Herbert Federer's 1969 text "Geometric Measure Theory" is a foundational, advanced work that established the rigorous framework for studying surfaces and area minimization through the lens of rectifiability and current theory. The text is renowned for providing the theoretical basis for solving the Plateau problem and establishing the regularity of area-minimizing surfaces. Access the text via Springer Link: Springer Nature . Geometric Measure Theory | Springer Nature Link

It introduced the modern, formal definition of currents, allowing for a rigorous treatment of boundary, convergence, and minimal surfaces.

Geometric Measure Theory: A Beginner's Guide by Frank Morgan (excellent for intuitive, visual understanding).