Prove
Lemma 8 ([Prescribing small curvature]). Let K_0∈C^∞((-1,1)^2) with K_0_∞≤10^…Lemma 7 ([Propagation to the pulse]). At θ=±δ/τ , on ξ≤Lr/2 , the functions…Proposition
Let u be the graph function and let Ω be the planar projection of the disk. This is a bounded domain with connected smooth boundary, and u is smooth on a neighborhood of Ω . Suppose the second fundamental form vanishes at every boundary point. The graph formula _ ij = u_ ij √1+Du^2 then gives D^2u=0 along ∂Ω . Differentiation along this connected curve shows that Du is a fixed vector a there. Hence u=a·x+c on the boundary for a constant c . The function v=u-a·x-c is zero on ∂Ω and has D^2v= D^2u<0 Ω. It cannot vanish identically. If it is nonzero, compactness gives either a positive interior maximum or a negative interior minimum. Its Hessian is semidefinite there, with nonnegative determinant, a contradiction. Thus the second fundamental form is nonzero at some boundary point. The Gauss equation and zero boundary curvature make its determinant zero, so its rank is one.