  We prove that the maximum number of edges in a 3-uniform linear hypergraph on
$n$ vertices containing no 2-regular subhypergraph is $n^{1+o(1)}$. This
resolves a conjecture of Dellamonica, Haxell, Luczak, Mubayi, Nagle, Person,
R\"odl, Schacht and Verstra\"ete. We use this result to show that the maximum
number of edges in a $3$-uniform hypergraph on $n$ vertices containing no
immersion of a closed surface is $n^{2+o(1)}$. Furthermore, we present results
on the maximum number of edges in $k$-uniform linear hypergraphs containing no
$r$-regular subhypergraph.
