BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Ben Sharp (University of Leeds) DTSTART:20210209T160000Z DTEND:20210209T170000Z DTSTAMP:20260423T021214Z UID:DGSTO/9 DESCRIPTION:Title: Ł ojasiewicz-type inequalities for the H-functional near simple bubble trees \nby Ben Sharp (University of Leeds) as part of Differential Geometry Seminar Torino\n\n\nAbstract\nThe H-functional E is a natural variant of t he Dirichlet energy along maps u from a closed surface S into R^3. Critica l points of E include conformal parameterisations of constant mean curvatu re surfaces in R^3. The functional itself is unbounded from above and belo w on H^1(S\,R^3)\, but all critical points have H-energy E at least 4π/3\ , with equality attained if and only if we are parametrising a round spher e (so S itself must be a sphere) - this is the classical isoperimetric ine quality.\n\nHere we will address the simple question: can one approach the natural lower energy bound by critical points along fixed surfaces of hig her genus? In fact we prove more subtle quantitative estimates for any (al most-)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 sh rinking disc on the torus that gets mapped to a larger and larger region o f the round sphere\, and away from the disc our maps converge to a constan t. 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 c hange 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 w hich bubbling maps u are becoming spherical in terms of the size of dE[u] - these are commonly referred to as Łojasiewicz-type estimates. \n\nThis is a joint work with Andrea Malchiodi (SNS Pisa) and Melanie Rupflin (Oxfo rd).\n LOCATION:https://researchseminars.org/talk/DGSTO/9/ END:VEVENT END:VCALENDAR