Consider a complete non-compact Riemannian manifold $M$. My question is, is it possible for a non-zero constant function $c$ to be in $H^1_0(M)$? My guess is, this should be possible when $M$ has finite volume, but I am not sure about this. A little insight would be highly appreciated.
2026-04-13 00:47:52.1776041272
$H^1_0(M)$ for non-compact $M$
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