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A brief description of my
Ph.D. Thesis
Thesis (Multi-valued graphs in
embedded constant mean curvature disks): (Thesis Work, Thesis Objectives, Future Directions)
Thesis Work. In my thesis I prove the following statement
which also appears in my paper
[Theorem 0.1., 3],
Let M be a
non zero CMC embedded disk with Gaussian curvature large at a point
then M contains a multi-valued graph around that point on the scale of
the norm squared of the second fundamental form.
Somewhat imprecisely, to contain a multi-valued graph [Definition 2.2.
3] means that M looks like a helicoid, Fig 1 (picture taken from
www.indiana.edu/~minimal/ maze/helicoid.html).
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| Fig 1. The Helicoid |
The helicoid is clearly not a graph, nonetheless, each half of the
helicoid minus the vertical axis can be viewed as a graph over the
universal cover of the punctured plane and this is, roughly speaking,
what it means to contain a multi-valued graph. The proof relies heavily
on two things: Knowing when a large CMC embedded geodesic ball is
stable, and once that is known, what that stability implies.
In my thesis, I also give examples of non zero CMC surfaces
containing arbitrary large multi-valued graphs [Appendix-A, 3].
Using the method of successive approximations, a sequence of normal
variations of the helicoid can be built which converges to a non zero
CMC embedded disk containing a multi-valued graph.
Thesis Objectives. The work on my thesis targets
mainly three objectives:
1) It generalizes Colding and Minicozzi's result for minimal
surfaces
[Theorem 0.4., 2].
2) It is a first step towards a classification of
singularities for sequences of embedded CMC disks; indeed, much more
needs to be done in this direction.
3) The proof by contradiction provides a new type of
compactness argument that does not require a bound on the area.
Future Directions. The work on my
thesis has
left open a few questions:
1) In the minimal case Colding and Minicozzi were able to extend the
multi-valued graph that forms locally, all the way up to the boundary
[1]. Is it possible for non zero CMC embedded disks to extend the
multi-valued graph to a larger scale?
2) In the minimal case Colding and Minicozzi were able to prove that an
embedded minimal disk can be broken up into two types of building
blocks. Where the Gaussian curvature is small we have graphical pieces,
where it is large it looks like a helicoid. Can the same global
structure theorem be proved for non zero CMC embedded disks?
3) Is it possible to prove global compactness results for complete non
zero CMC embedded surfaces? More precisely, given a sequence of CMC
embedded surfaces, is it true that there exists a subsequence
converging to a CMC lamination?
[1] T.H.
Colding and W.P. Minicozzi, The space of embedded minimal surfaces
of fixed genus in a 3-manifold I; Estimates off the axis for disks,
Annals of Math.
[2] T.H.
Colding and W.P. Minicozzi, The space of
embedded minimal surfaces of fixed genus in a 3-manifold II;
Multi-valued graphs in a disk, Annals of Math.
[3] G.
Tinaglia, Multi-valued graphs in embedded constant mean curvature
disks, Trans. Amer.
Math. Soc. 359
(2007), 143-164.
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