AI 中文总结
研究具有负高斯曲率的曲面到三维欧氏空间等距浸入的存在性，将高斯 - 科达齐方程转化为双曲守恒律，运用相关理论建立\(W^{2,p}\)-等距浸入存在性，构造出的浸入第二基本形式无界但\(L^p\)可积。
AI 中文摘要
本文关注具有负高斯曲率的曲面到三维欧几里得空间的等距浸入的存在性理论。我们将高斯 - 科达齐方程，即等距浸入的偏微分方程，重新表述为具有非零源项的查普利金气体流的双曲守恒律。然后，通过运用不变区域和补偿紧致性理论，我们建立了几个一般度量族的\(W^{2,p}\)-等距浸入的存在性，对于任意有限指标\(p\)且在任意大的矩形域上。这些度量包括各种经典极小曲面以及等温坐标或“倒数型”的度量。在等距浸入问题的流体动力学表述中，我们专注于相关双曲守恒律的两个黎曼不变量保持有界且具有不同符号的情况，并通过熵分析获得初边值问题的\(L^p\)解。本文构造的等距浸入可能具有无界但\(L^p\)可积的第二基本形式。
英文摘要
This paper is concerned with the existence theory of isometric immersions of surfaces with negative Gaussian curvature into the three-dimensional Euclidean space. We reformulate the Gauss--Codazzi equations, i.e., the partial differential equations for isometric immersions, into hyperbolic conservation laws for the flows of Chaplygin gas with nonzero source terms. Then, employing theories of invariant regions and compensated compactness, we establish for any given $p \in [2,\infty[$ the existence of $W^{2,p}_{\rm loc}$-isometric immersions of various general families of metrics over infinite strips $\mathbb{R}\times [0,T]$ with arbitrarily large $T$. Such metrics include those of various classical minimal surfaces: helicoid, catenoid, and Enneper surfaces, as well as metrics in isothermal coordinates or of the ``reciprocal-type''. In our fluid dynamical formulation of the isometric immersion problem, we specialise in the case that the two Riemann invariants for the associated hyperbolic conservation law remain bounded and of distinctive signs, and obtain $L^p$-solutions to the initial-boundary value problem via the method of relative entropy with respect to a nonstationary ODE background. The isometric immersions constructed in this paper have second fundamental forms belonging to $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$.
Comments39 pages. In V3, existence of isometric immersions over $\mathbb{R}\times[0,T]$ for arbitrarily large $T$ has been obtained via relative entropy arguments. We also construct (many) examples for isometrics in $W^{2,p}_{\rm loc}$ but with unbounded second fundamental form
