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Every smoothly bounded p-convex domain in R^n admits a p-plurisubharmonic defining function (2111.08113v2)

Published 15 Nov 2021 in math.CV and math.DG

Abstract: We show that every bounded domain $D$ in $\mathbb Rn$ 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 $\mathbb D*\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb D\to D$.

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