  A classical problem in real geometry concerns the representation of positive
semidefinite elements of a ring $A$ as sums of squares of elements of $A$. If
$A$ is an excellent ring of dimension $\geq3$, it is already known that it
contains positive semidefinite elements that cannot be represented as sums of
squares in $A$. The one dimensional local case has been afforded by Scheiderer
(mainly when its residue field is real closed). In this work we focus on the
$2$-dimensional case and determine (under some mild conditions) which local
excellent henselian rings $A$ of embedding dimension $3$ have the property that
every positive semidefinite element of $A$ is a sum of squares of elements of
$A$.
