奇数σ_k-曲率流的奇异旋转自相似环面
Singular Rotational Self-Similar Tori for Odd $σ_k$-Curvature Flows
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中文总结 AI 辅助
针对奇数k的σ_k-曲率流，构造了正则性低于经典要求的旋转自相似环面，通过多种技术求解退化轮廓方程组，证明该类环面是满足流方程的唯一解。
中文摘要 AI 辅助
对于每一对满足3≤k<n且k为奇数的整数，我们在ℝ^{n+1}中构造了一个紧致嵌入的旋转环面，其相似膨胀在索伯列夫几乎处处意义下满足未归一化的σ_k-曲率流。该环面的轮廓曲线具有赫尔德正则性C^{1,1/k}，且对所有1≤p<k/(k-1)具有索伯列夫正则性W^{2,p}。除两个奇异纬圈外，该环面是光滑的；整体上，流方程通过相关利普希茨边界的弱形状算子来解释。在旋转对称下，自相似方程〈X,ν〉=-σ_k（其中X为位置向量，ν为单位法向量）可约化为退化轮廓方程组。我们结合奇数次去奇异性、打靶论证、一致径向与轴向界，以及打靶参数与圆柱半径间的严格间隙求解该方程组。经典的C^2旋转环面无法满足孤子方程，因此在旋转环面类内正则性的损失是不可避免的。
英文摘要
For every pair of integers $3\leq k<n$ with $k$ odd, we construct a compact embedded rotational torus in $\mathbb{R}^{n+1}$ whose homothetic dilations satisfy the unnormalised $σ_k$-curvature flow in a Sobolev almost-everywhere sense. Its profile curve has Hölder regularity $C^{1,1/k}$ and Sobolev regularity $W^{2,p}$ for every $1\leq p<k/(k-1)$. Away from two singular latitudes the torus is smooth; globally, the flow equation is interpreted using the weak shape operator of the associated Lipschitz boundary. Under rotational symmetry, the self-similar equation $\langle X,ν\rangle=-σ_k$, where $X$ is the position vector and $ν$ is the unit normal, reduces to a degenerate profile system. We solve this system by combining an odd-power desingularisation, a shooting argument, uniform radial and axial bounds, and a strict gap between the shooting parameters and the cylindrical radius. No classical $C^2$ rotational torus can satisfy the soliton equation, so the loss of regularity is unavoidable within the rotational toroidal class.
发表机构
- Fudan University（复旦大学）
- South China Normal University（华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。
