On bending of a convex surface to a convex surface with prescribed spherical image (Q1916622)

Let \(F\) be a regular simply connected convex surface with a boundary \(\partial F\) in \(E^3\) such that \(\int_F K dS< 2\pi\) and \(k_g|_{\partial F}> 0\) where \(K\) is the Gaussian curvature of \(F\) and \(k_g\) is the geodesic curvature of \(\partial F\). Let \(G\) be a convex domain on the unit sphere bounded by a smooth curve \(\partial G\) and contained strictly in a hemisphere, and let \(mG\) be the area of \(G\). Let \(P\in \partial F\) be an arbitrary point of \(\partial F\) and let \(P^* \in \partial G\) be an arbitrary point of \(\partial G\). If \(mG= \int_F K dS\), then there exists a continuous bending of \(F\) to a convex surface \(F'\) such that the spherical image of \(F'\) coincides with \(G\) and \(P^*\) is the image of the point of the boundary \(\partial F'\) of \(F'\) corresponding to the point \(P\in \partial F\) under the isometry.