Abstract
We provide new elementary proofs of the following two results: every complex variety is locally the graph of a Dir-minimizing function, first proved by Almgren (Almgren’s big regularity paper, volume 1 of World Scientific Monograph Series in Mathematics, 2000); the gradients of Dir-minimizing functions, in principle square-summable, are p-integrable for some p > 2, proved by De Lellis and Spadaro (Higher integrability and approximation of minimal currents, 2009). In the planar case, we prove that our integrability exponents are optimal.
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Spadaro, E.N. Complex varieties and higher integrability of Dir-minimizing Q-valued functions. manuscripta math. 132, 415–429 (2010). https://doi.org/10.1007/s00229-010-0353-5
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DOI: https://doi.org/10.1007/s00229-010-0353-5