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dg.differential geometry – What does Colding-Minicozzi idea say about convergence with multiplicity? Answer

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dg.differential geometry – What does Colding-Minicozzi idea say about convergence with multiplicity?

Permit $ (M_j midj in mathbf {N}) $ breathe a sequence of compact minimal areas within the unit sphere $ B subset mathbf {R} ^ 3 $, with limits $ partial M_j subset partial B $. Assume that the sequence converges to the horizontal airplane $ mathbf {R} ^ 2 occasions {0 } $ with range $ Q geq 2 $:
commence {equation} lvert M_j rvert to Q cdot lvert mathbf {R} ^ 2 occasions {0 } rvert quad textual content {as $ j to infty $} aim { Equation}
within the various topology. Note that no restrict is assumed for the Morse index or the character of the surfaces: $ mathrm {genus} , M_j to infty $ and $ mathrm {index} , M_j to infty $
is mostly not prohibited.

Question. Does the idea developed by Colding and Minicozzi give extra details about convergence? If not, what kindly of Sheeting’s theorem is current if a uniform sure for the genus is assumed?

My (extraordinarily restricted) judgement of their labor is that it supplies decomposition outcomes. I’ve heard loosely the outcomes that stated minimal surfaces can breathe damaged down into elements that resemble catenoidal necks, multi-valued graphs alongside the strains of helicoids, and single-valued graphs. (I do not understand how precisely that’s, plane on a heuristic point.)

I may think about that if there was a uniform bond love
commence {equation} mathrm {genus} , M_j leq C quad textual content {for all $ j $}, aim {equation}
then this might breathe utilized to get some systematize of Sheeting’s theorem – maybe multi-valued – genuine exterior of the factors the place the necks are concentrated. If the genus goes to infinity this energy not breathe undoubted, however maybe another conclusions may breathe drawn concerning the organize.

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