On the isometric version of Whitney’s strong embedding theorem.(English)Zbl 07966471
Summary: We prove a version of Whitney’s strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any \(n\)-dimensional smooth compact manifold admits infinitely many global isometric embeddings into \(2n\)-dimensional Euclidean space, of Hölder class \(C^{1, \theta}\) with \(\theta < 1 / 3\) for \(n = 2\) and \(\theta < (n + 2)^{- 1}\) for \(n \geq 3\). The proof is performed by Nash-Kuiper’s convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 58A07 | Real-analytic and Nash manifolds |
Keywords:
isometric embedding;Nash-Kuiper theorem;absorbing higher order error;convex integration;corrugationCite
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