The symmetric \textquotedblleft doughnut \textquotedblright evolving by its mean curvature

Facet{breaking for Three{dimensional Crystals Evolving by Mean Curvature

We show two examples of facet{breaking for three{dimensional poly-hedral surfaces evolving by crystalline mean curvature. The analysis shows that creation of new facets during the evolution is a common phenomenon. The rst example is completely rigorous, and the evolution after the subdivision of one facet is explicitly computed for short times. Moreover, the constructed evolution is unique amon...

Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow

We study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are C3-close to round, but without assuming rotational symmetry or positive mean curvature, we show that mcf solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singularity formation. Our results show in a precise way that mcf solutions...

Symmetric curvature tensor

Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discu...

Symmetric Surfaces of Constant Mean Curvature in 3

symmetric curvature tensor

recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. using this machinery, we have defined the concept of symmetric curvature. this concept is natural and is related to the notions divergence and laplacian of vector fields. this concept is also related to the derivations on the algebra of symmetric forms which has been discus...