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These notes are intended to be a rather complete and self-con- tained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowl- edge of the subject. In particular, we improve known results regarding exis- tence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial dier- ential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial dierential equa- tions themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and reg- ularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unied summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some signicant analytical tools.

A tour of the theory of absolutely minimizing functions