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Theorem. Any $\omega$-nilpotent cover of a recurrent Riemannian manifold is Liouville. $\omega$-nilpotent ($\Gamma=\bigcup_{i=1}^{\infty}Z_{i}$, $Z_{i}$ normal in $\Gamma$, where $Z_{n+1}$ maps to th …

Let $L,\,\,J$ be two holomorphic line bundles over a compact Riemann surface $X$ of genus $g_X>0$ such that (1) $d_1:=\dim H^0\big(\operatorname{Hom}(L,J)\big)>0$ and $d_2:=\dim H^0\big(\operatorname{ …

Let $\bar{X}$ be a compact Riemann surface of genus $g$, and let $D:=\{p_{1},p_{2},\cdots,p_{n}\}$ be $n$ distinct points on $\bar{X}$. Define $X:=\bar{X}-D$ to be the punctured Riemann surface given …

Let $E$ be a rank two holomorphic vector bundle, consider the following extension of two holomorphic line bundles $$ \mathbb{E}:\ 0\rightarrow S\stackrel{j}{\rightarrow}E\stackrel{g}{\rightarrow} Q\ri …