Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function

Abstract

We show that every bounded domain $D$ in ${\mathbb{R}}^n$ with smooth $p$-convex boundary for $2\le p<n$ admits a smooth defining function $\rho$ which is $p$-plurisubharmonic on $\overline{D}$; if in addition $bD$ has no $p$-flat points then $\rho$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$ is $2$-convex then for any open connected conformal surface $M$ and conformal harmonic map $f:M\to \overline{D}$, either $f(M)\subset D$ or $f(M)\subset bD$. In particular, every conformal harmonic map ${\mathbb{D}}^{\ast }\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb{D}\to D$.

Bull. Sci. Math., in press

Cited by in corpus (1)

• Domains without parabolic minimal submanifolds and weakly hyperbolic domains