WorkshopsSoliton solutions for Lagrangian mean curvature flow
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In the last few years, mean curvature flow of higher codimension submanifolds has attracted special attention when the initial submanifold is Lagrangian in complex Euclidean space. The reasons for this growing interest are that the Lagrangian condition is preserved by mean curvature flow and, as the gradient flow of the volume functional, the Lagrangian mean curvature flow seems to be a potential approach to the construction of special Lagrangians, that play an important role in the T-duality formulation of mirror symmetry. The Lagrangian mean curvature flow fails to exist after a finite time and its singularities are modelled on soliton solutions. We are interested in two classes of solutions of the Lagrangian mean curvature flow that preserve the shape of the evolving submanifolds: the self-similar solutions and the translating solitons, for which the evolution is a homothety and a translation respectively. Using simple spherical and hyperbolic curves and certain solutions of the curve shortening flow, including self-shrinking and self- expanding curves or spirals, we construct and characterize many new examples of both classes in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones in dimension two. It is a hard and open problem to classify all the self-similar solutions or translating solitons for the mean curvature flow. Our contribution is the classification of the Hamiltonian stationary Lagrangian ones in complex Euclidean plane.
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Ildefonso Castro López
2011-07-07
15:15:00 - 16:15:00
R101 , Astronomy and Mathematics Building
In the last few years, mean curvature flow of higher codimension submanifolds has attracted special attention when the initial submanifold is Lagrangian in complex Euclidean space. The reasons for this growing interest are that the Lagrangian condition is preserved by mean curvature flow and, as the gradient flow of the volume functional, the Lagrangian mean curvature flow seems to be a potential approach to the construction of special Lagrangians, that play an important role in the T-duality formulation of mirror symmetry. The Lagrangian mean curvature flow fails to exist after a finite time and its singularities are modelled on soliton solutions. We are interested in two classes of solutions of the Lagrangian mean curvature flow that preserve the shape of the evolving submanifolds: the self-similar solutions and the translating solitons, for which the evolution is a homothety and a translation respectively. Using simple spherical and hyperbolic curves and certain solutions of the curve shortening flow, including self-shrinking and self- expanding curves or spirals, we construct and characterize many new examples of both classes in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones in dimension two. It is a hard and open problem to classify all the self-similar solutions or translating solitons for the mean curvature flow. Our contribution is the classification of the Hamiltonian stationary Lagrangian ones in complex Euclidean plane.
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