a standard bootstrapping argument.

Now we shall prove the long-time existence. Assume by contradiction that the maximal time interval [0, T) of existence of $\varphi$ is bounded. Then the achieved estimates and short-time existence would allow us to extend $\varphi$ past T, which is a contradiction, thus $T=\infty$. $\square$

Lemma 10

Assume $T=\infty$ and that $\Vert \nabla ^k\varphi \Vert _{C^0}$ is uniformly bounded for every $k\ge 1$. Then

$$\begin{aligned} \tilde{\varphi }:=\varphi -\int _M \varphi \Omega ^n \wedge \bar{\Theta }\end{aligned}$$

converges in $C^{\infty }$-topology to a smooth function $\tilde{\varphi }_\infty$. Moreover if

$$\begin{aligned} b:=\frac{\int _M\Omega ^n\wedge \bar{\Theta }}{\int _M\mathrm{e}^F\Omega ^n\wedge \bar{\Theta }}, \end{aligned}$$

then $(\tilde{\varphi }_\infty ,b)$ solves the quaternionic Monge–Ampère equation (2).

Proof

Let

$$\begin{aligned} f(t):=\int _M \varphi _t \,\Omega _{\varphi }^n\wedge \bar{\Theta }= \int _M\log \frac{\Omega _\varphi ^n}{\Omega ^n}\, \Omega _{\varphi }^n\wedge \bar{\Theta }-\int _{M} F\,\Omega _{\varphi }^n\wedge \bar{\Theta }. \end{aligned}$$

Using (6) we have

$$\begin{aligned} \begin{aligned} f'&=n\int _M \left( \Delta _{\varphi }\varphi _t+\log \frac{\Omega _\varphi ^n}{\Omega ^n}\Delta _{\varphi }\varphi _t-F\Delta _{\varphi }\varphi _t\right) \,\Omega _{\varphi }^n\wedge \bar{\Theta }\\&=n\int _M \varphi _t \Delta _{\varphi }\varphi _t\,\Omega _{\varphi }^n\wedge \bar{\Theta }=n\!\!\int _M \varphi _t \partial \partial _J\varphi _t \wedge \Omega _{\varphi }^{n-1}\wedge \bar{\Theta }=-n\!\int _M \partial \varphi _t \wedge \partial _J\varphi _t\,\wedge \Omega _{\varphi }^{n-1}\!\wedge \!\bar{\Theta }\\&= - \frac{1}{2} \int _M |\partial \varphi _t|^2_{g_{\varphi }} \Omega _\varphi ^n\wedge \bar{\Theta }. \end{aligned} \end{aligned}$$

Differentiating again we obtain

$$\begin{aligned} f''=- \frac{1}{2} \int _M \tfrac{\partial }{\partial t}|\partial \varphi _t|^2_{g_{\varphi }} \Omega _\varphi ^n\wedge \bar{\Theta }-\frac{n}{2} \int _M |\partial \varphi _t|^2_{g_{\varphi }}\Delta _{\varphi }\varphi _t\, \Omega _\varphi ^n\wedge \bar{\Theta }. \end{aligned}$$

Now

$$\begin{aligned} \frac{\partial }{\partial t} |\partial \varphi _t|^2_{g_{\varphi }}= -g_{\varphi }\left( \tfrac{\partial }{\partial t}g_{\varphi }, \partial \varphi _t\otimes \bar{\partial }\varphi _t\right) +2\mathrm {Re}\,g_{\varphi }(\partial \varphi _{tt},\bar{\partial }\varphi _{t}). \end{aligned}$$

(11)

