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Talks by Francesca Tripaldi

On the topology of surfaces with the simple lift property

Motivated by the work of Colding and Minicozzi and Hoffman and White on minimal laminations obtained as limits of sequences of properly embedded minimal disks, Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. Bernstein and Tinaglia prove that an embedded minimal surface \(\Sigma\subset\Omega\) with the simple lift property must have genus zero, if \(\Omega\) is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that \(\Omega\) cannot contain closed minimal surfaces. In this work, I generalize this result by taking an arbitrary orientable three-manifold \(\Omega\) and proving that one is able to restrict the topology of an arbitrary surface \(\Sigma\subset\Omega\) with the simple lift property. Among other things, I prove that the only possible compact surfaces with the simple lift property are the sphere and the torus in the orientable case, and the connected sum of up to four projective planes in the non-orientable case. In the particular case where \(\Sigma\subset\Omega\) is a leaf of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks, we are able to sharpen the previous result, so that the only possible compact leaves are the torus and the Klein bottle.

Sala de Conferencias, IEmath

Francesca Tripaldi

Number of talks
1

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