论文标题
Möbius-Invariant Willmore的奇异性和完全融合$ 3 $ -SPHERE
Singularities and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere
论文作者
论文摘要
在这里，我们继续对Möbius-Invariant Willmore Flow（MIWF）进行调查，开始以任意平滑且无脐带的初始浸入$ F_0 $进行移动，这些$ f_0 $分别映射了一些固定的紧凑型圆环$σ$中的$ \ m mathbb {r}^n $在这里，我们调查了MIWF的流动线$ \ {f_t \} $在$ \ mathbb {s}^3 $中以相对较低的willmore Energy开头，因为时间$ t $接近存在的最大时间$ t_ {max}（max max}（max}（f_0）$ $ \ \ \ {f_t {f_t {f_t \ t_t f_0）$。我们成功构建了发散的流线，并研究了MIWF的不同流动线和收敛流线的极限表面。至少可以在MIWF的某些通用流线$ \ {f_t \} $的限制表面上，可以在$ \ mathbb {r}^4 $中的积分$ 2 $ -2 $ -varifold $μ$的支持下识别$ \ {\ MATHCAL { $ 0 $或$ 1 $。在特殊情况下，$ spt（μ）$是属$ 1 $的紧凑型表面，可以通过均匀的保质性bi-lipschitz同构同构$ f $ f $ f $ f $的$ w^{2,2,2，2} \ cap w^{1，\ cap w^{1，\ infty} $，以及在某些其他条件下，在$ \ \ \ f _} $}的其他条件下类$ w^{4,2} $的差异性。最后，如果假定使用$ \ mathbb {s}^3 $的hopf-torus的初始沉浸式$ f_0 $ f _0 $的流量线$ \ {f_t \} $，使用willmore Energy不超过$8π$，那么我们就获得了有关流量$ \ \ \ \ {f _ f _ flof flow line of flow flow line of flow flow fline的说明，然后这种见解最终将产生一个标准，以使MIWF的这种流量线完全收敛到Clifford torus的参数化 - 直至$ \ Mathbb {s}^3 $ - as $ t \ aS $ t \ aS $ t \ nearrow \ nearrow \ introw \ infty $。
Here we continue the investigation of the Möbius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $Σ$ into $\mathbb{R}^n$ respectively $\mathbb{S}^n$. Here we investigate the behaviour of flow lines $\{F_t\}$ of the MIWF in $\mathbb{S}^3$ starting with relatively low Willmore energy, as the time $t$ approaches the maximal time of existence $T_{max}(F_0)$ of $\{F_t\}$. We succeed to construct divergent flow lines, and we investigate limit surfaces of both divergent and convergent flow lines of the MIWF. At least generically a limit surface of some general flow line $\{F_t\}$ of the MIWF can be identified with the support of an integral $2$-varifold $μ$ in $\mathbb{R}^4$, which is the weak limit of the sequence of varifolds $\{\mathcal{H}^2\lfloor_{F_{t_{l}}(Σ)}\}$, for an appropriately chosen sequence $t_{l} \nearrow T_{max}(F_0)$, and that $spt(μ)$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(μ)$ is a compact surface of genus $1$ it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism $f$ of class $W^{2,2}\cap W^{1,\infty}$, and under certain additional conditions on $\{F_{t_{l}}\}$ such a parametrization is a diffeomorphism of class $W^{4,2}$. Finally, if the initial immersion $F_0$ of a flow line $\{F_t\}$ is assumed to parametrize a Hopf-torus in $\mathbb{S}^3$ with Willmore energy not bigger than $8 π$, then we obtain more precise statements about the flow line $\{F_t\}$ as $t \nearrow T_{max}(F_0)$. This insight will finally yield a criterion for full convergence of such flow lines of the MIWF to parametrizations of the Clifford torus - up to Möbius-transformations of $\mathbb{S}^3$ - as $t \nearrow \infty$.