For the first term of (11) Cauchy–Schwarz inequality gives

$$\begin{aligned} -g_{\varphi }\left( \tfrac{\partial }{\partial t}g_{\varphi }, \partial \varphi _t\otimes \bar{\partial }\varphi _t\right) \le |\tfrac{\partial }{\partial t}g_\varphi |_{g_\varphi }|\partial \varphi _t|_{g_\varphi }^2 \le C|\partial \varphi _t|_{g_\varphi }^2 \end{aligned}$$

because $\Omega _\varphi$ and $g_{\varphi }$ are related by (1) and $\Omega _{\varphi }$ and $\tfrac{\partial }{\partial t}\Omega _{\varphi }$ are uniformly bounded in $C^{k}$-norm for every k. For the second term of (11) using (6) again we have

$$\begin{aligned}&-n\mathrm {Re}\int _M g_{\varphi }(\partial \Delta _\varphi \varphi _t,\bar{\partial }\varphi _{t}) \Omega _\varphi ^n\wedge \bar{\Theta }=- 2n^2 \mathrm {Re}\,\int _M \partial \Delta _\varphi \varphi _t \wedge \partial _J \varphi _{t} \wedge \Omega _\varphi ^{n-1}\wedge \bar{\Theta }\\&\quad = 2n^2 \mathrm {Re}\,\int _M \Delta _\varphi \varphi _t \, \partial \partial _J \varphi _{t} \wedge \Omega _\varphi ^{n-1}\wedge \bar{\Theta }= 2n^2 \int _M (\Delta _\varphi \varphi _t)^2\, \Omega _\varphi ^{n}\wedge \bar{\Theta }\end{aligned}$$

therefore

$$\begin{aligned} f'' \ge -C\int _{M}|\partial \varphi _t|_{g_\varphi }^2\,\Omega _\varphi ^n\wedge \bar{\Theta }. \end{aligned}$$

Thus we have a non increasing smooth function $f: [0,+\infty ) \rightarrow {\mathbb {R}}$ which is bounded from below and such that $f''(t) \ge C f'(t)$ for some positive constant C. This implies that $\lim _{t \rightarrow +\infty } f'(t) = 0$, i.e.

$$\begin{aligned} \lim _{t\rightarrow \infty } \int _M|\partial \varphi _t|^2_{g_\varphi }\Omega _\varphi ^n\wedge \bar{\Theta }=0. \end{aligned}$$

(12)

Now, $\tilde{\varphi }$ has a uniform $C^\infty$ bound and Ascoli-Arzelà theorem implies that there exists a sequence $\{t_k\}\subseteq \mathbb {R}$, $t_k\rightarrow \infty$ such that $\tilde{\varphi }(\cdot , t_k)$ converges to some $\tilde{\varphi }_\infty$ in $C^\infty$-topology. Since

$$\begin{aligned} \tilde{\varphi }_{t}=\log \frac{\Omega _{\tilde{\varphi }}^n}{\Omega ^n}-F-\int _M \left( \log \frac{\Omega _{\tilde{\varphi }}^n}{\Omega ^n}-F\right) \Omega ^n\wedge \bar{\Theta }, \end{aligned}$$

by (12) we get

$$\begin{aligned} 0= \lim _{t\rightarrow \infty } \int _M |\partial \tilde{\varphi }_t|^2_{g_{\tilde{\varphi }}}\Omega _{\tilde{\varphi }}^n\wedge \bar{\Theta }= \int _M \left| \partial \left( \log \frac{\Omega _{\tilde{\varphi }_\infty }^n}{\Omega ^n}-F\right) \right| ^2_{g_{\tilde{\varphi }}}\Omega _{\tilde{\varphi }_\infty }^n \wedge \bar{\Theta }. \end{aligned}$$

It follows that

$$\begin{aligned} \log \frac{\Omega _{\tilde{\varphi }_\infty }^n}{\Omega ^n}-F=C \end{aligned}$$

for some constant C, so that

$$\begin{aligned} \Omega _{\tilde{\varphi }_\infty }^n=\mathrm{e}^{F+C}\Omega ^n. \end{aligned}$$

