5.12 Kiselman's minimum principle53A48113C242

Proof. We first treat smooth functions with positive definite real Hessian in $t$ that tend to $+\infty $ at the boundary of the real fibers (Step 1). We then approximate $u$ by such functions on subdomains exhausting $\Omega $ (Step 2) and pass to the limit (Step 3).

Step 1: a smooth case. In this step, $\Omega $ may be any open set invariant under imaginary translations whose real fibers are convex. Suppose that $u$ is smooth and plurisubharmonic, invariant under imaginary translations, and that for every $z\in p(\Omega )$ the function $t\mapsto u(z,t)$ has positive definite real Hessian on $V_ z$ and tends to $+\infty $ at the boundary of $V_ z$ and, if $V_ z$ is unbounded, as $|t|\to \infty $. We show that $v$ is plurisubharmonic on each connected component of $p(\Omega )$.

Fix $z\in p(\Omega )$ and put $f(t)=u(z,t)$ for $t\in V_ z$. Since its Hessian is positive definite, $f$ is strictly convex on every segment in the open convex set $V_ z$, hence on $V_ z$. Its sublevel sets $\{ t\in V_ z:f(t)\leq c\} $ are compact: they are bounded, because $f\to +\infty $ as $|t|\to \infty $ when $V_ z$ is unbounded, and the limit in $\mathbb R^ k$ of a convergent sequence in such a set cannot lie on the boundary of $V_ z$, because $f\to +\infty $ there, so it lies in $V_ z$ and, since $f$ is continuous on $V_ z$, in the sublevel set. Hence $f$ attains its minimum on the nonempty set $V_ z$, and by strict convexity at a unique point $g(z)$.

This point is the only critical point of $f$: if $\nabla f(t_1)=0$, then the convex function $s\mapsto f(t_1+s(t_2-t_1))$ has derivative zero at $s=0$, hence $f(t_2)\geq f(t_1)$ for every $t_2\in V_ z$, and $t_1=g(z)$. Fix $z_0\in p(\Omega )$. Since the $t$-Hessian is invertible, the implicit function theorem gives a smooth solution $t=G(z)$ of $\partial _ tu(z,t)=0$ near $z_0$ with $G(z_0)=g(z_0)$, and $G=g$ near $z_0$ because $g(z)$ is the only critical point. Hence $g$ and $v(z)=u(z,g(z))$ are smooth on $p(\Omega )$.

At the point $(z,g(z))$, let $H=(\partial ^2u/\partial t_ a\partial t_ b)$, $B=(\partial ^2u/\partial z_ j\partial t_ a)$ and $C=(\partial ^2u/\partial z_ j\partial \bar z_ l)$. Thus $H$ is a real symmetric positive definite $k\times k$ matrix and $B$ is a complex $m\times k$ matrix; since $u$ is real, $(\partial ^2u/\partial t_ a\partial \bar z_ j)=B^*$, the conjugate transpose of $B$. Differentiating the identity $\partial _ tu(z,g(z))=0$ with respect to $\bar z$ gives $B^*+Hg_{\bar z}=0$, where $g_{\bar z}=(\partial g_ a/\partial \bar z_ j)$; hence $g_{\bar z}=-H^{-1}B^*$. By the chain rule and the same identity, $v_{z_ j}(z)=u_{z_ j}(z,g(z))$, and differentiating once more with respect to $\bar z$ gives

On the other hand, for functions of $z$ and $\operatorname{Re}w$ we have $\partial _{w_ a}=\partial _{\bar w_ a}=\frac12\partial _{t_ a}$. Hence the complex Hessian of $u$ with respect to $(z,w)$, which is positive semidefinite by proposition 5.9.2(1), is

Evaluating the associated Hermitian quadratic form at $(\xi ,-2H^{-1}B^*\xi )$, $\xi \in \mathbb {C}^ m$, gives $\xi ^*(C-BH^{-1}B^*)\xi \geq 0$. Thus $v_{z\bar z}\geq 0$, and the smooth function $v$ is plurisubharmonic on each component of $p(\Omega )$ by proposition 5.9.2(1).

