Global Nash-Kuiper theorem for compact manifolds

Abstract

We obtain global extensions of the celebrated Nash-Kuiper theorem for C 1,θ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent θ < 1/5. This extends previous results on embedding 2-discs as well as higher dimensional analogues.

The geometry of C1,α flat isometric immersions

“…This result follows from the convexity of the image of such surfaces. This convexity fails for C 1,α isometric immersions of convex surfaces if α < 1/5, in which regime the above mentioned h-principle for isometric immersions holds true, as shown in [9,16], improving on the results by [6,13]. On the other hand, following Borisov [5], Conti et al proved in [13,14] that when α > 2/3, the isometric image of a closed convex surface is convex (from which it follows that the immersion is a rigid motion), and that more generally the h-principle cannot hold true for isometric immersions of any elliptic 2-manifold with or without boundary.…”

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

“…Remark 1.2. The above conclusions are not true if α < 1/5, even for smooth metrics with positive curvature [9,5]. The question remains open for the range 1/5 ≤ α ≤ 2/3.…”

“…To put our results in their proper context, our results must be compared with the statements in [29,22,30,1,2,31,6,7,16,9,5,10,25] on isometric immersions, in [19,28,35,23] on the rigidity of Sobolev solutions to the Monge-Ampère equation, and in [3,20,21,13,26,25] on geometric or topological properties of weakly regular deformations. 1.2.…”