A Nash-Kuiper theorem for isometric immersions beyond Borisov's exponent

2025/03/18 by Cao, Wentao, Hirsch, Jonas, Inauen, Dominik · 2 citations
#53C21 #57N35 #58D10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.13867

Abstract

Given any short immersion from an n-dimensional bounded and simply connected domain into ℝⁿ⁺¹ and any Hölder exponent α<(1+n²-n)^-1, we construct a C^{1, α} isometric immersion arbitrarily close in the C⁰ topology. This extends the classical Nash--Kuiper theorem and shows the flexibility of C^{1, α} 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.

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• Isometric Immersions and Weak Solutions to the Darboux Equation
• A Nash-Kuiper theorem for isometric immersions in a high codimension

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