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Title:
A new decomposition of surfaces in the Heisenberg group
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Abstract:
The main new theorem (forthcoming joint work with Robert Young) that we will present in this talk is that the L_4 norm of the vertical perimeter of any measurable subset of the 3-dimensional Heisenberg group H is at most a universal constant multiple of its perimeter. This isoperimetric inequality is optimal, and its proof uncovers the following structural description of surfaces in H: They admit a multi-scale hierarchical decomposition into pieces that are close to ruled surfaces; these pieces can be long and narrow, sometimes giving the decomposition the appearance of a Venetian blind, with many narrow slats. Consequences include solutions of several longstanding questions in metric geometry.
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