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SUMMARY:The space of soap bubbles - Kusner\, R (University of Massachusett
 s)
DTSTART:20121120T113000Z
DTEND:20121120T123000Z
UID:TALK41569@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Understanding the space of all 'soap bubbles' - that is\, comp
 lete embedded constant mean curvatures (CMC) surfaces in $R^3$ - is a cent
 ral problem in geometric analysis. These CMC surfaces are highly transcend
 ental objects\; the topology and smooth structure of their moduli spaces a
 re understood only in some special cases. In this talk we will describe th
 e formal 'Lagrangian embedding' of CMC moduli space into the 'space of asy
 mptotes' \nand discuss where this is smooth\, namely\, at a surface with n
 o nontrivial square-integrable Jacobi fields. This nondegeneracy condition
  has now been established for all coplanar CMC surfaces of genus zero\; th
 is allows them to serve as 'building blocks' for more complicated CMC surf
 aces. There is also a surprising connection with complex projective struct
 ures and holomorphic quadratic differentials on $C$ obtained by taking the
  Schwarzian of the developing map for the projective structure. This assig
 ns each coplanar CMC surface a 'classifying' complex polynomial\, and lets
  us explicitly work out the smooth topology of their moduli spaces.\n
LOCATION:Seminar Room 1\, Newton Institute
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