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Article overview
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Hamiltonian Stationary Shrinkers and Expanders for Lagrangian Mean
Curvature Flows | Yng-Ing Lee
; Mu-Tao Wang
; | Date: |
2 Jul 2007 | Abstract: | We construct examples of shrinkers and expanders for Lagrangian mean
curvature flows. These examples are Hamiltonian stationary and asymptotic to
the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The
Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of
special Lagrangians or Lagrangian minimal surfaces in the variational approach.
It is known that these cone singularities cannot be resolved by any smooth
oriented Lagrangian submanifolds. The shrinkers and expanders that we found can
be glued together to yield solutions of the Brakke motion-a weak formulation of
the mean curvature flow. For any coprime pair $(p,q)$ other than $(2,1)$, we
construct such a solution that resolves any single Schoen-Wolfson cone
$C_{p,q}$. This thus provides an evidence to Schoen-Wolfson’s conjecture that
the $(2,1)$ cone is the only area-minimizing cone. Higher dimensional
generalizations are also obtained. | Source: | arXiv, arxiv.0707.0239 | Services: | Forum | Review | PDF | Favorites |
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