PomeLemma 8.10.6.2A6BE9847309 Let $E$ be a compact subset of a smooth real submanifold of codimension two contained in a coordinate chart. Then $\operatorname {Cap}_1(E)=0$.
Lemma 8.10.62A6BE9847309
Proof. It suffices to treat a compact subset of a submanifold chart, since finitely many such subsets cover $E$ and the capacity is subadditive. Choose real coordinates $(s,t)$ there, with $t\in \mathbb R^2$ and the submanifold given by $t=0$. Choose $0\leq \eta \leq 1$ smooth, supported in this chart and equal to one near the compact subset. Put $r=|t|$. For $0<\epsilon <1/2$ define
\[ h_\epsilon (r)= \begin{cases} 1,& 0\leq r\leq \epsilon ^2,\\ \displaystyle \frac{\log (\epsilon /r)}{\log (1/\epsilon )}, & \epsilon ^2<r<\epsilon ,\\ 0,& r\geq \epsilon . \end{cases} \]
The Sobolev function $f_\epsilon =\eta h_\epsilon $ equals one on an open neighborhood of that subset. On the fixed compact coordinate set the Sobolev norm is equivalent to the Euclidean one. Integration in the two normal variables gives
\[ \| f_\epsilon \| _{H^1}^2 \leq C\left(\epsilon ^2+ \int _{\epsilon ^2}^{\epsilon } \frac{\mathrm dr}{r\log (1/\epsilon )^2}\right) =C\left(\epsilon ^2+\frac1{\log (1/\epsilon )}\right) \longrightarrow 0. \]
The derivatives of $\eta $ contribute at most $C\epsilon ^2$. Taking the infimum over open neighborhoods proves the assertion. $\square$
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