Gang Z., Knopf D., Sigal I.M. Neckpinch Dynamics for Asymmetric Surfaces Evolving by Mean Curvature Flow

American Mathematical Society, 2018. — 90 p. — (Memoirs of the American Mathematical Society,Vol.253, №1210). — ISBN: 1470428407.The authors 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, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.The first bootstrap machine
Estimates of first-order derivatives
Decay estimates in the inner region
Estimates in the outer region
The second bootstrap machine
Evolution equations for the decomposition
Estimates to control the parameters a and b
Estimates to control the fluctuation φ
Proof of the Main Theorem
Appendixes
Mean curvature flow of normal graphs
Interpolation estimates
A parabolic maximum principle for noncompact domains
Estimates of higher-order derivatives