In the theory of PDEs, one central element is the Poincaré inequality, a special case of which is the constancy theorem: if the derivative vanishes, then the function must be constant on each connected component. To extend the theory to varifolds, it turns out that even the concept of components is non-trivial; such a theory was established by Menne. The present work was motivated by the need for a different notion of components to extend the previous theory to a multiple-valued setting. In this talk, we will first give a quick introduction to varifolds which gives a measure-theoretical generalization of manifolds, then discuss the notions of indecomposability of varifolds, and finally, present the idea of the proof of the existence theorem of integral decompositions.

 

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