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since question was bumped to the front page, I corrected the typo in the title
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Ricardo Andrade
Ricardo Andrade
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Extension of strictly plurisibharmonicplurisubharmonic functions on a Kähler manifold.

Changed Kahler to Kähler throughout, and added the 'kahler' tag.
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edit approved Aug 3, 2013 at 5:03
Michael Albanese
edit approved Aug 3, 2013 at 5:03
Michael Albanese
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Extension of strictly plurisibharmonic functions on a KahlerKähler manifold.

I was wondering, suppose I have a non-compact KahlerKähler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ such that $i\partial\bar{\partial}f>0$. Is it always possible for me to find a smooth function $h:M\longrightarrow\mathbb{R}$ such that $h|(M\backslash A)=f$$h|_{M\backslash A}=f$ and $i\partial\bar{\partial}h>0$ on the whole of $M$? If not in general, are there sufficient conditions on $M$ that will allow me to do this? Many thanks!

Extension of strictly plurisibharmonic functions on a Kahler manifold.

I was wondering, suppose I have a non-compact Kahler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ such that $i\partial\bar{\partial}f>0$. Is it always possible for me to find a smooth function $h:M\longrightarrow\mathbb{R}$ such that $h|(M\backslash A)=f$ and $i\partial\bar{\partial}h>0$ on the whole of $M$? If not in general, are there sufficient conditions on $M$ that will allow me to do this? Many thanks!

Extension of strictly plurisibharmonic functions on a Kähler manifold.

I was wondering, suppose I have a non-compact Kähler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ such that $i\partial\bar{\partial}f>0$. Is it always possible for me to find a smooth function $h:M\longrightarrow\mathbb{R}$ such that $h|_{M\backslash A}=f$ and $i\partial\bar{\partial}h>0$ on the whole of $M$? If not in general, are there sufficient conditions on $M$ that will allow me to do this? Many thanks!

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user4317
user4317
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Extension of strictly plurisibharmonic functions on a Kahler manifold.

I was wondering, suppose I have a non-compact Kahler manifold $M$ and suppose that outside some compact subset $A\subset M$, there exists a smooth function $f:M\backslash A\longrightarrow\mathbb{R}$ such that $i\partial\bar{\partial}f>0$. Is it always possible for me to find a smooth function $h:M\longrightarrow\mathbb{R}$ such that $h|(M\backslash A)=f$ and $i\partial\bar{\partial}h>0$ on the whole of $M$? If not in general, are there sufficient conditions on $M$ that will allow me to do this? Many thanks!