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Mean curvature flow is classically understood as the $L^2$ -gradient flow of the volume functional. In the non-compact setting, however, this variational interpretation degenerates, since the volume is infinite and the associated energy identity becomes vacuous. In this talk, we present a new variational framework for non-compact mean curvature flow by interpreting it as the gradient flow of a direction energy. Although this functional differs from the volume by a formal null-Lagrangian, it can be finite for non-compact hypersurfaces, and enables us to study the behavior of global solutions. This is joint work with T. Miura (Kyoto).

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