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SUMMARY:Rob Kusner (University of Massachusetts at Amherst)
DTSTART:20210629T170000Z
DTEND:20210629T180000Z
DTSTAMP:20260423T035021Z
UID:OSGA/77
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OSGA/77/">ON
  THE CANHAM PROBLEM: BENDING ENERGY MINIMIZING  SURFACES OF ANY GENUS AND 
 ISOPERIMETRIC RATIO</a>\nby Rob Kusner (University of Massachusetts at Amh
 erst) as part of Online Seminar "Geometric Analysis"\n\n\nAbstract\nIn 197
 0\, the biophysicist Peter CANHAM proposed that the shapes of red blood ce
 lls could be described variationally\, leading to the Canham problem: find
  the surfaces of genus $g$ embedded in $\\R^3$ that minimize their Willmor
 e bending energy $W=\\frac14 \\int H^2$ with given area and enclosed volum
 e\, or equivalently (since $W$ is scale invariant) with given isoperimetri
 c ratio $v \\in (0\, 1)$. Building on very recent work of Andrea MONDINO &
  Christian SCHARRER\, we solve the “existence” part of the problem\; i
 t suffices to find a comparison surface of genus $g$ with arbitrarily smal
 l isoperimetric ratio $v$ and $W < 8π$\, which we construct by gluing $g+
 1$ small catenoidal bridges to the bigraph of a singular solution for the 
 linearized Willmore equation $∆(∆+2)φ = 0$ on the $(g+1)$-punctured s
 phere.  (If time permits\, we may discuss our ongoing work to understand t
 he “small $v$” limit\, as well as “uniqueness” aspects of the Canh
 am problem.)\n\n— Rob KUSNER\, UMassAmherst & CoronavirusU\n\n[joint wor
 k with Peter MCGRATH\, NorthCarolinaStateU]\n
LOCATION:https://researchseminars.org/talk/OSGA/77/
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