Hardt, Robert M.2009-06-042009-06-042006Peterson, James Ernest. "Coordinate scans, compactness properties, and area minimization." (2006) Diss., Rice University. <a href="https://hdl.handle.net/1911/18957">https://hdl.handle.net/1911/18957</a>.https://hdl.handle.net/1911/18957In 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.58 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisTHESIS MATH. 2006 PETERSON