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We consider the three-dimensional free boundary incompressible idealmagnetohydrodynamics (MHD) equations with surface tension in a bounded domain.The moving closed surface is parameterized by the height function, which isdefined on a smooth and compact reference hypersurface. In Euleriancoordinates, we establish a priori estimates for smooth solutions in the$C^\infty$-class without losing any regularity. This approach allows us toavoid dealing with the spatial regularity of the flow map in Lagrangiancoordinates, which may lack maximal regularity and the geometriccharacteristics, such as the curvature and normal velocity, of the evolutionarydomain. The scaling $\partial_t\sim\nabla^{\frac 32}$ motivates us to formulatethe energy functional, and the regularity estimates are driven by the curvaturebound and the regularity of the pressure. In the spirit of the Beale-Kato-Majdacriterion, we propose a set of the a priori assumptions to guarantee thepossibility of extending the solution for a short period while preserving theinitial regularity within the time interval of existence. As far as we know,this is the first result involving the blow-up of the free boundaryincompressible ideal MHD equations with surface tension. It is worth notingthat the velocity and magnetic fields remain bounded in Sobolev's space $H^6$,while the second fundamental form remains bounded in $H^5$, throughout the timeinterval $[0,T_0]$. The value of $T_0$ depends only on the $H^6$ norm of theinitial velocity and magnetic fields, the initial domain, and the $H^5$ norm ofthe mean curvature.

On the motion of the closed free surface in three-dimensional incompressible ideal MHD with surface tension