What does the term "generic" mean for a hypersurface in $\mathbb{R}^3$? Please give me the definition of "$M$ is generic" , where $M$ is a hypersurface in $\mathbb{R}^3$.
I have looked at the introduction of the paper. There may be no explanation of the definition of the term "generic".
What it is not: If you read the whole paper, you will understand that the term "generic" is used only in a non-rigorous way. Definitely not the one we see in differential topology/algebraic geometry.
What it is: In the paper they introduced the notion of entropy $\lambda$ of an immersion. One important property is that if $\{\Sigma_t\}$ is the mean curvature flow starting at $\Sigma$, then $\lambda(\Sigma_t)$ is nonincreasing in $t$. When $\{\Sigma_t\}$ becomes singular at time $T$ and $S$ is a self-shrinker which models the singularity, then we also have $$\lambda (S)\le \lim_{t\to T} \lambda(\Sigma_t),$$ (mainly because $\lambda$ is translation and scaling invariant). This is simple yet profound as now we have a quantity which is momotonic along the flow, and the monotonicity actually passes to the "limit" as $t\to T$, when the flow is no longer smooth.
Now they ask: Can we slightly perturb $\Sigma_t$ to a new $\hat \Sigma$ for some $t<T$ closed to $T$, so that $\lambda (\hat \Sigma)<\lambda (S)$ (Remark: In some nice case a rescaling of $\Sigma_{t}$ is closed to $S$, so we are looking for a hypersurface closed to $S$)? If that can be done, then we restart the flow at $\hat \Sigma$:
$$ \Sigma \overset{flow}{\to } \Sigma_{t_0} \overset{perturb}{\to } \hat \Sigma \to \text{flow again}$$
The point is that when in the restart flow $\hat \Sigma_t$ hits a singularity which is modeled by a self-shrinker $\hat S$, this $\hat S$ cannot be $S$ as $\lambda (\hat S) <\lambda (S)$. This could be very important as the classification of self-shrinkers seems to be a hard problem. By doing this perturbation we hope to "ignore" a lot of self-shrinkers, bypassing the hard analysis question.
Of course, whether or not such a perturbation can be done depend on the self-shrinker $S$. For example, such a perturbation cannot be found when $S = \mathbb S^n$ is the $n$-sphere in $\mathbb R^{n+1}$: If $\Sigma$ is a hypersurface which is $C^2$-closed to the sphere, then it is also convex and Huisken (JDG, 84) shows that the mean curvature flow starting at $\Sigma$ must have $\mathbb S^n$ as a singularity. In terms of monotonicity of $\lambda$, this shows that $\lambda (\Sigma) \ge \lambda (\mathbb S^n)$ for all $\Sigma$ closed to $\mathbb S^n$!
Any self-shrinker $S$ which is a local minimum of $\lambda$ (like $\mathbb S^n$) is called entropy stable.
One main theorem in the paper is
Theorem The only entropy stable self-shrinking hypersrufaces in $\mathbb R^{n+1}$ are the sphere $\mathbb S^n$ and the generalized cylinders $\mathbb R^k \times \mathbb S^{n-k}$.
Thus the authors called these generic singularities (so they did not define generic hypersurfaces in general).
Parts of the analysis in the paper are generalized to higher codimensions, but there isn't a result characterizing entropy stable self-shrinkers (even for compact self-shrinkers) in higher codimensions which is as precise as the above theorem in codimension one.