The geometry of \(m\)-hyperconvex domains
The authors study the geometry of \(m\)-regular domains. Let \(\Omega\subset\mathbb C^n\) be a bounded domain, \(n\geq2\), let \(1\leq m\leq n\), and let \(\beta:=dd^c|z|^2\). Denote by \(\mathcal{SH}_m(\Omega)\) the space of all \(m\)-subharmonic functions on \(\Omega\), i.e., the set of all \(u\in\mathcal{SH}(\Omega)\) for which \(dd^cu\wedge\alpha_1\wedge\dots\wedge\alpha_{m-1}\wedge\beta^{n-m}\geq0\) in the sense of currents for all \((1,1)\)-forms \(\alpha_1,\dots,\alpha_{m-1}\) with constant coefficients such that \(\alpha_j^k\wedge\beta^{n-k}\geq0\), \(k=1,\dots,m\), \(j=1,\dots,m-1\). Among others the authors present the following two important theorems. --- The following conditions are equivalent: (1) \(\Omega\) is \(m\)-hyperconvex, i.e., \(\Omega\) possesses an \(m\)-subharmonic negative exhaustion function. (2) For every \(a\in\partial\Omega\) there exists a weak \(m\)-subharmonic barrier, i.e., an \(m\)-subharmonic function \(u:\Omega\longrightarrow(-\infty,0)\) such that \(\lim_{\Omega\ni z\to a}u(z)=0\). (3) \(\Omega\) admits a negative, smooth, and strictly \(m\)-subharmonic exhaustion function. (4) For any \(z\in\partial\Omega\) and \(\mu\in\mathcal J^m_z\) we have \(\operatorname{supp}\mu\subset\partial\Omega\), where \(\mathcal J^m_z\) denotes the se of all non-negative Borel measures on \(\overline\Omega\) such that \(u(z)\leq\int ud\mu\) for all \(u\in\mathcal{SH}_m(\Omega)\cap\mathcal C(\overline\Omega)\). --- The following properties are equivalent: (1) For every \(a\in\partial\Omega\) the set \(\partial\Omega\) is \(B_m\)-regular at \(a\), i.e., \(\lim_{\Omega\ni z\to a}\operatorname{BP}^m_f(z)=f(a)\) for every \(f\in\mathcal C(\partial\Omega,\mathbb R)\), where \(\operatorname{BP}^m_f(z):=\sup\{v(z): v\in\mathcal{SH}_m(\Omega): \forall_{\xi\in\partial\Omega}: \limsup_{\Omega\ni\zeta\to\xi}v(\zeta)\leq f(\xi)\}\). (2) For every \(a\in\partial\Omega\) there exists a strong \(m\)-subharmonic barrier, i.e., an \(m\)-subharmonic function \(u:\Omega\longrightarrow\mathbb R\) such that \(\lim_{\Omega\ni z\to a}u(z)=0\) and \(\limsup_{\Omega\ni z\to w}u(z)<0\) for all \(w\in\overline\Omega\setminus\{a\}\). (3) \(\Omega\) admits a negative smooth function \(\varphi\in\mathcal{SH}_m(\Omega)\) such that \(\varphi-|z|^2\in\mathcal{SH}_m(\Omega)\). (4) \(\partial\Omega=\{z\in\overline\Omega: \mathcal J^m_z=\}\delta_z\}\}\).
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