Min-max minimal surfaces in 3-manifolds

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.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2014

Author dc:creator

• Ketover, Daniel

Advisor dc:contributor.advisor

• Tobias H. Colding.

• Mathematics.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/90188

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/90188