Rigidity and Flexibility of Isometric Extensions

Abstract

In this paper we consider the rigidity and flexibility of C 1,θ isometric extensions and we show that the Hölder exponent θ 0 = 1 2 is critical in the following sense: if u ∈ C 1,θ is an isometric extension of a smooth isometric embedding of a codimension one submanifold Σ and θ > 1 2 , then the tangential connection agrees with the Levi-Civita connection along Σ. On the other hand, for any θ < 1 2 we can construct C 1,θ isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for C 1,θ isometric embeddings, θ < 1 2 , of compact Riemannian manifolds with C 1 metrics and sharper amount of codimension.

Very weak solutions to the two-dimensional Monge-Ampére equation

“…The idea of absorbing the error terms coming from the oscillations into the decomposition first appears in [13] in the context of isometric embeddings into Euclidean space with high codimension, where the author makes use of the freedom of high codimension to absorb such errors with arbitrary precision. This allows for the construction of isometric embeddings of class C 1,θ for all θ < 1, given that the metric g is regular enough and the codimension high enough (this idea is also used for less regular metrics and smaller codimension in [2,5,7]).…”

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

“…As it is suggested in Remark 1.2, the main challenging problem is to extend the results of this paper to the case α > 1/2. Apart from partial results regarding isometric extensions [8,4] for this regime, the problem is still largely open and cannot be answered using the current methods. Here, we conjecture that slight improvements are possible regarding the 2/3 regime, in the line of analysis in [26,25].…”