Can Mean-Curvature Flow Be Made Non-Singular?

Details Can Mean-Curvature Flow Be Made Non-Singular? Can Mean-Curvature Flow Be Made Non-Singular?

2012-03-30

arxiv.org/abs/1203.6819v4

Mathematics

Differential Geometry

Scientific paper

This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the numerical instability in the discretization and show that this modification results in a simpler formulation for both the discretized and continuous formulations. We discuss the properties of the modified flow and present empirical evidence that not only does it define a stable surface evolution for genus-zero surfaces, but that the evolution converges to a conformal parameterization of the surface onto the sphere.

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