On the non-degenerate and degenerate generic singularities formed by mean curvature flow

We study a neighborhood of generic singularities formed by mean curvature flow (MCF). For various possibilities when the singularities are modeled on $S^3\times R$, we provide a detailed description for a small, but fixed, neighborhood of singularity, including proving that a small neighborhood is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidence to support it. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead of a standard straight one.