This self-contained book lays the foundations for a systematic understanding of potential theoretic and uniformization problems on fractal Sierpiński carpets, and proposes a theory based on the latest developments in the field of analysis on metric spaces. The first part focuses on the development of an innovative theory of harmonic functions that is suitable for Sierpiński carpets but differs from the classical approach of potential theory in metric spaces. The second part describes how this theory is utilized to prove a uniformization result for Sierpiński carpets. This book is intended for researchers in the fields of potential theory, quasiconformal geometry, geometric group theory, complex dynamics, geometric function theory and PDEs.

Dimitrios Ntalampekos is a Milnor Lecturer at Stony Brook University, working in the field of analysis on metric spaces. He completed his PhD degree at the University of California, Los Angeles under the supervision of Mario Bonk. He holds a MS in Mathematics from the same university, and pursued his undergraduate studies at the Aristotle University of Thessaloniki.