具有有界各向异性一阶变差的varifolds的正则性

中文摘要 AI 辅助

我们证明了一个关于$m$-varifolds的$\u03b5$-正则性定理，其中平均曲率属于$L^p$（$p>m$），且相对于满足Michael-Simon不等式和二次暴露条件的各向异性被积函数：在足够平坦的密度为1的点附近，此类varifolds可表示为$C^{1,\alpha}$图。结合最近各向异性Michael-Simon不等式的证明，这为一大类各向异性被积函数（包括接近面积泛函的那些）在任意余维数下建立了各向异性Allard正则性定理。我们还首次展示了满足均匀标量原子条件和Michael-Simon不等式但不接近任何椭球范数的各向异性例子。这些包括在每一维和余维数下，对于明确的$q$范围的$\ell^q$范数，以及一类新的轴对称各向异性。

英文摘要

We prove an $\varepsilon$-regularity theorem for $m$-varifolds with mean curvature in $L^p$, $p>m$, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as $C^{1,α}$ graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the $\ell^q$ norms in every dimension and codimension, for explicit ranges of $q$, and a new class of axisymmetric anisotropies.

发表机构

• Bocconi University（博科尼大学）
• MIT（麻省理工学院）
• Columbia University（哥伦比亚大学）

机构由 AI 辅助整理，请以论文原文为准。