James Tissot – The no-touching principle of minimal surfaces and applications

Abstract: The standard definition of a minimal surface is variational: a surface is minimal if nearby surfaces have the same area to first order. What kind of properties can we deduce from such a definition? In the seminar, I will show how this variational definition implies that distinct minimal surfaces cannot touch tangentially. I will then apply this to deduce a couple of important results: (1) there is no closed minimal surfaces in Euclidean space and more generally a compact minimal surface is contained in the convex hull of its boundary; (2) there are no solutions to Plateau’s problem for insufficiently curved boundary curves. I hope to make the talk self-contained and focus on the main ideas rather than technical details.