Coordinate scans, compactness properties, and area minimization

• Main Author: Peterson, James Ernest
• Other Authors: Hardt, Robert M.
• Format: Others
• Language: English
• Published: 2009
• Subjects: Mathematics
• Online Access: http://hdl.handle.net/1911/18957

In the framework of geometric measure theory, we investigate compactness results and possible solutions of area-minimizing problems on surfaces using area functionals other than the traditional mass norm. These solutions will be of a relatively new class of objects called rectifiable coordinate scans. We begin by reviewing traditional theory and motivating problems for using non-mass area functionals. Next we set up the basic definitions and theorems, mostly in analogy to the classical theory of currents. Our major result is a rectifiable compactness theory which leads to solutions of Plateau-type problems for scans. Finally, we use our compactness results to construct a Holder continuous area-decreasing flow.