Existence of smooth Isometric immersions

For generalized crumpling problem (GC), we need to characterize all W^{2, 2} isometric immersions immersions of a subset of $\mathbb {R}$^m into a small ball in $\mathbb {R}$^d. We investigated the associated geometric problem of characterizing the smooth/piecewise smooth isometric immersions in this situation. We showed following ``rigidity'' result [VWKG00] about the existence of immersions $\phi$ : $\mathcal {S}$ $\subset$ $\mathbb {R}$^m $\rightarrow$ $\Omega$ $\subset$ $\mathbb {R}$^d :

Theorem 4   If d $\geq$ 2m, we always have a smooth isometric immersion $\phi$ : $\mathcal {S}$ $\rightarrow$ $\Omega$.

If d < 2m, for every S there exists an r > 0 such that there exists no smooth (or even C³) isometric immersion $\phi$ : $\mathcal {S}$ $\rightarrow$ B_r^d, a d dimensional ball with radius r.

In contrast, classical results of Nash and Kuiper [Nas54,Kui55] show that a sufficient condition for the existence of C¹ isometries is d $\geq$ m + 1.

Refining Thm. 4, we conjecture a lower bound of the dimension of a minimal obstructions for periodic boundary conditions [DWVK01] -

Conjecture 5   The minimal obstruction has dimension at least 2m - d - 1. If d = m + 1, m $\geq$ 2, the minimal obstruction has dimension m - 1.

Numerical simulations of (GC) showed energy concentration on singular sets consistent with predictions this conjecture. Numerics also suggest that the lower bounds in this conjecture are sharp [DWVK01]. If this is indeed true, it will imply many interesting geometric results, e.g.

Conjecture 6   There is a countable collection of isolated points C $\in$ $\mathbb {R}$³ such that, there is a smooth isometric immersion of $\mathbb {R}$³ - C into the unit ball in $\mathbb {R}$⁵.

Cuurently, I am examining similar questions for isometric immersions of portions of the hyperbolic plane $\mathbb {H}$² into $\mathbb {R}$³, with certain curvature restrictions. This is the relevant geometric problem for the variational problem governing the multiple scale buckling in thin elastic sheets, seen in experiments by Sharon and coworkers at the university of Texas.