I faced this sentence in my studies on Ricci flow:
The Bryant soliton is a singularity model for the degenerate neckpinch.
Q1: What is the definition and meaning of singularity model? Can one model all the singularities of the Ricci flow by solitons?
Q2: What is intuition of the degenerate neckpinch?
I need the answers of these questions Immediate. Can anyone help me?
Thanks in advance.
There is an answer I have from earlier lecture notes, it helped me to understand.
Let $(\mathcal{M},h)$ a closed Riemannian manifold. One can develop the metric $h$ by: $$\frac{\partial}{\partial t}h=-2\; \mathfrak{Rc}(h)$$ a weak parabolic system known as Ricci flow.
Now imagine under certain curvature conditions on initial $h$, it is possible to prove normalized convergence to a round metric, hence say a lot about the topology of $\mathcal{M}$. But for a large set of initial metrics the flow will become singular in finite time before converging to any smooth limiting metric (exciting!).
Apparently this happens when the curvature blows up on certain regions of the manifold $\mathcal{M}$ and the standard short-time existence result for Ricci flow says that if the flow becomes singular at some finite time $T<\infty$ then $$\lim \sup_{t\rightarrow T}\ \max_{x\in $\mathcal{M})}\vert \mathfrak{Rc}(x,t) \vert=\infty$$
In order to analyze these singularities, one follows the conventional singular analysis from non-linear PDEs and does a blow-up at the singularity using the scaling symmetry of the equation. Depending on how much compactness is at hand, one can extract a singularity model (re. your question) from a sequence of such blow-ups, which will usually have a better geometry than the original Ricci flow.
Indeed, if one has a good knowledge about the possible singularity models, then one is able to understand the structure of the singularity formation and see how to do surgery while controlling the geometry and the topology of the manifold, thus arriving at the so-called Ricci flow with surgeries.
It would take very long continuing on this and also neckpinching of Ricci flow go here.