Ciao, 

I am a graduate student in the Department of Mathematics, at Johns Hopkins University.

I received my "Laurea in Matematica" at University of Bologna, Italy.

My primary fields of interest are Minimal Surfaces, Geometric Analysis and P.D.E..

My advisor is Professor Minicozzi.

CV,   Thesis,   Current Work,   Papers

Minimal Surfaces and Constant Mean Curvature Surfaces

Minimal surfaces are defined as surfaces which are critical points for the area functional. It so happens that the mean curvature of a minimal surface, the average of the two principal curvatures, is identically zero. Surfaces which are critical points for the area functional under a volume constraint are instead called constant mean curvature (CMC) surfaces and in fact the average of the two principal curvatures is constant. Not only are there plenty of mathematical examples for both of these surfaces (for instance planes, helicoids and catenoids are minimal surfaces while spheres, cylinders and Delauney surfaces are non zero CMC surfaces), but they can easily be realized and observed in the real world. In nature, the shape of a soap film approximates with great accuracy that of a minimal surface while soap bubbles provide the analogous approximation for CMC surfaces. In other words, a soap bubble is the least-area surface that encloses the fixed volume inside.

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).

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, to appear, math.AP/0210106.

[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, to appear, math.AP/02100086.

[3]    G. Tinaglia, Multi-valued graphs in embedded constant mean curvature disks, Transactions of the AMS, to appear, math.DG/0409184.

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Current Work:

The famous Plateau problem is the following;

given a Jordan curve in three-dimensional space, does there exist a surface of the type of a disk, whose boundary is that curve and has the smallest possible area?

If we take a piece of flexible wire and dip it into a soap solution, to form a soap film across the hoop, of all possible surfaces whose boundary is that wire, the soap film has the smallest area. However, this experimental evidence of the existence of a solution for the Plateau problem clearly cannot replace a mathematical proof. The Plateau problem was solved for many special contours but it was in 1930 that Douglas and Rado independently and almost simultaneously showed there exists at least one solution to it for a general Jordan curve. After this major breakthrough, many questions are still unanswered--questions on the number, shape and properties of such solutions, depending on the boundary.

Rado proved that if a Jordan curve admits a one-to-one orthogonal projection onto a convex curve in a plane, then the solution to the Plateau problem is unique, it is free of branch points and it can be expressed as the graph of a function.

I am currently trying to prove similar structure theorems for embedded solutions of the Plateau problem given some restrictions on the boundary. (More details in my research statement.)

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A generalization of Rado's Theorem for almost graphical boundaries, joint with B. Dean., Math. Zeit.,to appear, LANL link.

Multi-valued graphs in embedded constant mean curvature disks, Transactions of the AMS, to appear, LANL link.

In this paper we prove that an embedded and simply connected constant mean curvature surface with Gaussian curvature large at a point contains a multi-valued graph around that point on the scale of the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for minimal surfaces.

Local behavior of embedded constant mean curvature disks, Seminari di Geometria 2001-2004, Universita' di Bologna, Bologna, (73-80).

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