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Register a new userEffectiveness of strong openness property in $L^p$
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In this article, we obtain an effectiveness result of strong openness property in $L^p$ with some applications.
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In this article, we present a twisted version of strong openness property in $L^p$ with applications.
In this article, we establish a sharp effectiveness result of Demaillys strong openness conjecture. We also establish a sharp effectiveness result related to a conjecture posed by Demailly and Kollar.
In the present article, we obtain an optimal support function of weighted $L^2$ integrations on superlevel sets of weights of multiplier ideal sheaves, which implies the strong openness property of multiplier ideal sheaves.
In this note, we reveal that our solution of Demaillys strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll{a}r and Jonsson-Mustatu{a} implies the truth of twist
Teichmullers problem from 1944 is this: Given $xin [0,1)$ find and describe the extremal quasiconformal map $f:IDtoID$, $f|partial ID=identity$ and $f(0)=-xleq 0$. We consider this problem in the setting of minimisers of $L^p$-mean distortion. The classical result is that there is an extremal map of Teichmuller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x eq 0$. For the $L^p$-norm, when $p=1$ it is known that there can be no locally quasiconformal minimiser unless $x=0$. Here we show that for $1leq p<infty$ there is a minimiser in a weak class and an associated Ahlfors-Hopf holomorphic quadratic differential with a pole of order $1$ at $f(0)=r$. However, this minimiser cannot be in $W^{1,2}_{loc}(ID)$ unless $r=0$ and $f=identity$. Hence there is no locally quasiconformal minimiser. A similar statement holds for minimsers of the exponential norm of distortion. We also use our earlier work to show that as $ptoinfty$, the weak $L^p$-minimisers converge locally uniformly in $ID$ to the extremal quasiconformal mapping, and that as $pto 1$ the weak $L^p$-minimisers converge locally uniformly in $ID$ to the identity.
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