  We consider rigidity properties of steady Euler flows in two-dimensional
bounded domains. We prove that steady Euler flows in a disk with exactly one
interior stagnation point and tangential boundary conditions must be circular
flows, which confirms a conjecture proposed by F. Hamel and N. Nadirashvili in
[J. Eur. Math. Soc., 25 (2023), no. 1, 323-368]. Moreover, for steady Euler
flows on annuli with tangential boundary conditions, we prove that they must be
circular flows provided there is no stagnation point inside, which answers
another open problem proposed by F. Hamel and N. Nadirashvili in the same
paper. We secondly show that the no-slip boundary conditions would result in
absolute rigidity in the sense that except for the disks (\emph{resp}. annuli),
there is no other smooth simply (\emph{resp}. doubly) connected bounded domain
on which there exists a steady flow with only one (\emph{resp}. no) interior
stagnation point and no-slip boundary conditions, and if present on the other
hand, the flow must be circular. The arguments are based on the geometry of
streamlines and 'local' symmetry properties for the non-negative solutions of
semi-linear elliptic problems.
