A generalization due to Sorin Mardare of the fundamental theorem of surface theory for surfaces with little regularity asserts that, if, for any $p>2$, the components of a positive-definite $2\times 2$ symmetric matrix field in the space $W_{\mathrm{loc}}^{1,p}$ and the components of another $2\times 2$ symmetric matrix field in the space $L_{\mathrm{loc}}^p$ satisfy together the Gauss and Codazzi-Mainardi equations in a simply-connected open subset of ${\mathbb{R}}^2$, then there exists a surface defined in the three-dimensional Euclidean space ${\mathbb{E}}^3$ by an immersion with components in the space $W_{\mathrm{loc}}^{2,p}$, whose first and second fundamental forms are precisely the given matrix fields; besides, this surface is uniquely determined up to isometries in ${\mathbb{E}}^3$.

We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms for several Fréchet topologies, which include in particular the above spaces $W_{\mathrm{loc}}^{1,p}$ for the first fundamental form and $L_{\mathrm{loc}}^p$ for the second fundamental form, for any $p>2$.

Sorin Mardare对表面表面基本定理的一般性定理的推论认为，对于几乎没有规则性的表面， $p>2$，正定的组成部分 $2\times 2$ 空间中的对称矩阵场 ${\mathrm{w\; ^}}_{\mathrm{位置}}^{\mathrm{1个}，p}$ 和另一个的组成部分 $2\times 2$ 空间中的对称矩阵场 ${\mathrm{大号}}_{\mathrm{位置}}^p$ 在一个简单连通的的子集中满足Gauss和Codazzi-Mainardi方程 ${\mathbb{[R}}^2$，则存在一个在三维欧几里得空间中定义的曲面 ${\mathbb{Ë}}^3$ 通过将组件浸入空间中 ${\mathrm{w\; ^}}_{\mathrm{位置}}^{2，p}$，其第一和第二基本形式正是给定的矩阵字段；此外，该表面是唯一确定的，直到${\mathbb{Ë}}^3$。

我们在此确定，以这种方式定义的表面根据几种Fréchet拓扑的两种基本形式而不断变化，其中特别包括上述空间 ${\mathrm{w\; ^}}_{\mathrm{位置}}^{\mathrm{1个}，p}$ 对于第一个基本形式 ${\mathrm{大号}}_{\mathrm{位置}}^p$ 对于第二种基本形式，对于任何 $p>2$。