Min-max minimal surfaces in 3-manifolds
Massachusetts Institute of Technology. Department of Mathematics.
Tobias H. Colding.
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A Heegaard splitting of a 3-manifold gives rise to a natural set of sweepouts which by the Almgren-Pitts and Simon-Smith min-max theory generates a min-max sequence converging as varifolds to a smooth minimal surface (possibly disconnected, and with multiplicities). We prove a conjecture of Pitts-Rubinstein about how such a min-max sequence can degenerate; namely we show that after doing finitely many disk surgeries and isotopies on the sequence, and discarding some components, the remaining components are each isotopic to one component (or a double cover of one component) of the min-max limit. This convergence immediately gives rise to new genus bounds for min-max limits. Our results can be thought of as a min-max analog to the theorem of Meeks-Simon-Yau on convergence of a minimizing sequence of surfaces in an isotopy class.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.35Cataloged from PDF version of thesis.Includes bibliographical references (pages 51-54).
DepartmentMassachusetts Institute of Technology. Department of Mathematics.; Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology