Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form

[2607.18792] Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2607.18792 (math) [Submitted on 21 Jul 2026 ( v1 ), last revised 15 Aug 2026 (this version, v3)] Title: Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form Authors: Siran Li View a PDF of the paper titled Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form, by Siran Li View PDF HTML (experimental) Abstract: 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 $L^p_{\rm loc}\setminus L^\infty_{\rm loc}$. Comments: 39 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 Subjects: Differential Geometry (math.DG) ; Analysis of PDEs (math.AP) MSC classes: Primary: 53C42, 35L65, 35D30, Secondary: 35L60, 35Q31, 76N10 Cite as: arXiv:2607.18792 [math.DG] (or arXiv:2607.18792v3 [math.DG] for this version) https://doi.org/10.48550/arXiv.2607.18792 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Siran Li [ view email ] [v1] Tue, 21 Jul 2026 07:13:18 UTC (36 KB) [v2] Sun, 26 Jul 2026 00:21:01 UTC (40 KB) [v3] Sat, 15 Aug 2026 02:57:21 UTC (48 KB) Full-text links: Access Paper: View a PDF of the paper titled Existence of semiglobal $W^{2,p}$-isometric immersions for negatively curved surface metrics with unbounded second fundamental form, by Siran Li View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-07 Change to browse by: math math.AP References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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