Neckpinch singularities in fractional mean curvature flows

非退化颈缩的形成与结构稳定性

Formation and structural stability of nondegenerate neckpinches

本文证明平均曲率流中非退化颈缩奇点在有限时间内形成，给出其精确渐近轮廓，并证明其在C^2扰动下结构稳定，结合密度定理表明该现象在三维中普遍且稳定。

Minimally invasive surgery for Ricci flow singularities

In this paper, we construct smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches on S^(n+1), without performing an intervening surgery. In the restrictive context of rotational symmetry, this construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities".

On Weak Super Ricci Flow through Neckpinch

In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion components. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forwa…

Appearance of stable minimal spheres along the Ricci flow in positive scalar curvature

We construct examples of spherical space forms $(S^3/Γ,g)$ with positive scalar curvature and containing no stable embedded minimal surfaces, such that the following happens along the Ricci flow starting at $(S^3/Γ,g)$: a stable embedded minimal two-sphere appears and a non-trivial singularity occurs. We also give in dimension $3$ a general contruction of Type I neckpinching and clarify the relationship between stable spheres and non-trivial Type I singularities of the Ricci flow. Some symmetry assumptions prevent the appearance of stable spheres, and this has consequences on the types of sing…

Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological…

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that f…

Fractional combinatorial Calabi flow on surfaces

Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings o…

穿孔球面上具有常分数阶高阶 Q-曲率的完备度量

Complete metrics with constant fractional higher order $Q$-curvature on the punctured sphere

本文在穿孔球面上构造具有常高阶分数曲率的完备度量，通过粘合方法与半气泡塔解逼近，将问题转化为无穷维 Toda 型系统，并证明其可解性等价于平衡构型的存在性。

The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension $1$. More precisely, given a sequence $u_k :\mathbb{R} \to \mathbb{R}$ of solutions to \begin{equation} (-Δ)^\frac{1}{2} u_k =K_ke^{u_k}\quad \text{in }\mathbb{R}, \end{equation} with $K_k$ bounded in $L^\infty$ and $e^{u_k}$ bounded in $L^1$ uniformly with respect to $k$, we show that up to extracting a subsequence $u_k$ can blow-up at (at most) finitely many points $B=\{a_1,\dots, a_N\}$ and either (i) $u_k\to u_\infty$ in $W^{1,p}_{loc}(\mathbb{R}\setminus B)$ and $K_ke^{u_k} \stac…