Plateau-rayleigh instability of singular minimal surfaces

Abstract

<p style='text-indent:20px;'>We prove a Plateau-Rayleigh criterion of instability for singular minimal surfaces, providing explicit bounds on the amplitude and length of the surface. More generally, we study the stability of <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-singular minimal hypersurfaces considered as hypersurfaces in weighted manifolds. If <inline-formula><tex-math id="M2">\begin{document}$ \alpha&lt;0 $\end{document}</tex-math></inline-formula> and the hypersurface is a graph, then we prove that the hypersurface is stable. If <inline-formula><tex-math id="M3">\begin{document}$ \alpha&gt;0 $\end{document}</tex-math></inline-formula> and the surface is cylindrical, we give numerical evidences of the instability of long cylindrical <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-singular minimal surfaces.</p>