Step 2: regularization. Now let $u$ be as in the theorem. For $\varepsilon >0$ let $u_\varepsilon =u*\rho _\varepsilon $ and $\psi _\varepsilon =\psi *\rho _\varepsilon $ on $\Omega _\varepsilon =\{ x\in \Omega :\operatorname {dist}(x,\mathbb {C}^{m+k}\setminus \Omega )>\varepsilon \} $, where $\rho _\varepsilon $ is the radial mollifier of section 2.5. Imaginary translations are isometries preserving $\Omega $, and convolution commutes with translations, so $\Omega _\varepsilon $, $u_\varepsilon $ and $\psi _\varepsilon $ are invariant under imaginary translations. By proposition 5.9.2, the functions $u_\varepsilon $ and $\psi _\varepsilon $ are smooth and plurisubharmonic on each connected component of $\Omega _\varepsilon $, so their complex Hessians are positive semidefinite at every point, and they decrease to $u$ and $\psi $ as $\varepsilon \downarrow 0$; in particular $u_\varepsilon \geq u$ and $\psi _\varepsilon \geq \psi $.

The real fibers of $\Omega _\varepsilon $ are convex. Indeed, $(z,t)\in \Omega _\varepsilon $ if and only if the closed ball of radius $\varepsilon $ about $(z,t)$ lies in $\Omega $ (if it does, its distance to the closed complement of $\Omega $ is positive, because it is compact); by invariance this means that $(z+y,t+y')\in \Omega _{\mathbb R}$ for all $(y,y')\in \mathbb {C}^ m\times \mathbb R^ k$ with $|y|^2+|y'|^2\leq \varepsilon ^2$. Hence the real fiber of $\Omega _\varepsilon $ over $z$ is the intersection of the convex sets $V_{z+y}-y'$ over these $(y,y')$. Moreover, on each real fiber of $\Omega _\varepsilon $ the function $t\mapsto \psi _\varepsilon (z,t)$ is convex, since by invariance its real Hessian is four times the complex Hessian of $\psi _\varepsilon $ in $w$, which is positive semidefinite, and a smooth function with positive semidefinite Hessian on an open convex set is convex along every segment.

For $c\in \mathbb R$ let $K_ c=\{ (z,t)\in \Omega _{\mathbb R}:\psi (z,t)\le c\} $; it is a compact subset of $\Omega $. Choose $\varepsilon _ j\downarrow 0$, $j\geq 1$, such that $\operatorname {dist}(K_{j+1},\mathbb {C}^{m+k}\setminus \Omega )>4\varepsilon _ j$ whenever $K_{j+1}\neq \varnothing $. Since $\psi $ and the distance to the complement of $\Omega $ are invariant under imaginary translations, the same bound holds for the set $\{ x\in \Omega :\psi (x)\leq j+1\} =K_{j+1}+\mathrm{i}\mathbb R^ k$. Define

The open set $G_ j$ is invariant under imaginary translations, and its real fiber over $z$ is a sublevel set of the convex function $\psi _{\varepsilon _ j}(z,\, \cdot \, )$ on the convex real fiber of $\Omega _{\varepsilon _ j}$, hence convex. Since $\psi \leq \psi _{\varepsilon _ j}$, the real slice of $G_ j$ is contained in the compact set $K_ j\subset K_{j+1}$, which lies in $\Omega _{\varepsilon _ j}$ at positive distance from its complement. Consequently, for each $z$, the closure of the real fiber of $G_ j$ over $z$ is a compact subset of the real fiber of $\Omega _{\varepsilon _ j}$, and $\psi _{\varepsilon _ j}(z,\, \cdot \, )=j$ on its boundary.

