Fine Level Set Structure of Flat Isometric Immersions
Fine Level Set Structure of Flat Isometric Immersions
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平面等距浸没的精细水平集结构
DOI：
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发表时间：
2011
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通讯作者：
Peter Hornung
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作者：
Peter Hornung
Pogorelov的一个结果断言，有界区域$${S子集mathbb {R}^2}$$到$${mathbb {R}^3}$$的C1等距浸入u，其法线取值于一个零区域集合，具有如下正则性：梯度$${f：=
在f的非退化水平集由在两个端点处与S的边界相交的直线段组成的意义上，f是“可展的”。受非线性弹性力学应用的启发，我们研究了当S是一个任意有界Lipschitz域时f的水平集结构。我们表明，f可以近似一致有界的映射与简化的水平集结构。我们还表明，域S可以被分解（控制剩余）成100多个子域，其中每个承认一个全球性的曲率线参数化。
A result by Pogorelov asserts that C1 isometric immersions u of a bounded domain $${S subset mathbb R^2}$$ into $${mathbb {R}^3}$$ whose normal takes values in a set of zero area enjoy the following regularity property: the gradient $${f :=
abla u}$$ is ‘developable’ in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.