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A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent
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abstract
Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any H\"older exponent $\alpha<(1+n^2-n)^{-1}$, we construct a $C^{1, \alpha}$ isometric immersion arbitrarily close in the $C^0$ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of $C^{1, \alpha}$ isometric immersions beyond Borisov's exponent. In particular, for $n=2$, the regularity threshold aligns with the Onsager exponent $1/3$ for the incompressible Euler equations. Our proof relies on three novelties that allow for the cancellation of leading-order error terms in the convex integration scheme: a new corrugation ansatz, an integration by parts procedure, and an adapted algebraic decomposition of these errors.
Forward citations
Cited by 3 Pith papers
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Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films
Every short immersion into R^{d+k} is uniformly approximable by C^{1,α} isometric immersions for α < min{(r+β)/2, 1/(1+d(d+1)/k)}.
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Isometric Immersions and Weak Solutions to the Darboux Equation
For Holder exponents greater than 1/2, weak C^{1,theta} solutions of the Darboux equation are equivalent to C^{1,theta} isometric immersions, via a new distributional Gaussian curvature and flatness criterion.
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A Nash-Kuiper theorem for isometric immersions in a high codimension
For n≥3, any short immersion of an n-dimensional domain into R^{2n} can be uniformly approximated by C^{1,θ} isometric immersions with θ close to 1/n in odd dimensions and 1/(n+1) in even dimensions.
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