Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the space of constant functions. Well, by the existence of nonparabolic end $E$, i can build a harmonic $g$ function in $M- \Omega$( where $\Omega$ is subdomain compact of $M$ such that $E$ is a nonparabolic end of $M$). The function $g$ is limited and has finite Dirichlet integral. But i don't know as to use the fact which has only one nonparabolic end of $M$.
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$\begingroup$ Do you want to prove that if a manifold has only one non-parabolic end, then it has no bounded harmonic functions of finite energy? This is false. $\endgroup$– R WCommented May 30, 2019 at 0:46
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$\begingroup$ No, i would like to prove that : If the manifold has only one nonparabolic end then there exists a subspace of space of bounded harmonic functions with finite Dirichlet integral which is the space of constant functions. $\endgroup$– Cézar BezerrCommented May 30, 2019 at 1:22
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3$\begingroup$ What do you mean? The space of constant functions is a subspace of the space of bounded harmonic functions with finite Dirichlet integral. $\endgroup$– R WCommented May 30, 2019 at 2:18
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