This means that $(\tilde{\varphi }_\infty ,\mathrm{e}^C)$ solves the quaternionic Calabi–Yau equation. Finally, we prove that $\lim _{t\rightarrow \infty }\tilde{\varphi }=\tilde{\varphi }_\infty$. Assume by contradiction that there exists $\varepsilon >0$ and a sequence $t_k\rightarrow \infty$ such that

$$\begin{aligned} \Vert \tilde{\varphi }(\cdot , t_k)-\tilde{\varphi }_{\infty }\Vert _{C^{\infty }}\ge \varepsilon \end{aligned}$$

for every $t_k$. We may assume that $\tilde{\varphi }(\cdot , t_k)$ converges in $C^{\infty }$-topology to $\tilde{\varphi }_{\infty }'$. Hence

$$\begin{aligned} \Omega _{\tilde{\varphi }'_\infty }^n=\mathrm{e}^{F+C'}\Omega ^n. \end{aligned}$$

Since $\tilde{\varphi }_\infty$ and $\tilde{\varphi }'_\infty$ solve the same quaternionic Calabi–Yau equation, from uniqueness follows $\tilde{\varphi }_\infty =\tilde{\varphi }'_\infty$ and the lemma is proved. $\square$

Proof of Theorem 1

We put together Lemmas 5–10 proved in this section. Lemmas 5, 6, 7 imply that if $\varphi$ solves (3), its quaternionic Laplacian $\Delta _{{\hat{g}}}\varphi$ with respect to the background hyperkähler metric ${\hat{g}}$ has a uniform upper bound. Hence Lemmas 8 and 9 can be applied and (3) has a long-time solution $\varphi$ such that $\Vert \nabla ^k\varphi \Vert _{C^0}$ is bounded for every $k\ge 1$. Therefore, taking $\Theta =\Omega _{{\hat{g}}}^n$, Lemma 10 implies the last part of the statement. $\square$

4 Further developments

On a hypercomplex manifold (M, I, J, K) with a HKT form $\Omega$ a (1, 0)-form $\vartheta$ is defined by the relation

$$\begin{aligned} {\bar{\partial }}\Omega ^{n}={\bar{\vartheta }}\wedge \Omega ^{n}. \end{aligned}$$

If the canonical bundle is holomorphically trivial then we can take $h\in C^{\infty }(M)$ such that $\partial J{\bar{\vartheta }}=\partial \partial _J h.$ Now the proof of Lemma 10 suggests to consider on a $\mathrm{SL}(n,\mathbb {H})$-manifold with holomorphic q-real volume form $\Theta$ the operator ${\mathcal {M}}$ acting on HKT forms in the $\partial \partial _J$-class of a fixed HKT form $\Omega _0$ as

$$\begin{aligned} {\mathcal {M}}(\Omega _{\varphi }){:}{=}\int _{M}\log \frac{\Omega _{\varphi }^{n}}{\Omega ^{n}_{0}}\,\Omega _{\varphi }^{n}\wedge {\bar{\Theta }}-\int _{M} h\,\Omega _{\varphi }^{n}\wedge {\bar{\Theta }}. \end{aligned}$$

This is related to the following geometric flow of HKT forms

$$\begin{aligned} \partial _t\Omega =-\partial J{\bar{\vartheta }},\quad \Omega (0)=\Omega _0. \end{aligned}$$

(13)

Indeed working as in the proof of Lemma 10 one can observe that ${\mathcal {M}}$ is decreasing along flow (13), thus ${\mathcal {M}}$ plays a role similar to that of the Mabuchi functional in Calabi–Yau geometry. It is easy to prove that the gradient flow of ${\mathcal {M}}$ can be expressed in terms of the quaternionic potential $\varphi$ as

$$\begin{aligned} \varphi _{t}=\frac{J {\bar{\partial }}_J{\bar{\vartheta }}_\varphi \wedge \Omega _\varphi ^{n-1}}{\Omega ^n_\varphi },\quad \varphi (0)=0 \end{aligned}$$

(14)

and the fixed points of ${\mathcal {M}}$ are the HKT forms in the $\partial \partial _J$-class of $\Omega _0$ which are balanced, i.e. which induce a balanced metric.

From this perspective we believe that the operator ${\mathcal {M}}$ and the flow (14) could give new insights in the search of canonical HKT metrics and this will be the subject of a future work.