Rigidity and flexibility of isometric extensions
Wentao Cao, Dominik Inauen · Commentarii Mathematici Helvetici · 2023
In this paper we consider the rigidity and flexibility of C^{1, \theta} isometric extensions. We show that the Hölder exponent \theta_0=\frac12 is critical in the following sense: if u\in C^{1,\theta} is an isometric extension of a smooth isometric embedding of a codimension one submanifold \Sigma and \theta> \frac12 , then the tangential connection agrees with the Levi-Civita connection along \Sigma . On the other hand, for any \theta<\frac12 we can construct C^{1,\theta} isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for C^{1, \theta} isometric embeddings, \theta<\frac12 , of compact Riemannian manifolds with C^1 metrics and sharper amount of codimension.