In this thesis, we discuss results on complete embedded minimal surfaces in R3 with finite topology and one end. Using the tools developed by Colding and Minicozzi in their lamination theory [4, 9, 10, 11, 12], we provide a proof of the uniqueness of the helicoid. We then extend these techniques to show that any complete, embedded minimal surface with one end and finite topology is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. Using a result of Hauswirth, Perez, and Romon [19], as an immediate corollary we get that these surfaces are actually asymptotic to a helicoid (in a C0 sense). In the final chapter, we move away from complete surfaces and consider local results on embedded disks. We sharpen results of Colding and Minicozzi on the shapes of embedded minimal disks in R3 , giving a more precise scale on which minimal disks are "helicoidal".
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