Luca Seemungal

Profile

I am a first year PhD student in geometric analysis. Before coming to Leeds, I spent some time at Warwick.

Research interests

I’m currently thinking about immersed minimal and CMC surfaces, both their stability and their behaviour near branch points.

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Take a loop of wire, bend it in some interesting way, dip it in a tub of soapy water, and carefully withdraw it. The soap film that spans the wire is a physical realisation of a minimal surface. Now, again with great care, blow on the soap film and form a bubble. This bubble is a physical realisation of a surface of constant mean curvature (a CMC surface).

Mathematically speaking, we define minimal surfaces to be critical points of the area functional. This is a mathematically natural concept (regardless of the physical motivation) because minimal surfaces are the natural generalisation of geodesics to two dimensions. Beware: while we call these minimal surfaces “minimal”, they are only critical points of the area functional, and it may not be the case that they are actually minimisers. (For example, the point x=0 is a critical point of f(x)=-x^2 but is not a minimiser.) Thus in the study of minimal surfaces one is immediately confronted with the fact that often, minimal surfaces are not minimisers of the area functional. It is therefore natural to study the “stability” of these minimal surfaces. That is, we count the number of ways in which a minimal surface can be perturbed so that its area decreases, up to second order. This number is called the (Morse) index of the minimal surface.

Closely related to minimal surfaces are CMC surfaces. From the geometric point of view, it turns out that minimal surfaces have zero mean curvature; surfaces of constant mean curvature (with mean curvature not necessarily equal to zero) are therefore a generalisation of minimal surfaces. On the other hand, from the variational point of view, CMC surfaces are constrained critical points of the area functional, where we only admit variations that fix enclosed volume. Examples include soap bubbles: a bubble minimises area while taking account of the restriction that the internal volume of air cannot change. Bubbles are actually minimisers, but again, we must confront the reality that CMC surfaces may not actually be minimising, and so we study their Morse index also.

In case anyone has any funny ideas, I’d like to stress that CMC surfaces are mathematically natural objects, and are not just arbitrary generalisations of minimal surfaces. Indeed: CMC surfaces solve a “linearisation” of the isoperimetric problem, a classical problem in geometry and analysis. The isoperimetric problem asks us to find, for a given fixed volume, the shape that encloses this volume with least area. CMC surfaces solve the linearised isoperimetric problem: find a shape that extremises the area when considering perturbations that fix enclosed volume. Considering the Morse index of a CMC surface makes headway into solving the isoperimetric problem up to second order: does our CMC surface actually minimise area up to second order? As one expects, spheres in ℝ³ (ordinary Euclidean 3-space) are CMC surfaces and are also solutions to the isoperimetric problem.

The isoperimetric problem is an ancient problem and has had various interpretations and relations. For example in antiquity, the Greeks considered themselves to having had solved the problem, where “shape” means “polygon” or “circle”. But with the differential geometry of curves being discovered much later in the 17th century, the problem suddenly took a new meaning, as “shape” means “any piece-wise closed C² curve”. This shows that ancient ideas in mathematics are constant sources of insipration, are never fully explored, and always have more to give. Moreover, the tools to study CMC surfaces only appeared in the 18th century, and matured in the 20th century with geometric measure theory. We take the attitude that CMC surfaces are in a sense a contemporary interpretation of an ancient problem.

Qualifications

• MMath University of Warwick