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Van Abel June 8, 2023ArXivconvergence, Minkowski inequality, optimal constant Leave a comment

On the Minkowski inequality near the sphere

Title: On the Minkowski inequality near the sphere Authors: Otis Chodosh, Michael Eichmair, Thomas Koerber Categories: math.DG Comments: All comments welcome \\ We construct a sequence $\{\Sigma_\ell\}_{\ell=1}^\infty$ of closed, axially symmetric surfaces $\Sigma_\ell\subset \mathbb{R}^3$ that converges to the unit sphere in $W^{2,p}\cap C^1$ for every $p\in[1,\infty)$ and such that, for every $\ell$, $$ \int_{\Sigma_{\ell}}H_{\Sigma_\ell}-\sqrt{16\,\pi\,|\Sigma_{\ell}|}

Van Abel June 8, 2023ArXivbranch points, bubbles, conformally invariant Lagrangians, harmonic maps, Morse index, upper semi-continuity results, Willmore immersions Leave a comment

Morse Index Stability of Willmore Immersions I

Title: Morse Index Stability of Willmore Immersions I Authors: Alexis Michelat and Tristan Rivi\`ere Categories: math.DG math.AP Comments: 96 pages MSC-class: 35J35, 35J48, 35R01, 49Q10, 53A05, 53A10, 53A30, 53C42 \\ In a recent work, F. Da Lio, M. Gianocca, and T. Rivi\`ere developped a new method to show upper semi-continuity results in geometric analysis, which they applied to conformally invariant Lagrangians in dimension $2$ (that include harmonic maps). In this article, we apply this method to show that the sum … Continue reading “Morse Index Stability of Willmore Immersions I”

Van Abel June 5, 2023MATHharmonic maps, Pohozaev Leave a comment

调和映射的Pohozaev恒等式

$\newcommand{\div}{\mathrm{div}\,}$ 我们知道调和映射的方程为 $$ \Delta u+A(u)(\nabla u,\nabla u)=0, $$ 其中$u:M^2\to N$是黎曼流形间的映射而$A(u)$是$N\hookrightarrow \mathbb{R}^n$在$u$处的第二基本形式。我们将用两种办法来证明如下的Pohozaev恒等式。 Theorem 1 (Pohozaev恒等式). 假设$u:M^2\to N$是光滑调和映射，则有 \[ \int_{\partial B_\rho} \lvert u_r \rvert^2=\int_{B_\rho}r^{-2} \lvert u_\theta \rvert^2. \]

Van Abel June 4, 2023MATHmetapost, 平均曲率流, 离散 Leave a comment

离散平均曲率流的一种数值模拟

给定一个$n$-多边形, 假设其顶点满足方程 \[ \dot v_i(t)=\frac{\nu_i(t)}{\| \nu_i(t) \|^2}, \] 其中$\nu_i(t)$是顶点$v_{i-1},v_i,v_{i+1}$构成的三角形之外接圆心。它可以视为连续情形下的曲线平均曲率流的一种离散推广。我们知道连续情形下，平均曲率流有所谓的Gage-Hamilton-Grayson定理，它表明平均曲率流保持简单曲线为简单曲线。但下面的数值模拟表面，这个平均曲率流不一定保持曲线的简单性。

Van Abel June 1, 2023MATH三角剖分, 代数拓扑 Leave a comment

关于代数拓扑曲面分类定理:I

曲面分类定理的第一步是使用三角剖分，将曲面转化为简单多边形。Massey的书上列举了正方体的三角剖分转换为多边形的例子。这里，我们来看另一些例子。其基本想法是，通过对给定的剖分三角形重新编号$T_1,T_2,\ldots, T_n$, 使得$T_i$与$T_1,\ldots, T_{i-1}$至少有一条公共边, $i=2,3,\ldots, n$. Example 1 ([1,Ex.~7.1, P.~21]). 三角剖分为 124 236 134 246 367 347 469 459 698 678 457 259 289 578 358 125 238 135 References W. Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics, Vol. 56, Springer-Verlag, New York-Heidelberg, 1977. 0—387. Reprint of the 1967 edition. MR0448331

Van Abel June 1, 2023杂记metapost, 代数拓扑, 曲面的分类 Leave a comment

代数拓扑里面简化多边形的作图程序

在代数拓扑里，我们将曲面视为将多边形的对应边粘贴而成的图形。当然一个重要的问题： 1. 如何将一个闭曲面三角剖分； 2. 如何从给定的三角剖分粘贴成多边形，使得三角剖分中公共的边作为多边形的内部的边、非公共边作为多边形真正的边。一个操作过程可以参考[1,Chap.~1, Sec.~6]. References W. Massey, Algebraic topology: an introduction, Graduate Texts in Mathematics, Vol. 56, Springer-Verlag, New York-Heidelberg, 1977. 0—387. Reprint of the 1967 edition. MR0448331