The H-functional E is a natural variant of the Dirichlet energy along maps u from a surface S into R^3. Critical points of E include conformal parameterisations of constant mean curvature surfaces in R^3. When S is closed (compact, empty boundary) all critical points have H-energy E at least 4π/3, with equality attained if and only if we are parametrising a round sphere (so S itself must be a sphere) – this is the classical isoperimetric inequality.

Here we will address the simple question: can one approach the natural lower energy bound by critical points along fixed surfaces of higher genus? In fact we prove more subtle quantitative estimates for any (almost-)critical point whose energy is close to 4π/3. Standard theory tells us that a sequence of (almost-)critical points on a fixed torus T, whose energy approaches 4π/3, must bubble-converge to a sphere: there is a shrinking disc on the torus that gets mapped to a larger and larger region of the round sphere, and away from the disc our maps converge to a constant. Thus the limiting object is really a map from a sphere to R^3, and the challenge is to compare maps from a torus with the limiting map (i.e. a change of topology in the limit). In particular we can prove a gap theorem for the lowest energy level on a fixed surface and estimate the rates at which bubbling maps u are becoming spherical in terms of the size of dE[u] – these are commonly referred to as Łojasiewicz-type estimates. 

This is a joint work with Andrea Malchiodi (SNS Pisa) and Melanie Rupflin (Oxford).
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