The function $U_ j$ is smooth, plurisubharmonic and invariant under imaginary translations. Indeed, by the chain rule the complex Hessian of $1/(j-\psi _{\varepsilon _ j})$ has the entries

where the subscripts denote derivatives with respect to the coordinates of $(z,w)$. Since $j-\psi _{\varepsilon _ j}>0$ on $G_ j$ and $(\psi _{\varepsilon _ j})_{\bar b}=\overline{(\psi _{\varepsilon _ j})_ b}$, this matrix is the sum of positive multiples of the complex Hessian of $\psi _{\varepsilon _ j}$ and of a positive semidefinite matrix of rank at most one, hence positive semidefinite. Moreover, $|\operatorname{Re}w|^2$ has the positive semidefinite complex Hessian in $(z,w)$ given by the diagonal matrix with entries $0$ in the $z$-directions and $\frac12$ in the $w$-directions. Hence $U_ j$ is plurisubharmonic on each connected component of $G_ j$, by proposition 5.9.2(1).

The real $t$-Hessian of $U_ j$ is positive definite: the $t$-Hessians of $u_{\varepsilon _ j}$ and $1/(j-\psi _{\varepsilon _ j})$ are positive semidefinite, as for $\psi _\varepsilon $ above, and that of $|t|^2/j$ is $(2/j)I$. On each real fiber of $G_ j$, which is bounded, $U_ j(z,t)\to +\infty $ at the boundary, because $1/(j-\psi _{\varepsilon _ j})\to +\infty $ there while the other two terms are bounded below on the compact closure of the fiber. Hence Step 1 applies to $U_ j$ on $G_ j$, and

is plurisubharmonic on each connected component of $p(G_ j)$.

Step 3: passage to the limit. The sets $G_ j$ increase to $\Omega $. Indeed, $\varepsilon _{j+1}\leq \varepsilon _ j$ gives $\Omega _{\varepsilon _ j}\subset \Omega _{\varepsilon _{j+1}}$ and $\psi _{\varepsilon _{j+1}}\leq \psi _{\varepsilon _ j}<j<j+1$ on $G_ j$. Every point $x\in \Omega $ lies in $\Omega _{\varepsilon _ j}$ for large $j$, and $\psi _{\varepsilon _ j}(x)\to \psi (x)<\infty $, so $x\in G_ j$ for large $j$. On $G_ j$ we have $U_{j+1}\leq U_ j$, since $u_{\varepsilon _{j+1}}\leq u_{\varepsilon _ j}$, $|\operatorname{Re}w|^2/(j+1)\leq |\operatorname{Re}w|^2/j$ and $0<j-\psi _{\varepsilon _ j}\leq j+1-\psi _{\varepsilon _{j+1}}$. Hence the functions $v_ j$ decrease on the increasing open sets $p(G_ j)$. They satisfy $v_ j\geq v$, since $U_ j\geq u_{\varepsilon _ j}\geq u$. Conversely, let $z\in p(\Omega )$ and $t\in V_ z$. Then $(z,t)\in G_ j$ for large $j$, and $U_ j(z,t)\to u(z,t)$, also when $u(z,t)=-\infty $, because $u_{\varepsilon _ j}(z,t)\to u(z,t)$ and the other two terms tend to zero. Hence $\lim _ jv_ j(z)\leq u(z,t)$ for every $t\in V_ z$, and therefore $v_ j\downarrow v$ on $p(\Omega )$.

Let $B'\Subset p(\Omega )$ be a ball. Since $\overline{B'}$ is compact and covered by the increasing open sets $p(G_ j)$, we have $B'\subset p(G_ j)$ for large $j$, and the $v_ j$ with such $j$ form a decreasing sequence of plurisubharmonic functions on $B'$. By theorem 5.10.1, $v$ is plurisubharmonic or identically $-\infty $ on $B'$. Since every point of the connected open set $p(\Omega )$ lies in such a ball, lemma 5.9.4 shows that $v$ is plurisubharmonic or identically $-\infty $ on $p(\Omega )$; this proves the theorem. $\square$

Proof. It suffices to prove the assertion on every ball $B(z_0,r)\Subset D$: since $D$ is connected, lemma 5.9.4 then gives it on $D$. We construct a function $\psi $ as in theorem 5.12.1 for the domain $B(z_0,r)\times (V+\mathrm{i}\mathbb R^ k)$, which is invariant under imaginary translations and has real fibers $V$.

