Varifolds: a Modern Approach to Classical Results

The thesis focuses on the theory of varifolds, starting with the introduction of the notions of varifold, rectifiable and integral varifold, and generalized mean curvature. Classical results will be presented, including monotonicity formulas, rectifiability criteria, a precompactness theorems for integral varifolds and Allard's regularity theorem. Modern techniques will be employed, inspired by recent developments for varifolds with locally bounded variation with respect to anisotropic functionals. The work will conclude with an application of the theory of varifolds concerning the existence and regularity of solutions to the Plateau problem.