Complexities of the Second Variation of Minimal Surfaces of codimension greater than one, pt. 1

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The nontriviality of the normal bundle of a minimal submanifold of codimension greater than one makes the second variation difficult to study because it is not clear how to choose variations that reduce the area of the submanifold. For a 2-D minimal surface, a partial averaging technique is encoded by a complex-valued version of the second variation formula and this has the benefit of bringing complex analytic techniques into play. I will survey the achievements of this technique and discuss open problems in this area. Other results, including ones for minimal hypersurfaces, that place this work in a broader context will also be mentioned.

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