We first need a continuous convex function $b\geq 0$ on $V$ with compact sublevel sets. If $V=\mathbb R^ k$, take $b(t)=|t|^2$. Otherwise let $\delta (t)=\operatorname {dist}(t,\mathbb {R}^ k\setminus V)$, which is continuous and positive on $V$. It is concave on $V$: if the balls $B(t_1,r_1)$ and $B(t_2,r_2)$ lie in $V$ and $0\leq \lambda \leq 1$, then by convexity $V$ contains $\lambda B(t_1,r_1)+(1-\lambda )B(t_2,r_2)=B(\lambda t_1+(1-\lambda )t_2,\lambda r_1+(1-\lambda )r_2)$, so $\delta (\lambda t_1+(1-\lambda )t_2)\geq \lambda r_1+(1-\lambda )r_2$, and letting $r_ i\uparrow \delta (t_ i)$ gives concavity. Consequently $1/\delta $ is convex: since $x\mapsto 1/x$ is decreasing and convex on $(0,\infty )$,

Put $b(t)=|t|^2+1/\delta (t)$. For $c>0$ the set $\{ b\leq c\} $ is closed in $V$ and contained in the compact set $\{ t:|t|^2\leq c,\ \delta (t)\geq 1/c\} \subset V$, hence compact.

A continuous convex function $f$ on $V$ defines a plurisubharmonic function $f(\operatorname{Re}w)$ on every domain of the form $D'\times (V+\mathrm{i}\mathbb R^ k)$. Indeed, $f(\operatorname{Re}w)$ is continuous, and its restriction to an affine complex line is $f\circ A$ for a real affine map $A$ from an open subset of $\mathbb {C}$ to $V$. If the closed disc of radius $r$ about $\zeta _0$ lies in the domain of $A$, then, since $A$ is affine, $A(\zeta _0)$ is the midpoint of $A(\zeta _0+r\mathrm{e}^{\mathrm{i}\theta })$ and $A(\zeta _0-r\mathrm{e}^{\mathrm{i}\theta })$ for every $\theta $, and the convexity of $f$ gives

Averaging over $\theta \in [0,2\pi ]$, we obtain the submean inequality for the continuous function $f\circ A$ on this circle.

Now put $\psi (z,w)=(r^2-|z-z_0|^2)^{-1}+b(\operatorname{Re}w)$. This function is continuous, positive and invariant under imaginary translations. It is plurisubharmonic as a sum of two plurisubharmonic functions (proposition 5.9.5). The second term is plurisubharmonic by the preceding paragraph. For the first, $x\mapsto (r^2-x)^{-1}$ is convex and increasing on $(-\infty ,r^2)$ and $|z-z_0|^2$ has the complex Hessian $I$, so by the chain rule the complex Hessian of $(r^2-|z-z_0|^2)^{-1}$ is positive semidefinite, and proposition 5.9.2(1) applies. For $c>0$, the set $\{ (z,t)\in B(z_0,r)\times V:\psi (z,t)\leq c\} $ is closed in $B(z_0,r)\times V$ and, since both terms of $\psi $ are nonnegative, contained in the compact set $\{ |z-z_0|^2\leq r^2-1/c\} \times \{ b\leq c\} $; it is therefore compact, and for $c\leq 0$ it is empty. Hence theorem 5.12.1 applies on $B(z_0,r)\times (V+\mathrm{i}\mathbb R^ k)$, and shows that $\inf _{t\in V}u(z,t)$ is plurisubharmonic or identically $-\infty $ on $B(z_0,r)$.

Finally, if $\ell $ is real affine, then $\ell (\operatorname{Re}w)$ is the real part of a complex affine function of $w$. Hence $-\ell (\operatorname{Re}w)$ is smooth with vanishing complex Hessian, so it is plurisubharmonic by proposition 5.9.2(1), and $u-\ell (\operatorname{Re}w)$ is plurisubharmonic by proposition 5.9.5. It is invariant under imaginary translations, and the first part applies to it. $\square$