An Introduction of Infinity Harmonic Functions

This note serves as a basic introduction on the analysis of infinity harmonic functions, a subject that has received considerable interests very recently. The author discusses its connection with absolute minimal Lipschitz extension, present several equivalent characterizations of infinity harmonic functions. He presents the celebrated theorem by R. Jensen [17] on the uniqueness of infinity harmonic functions, the linear approximation property of infinity harmonic functions by Crandall and Evans [9] and the asymptotic behavior near an isolated singularity of infinity harmonic functions by Savin, Wang, and Yu [25].