The Infinitesimal Rigidity Of Convex Surfaces

The isometric embedding is a hot issue in differential geometry.After Riemann proposed Riemann manifolds,a question was naturally raised:whether a Riemann manifold is a submanifold in some Euclidean space,that is,the existence of isometric embedding.Isometric embedding problems have important applications in physics and geometry.The existence and uniqueness of isometric embedding problem is the focus of research in differential geometry.The uniqueness problem of isometric embedding is the rigidity of the surface,and its linearization is called infinitesimal rigidity.It means that there is no nontrivial continuous motion for isometric embedding to maintain rigidity under proper metric.This is crucial for solving the linearized isometric embedding problem.This thesis mainly studies the infinitesimal rigidity of two types of convex surfaces.The first type is the infinitesimal rigidity of convex surfaces with planar boundary.A new proof is given by using the dual relation of linearized isometric embedding problem.The key point is to linearize the isometric embedding problem first,and prove the dual relation between the linearized isometric embedding system and the homogeneous linearized Gauss-Codazzi system.And then we find the dual condition for the boundary condition.The infinitesimal rigidity is transformed into the uniqueness of the solution of the dual problem.Based on the idea of Li^[2],it is proved that this kind of convex surface has non-infinitesimal rigidity in three-dimensional Euclidean space when its Gauss map covers hemisphere.The second type is the convex surface constructed in the Minkowski problem.On the basis of reviewing the Minkowski problem,it proves a special case that is the infinitesimal rigidity ofS² with planar boundary.It linearized the equation of Minkowski problem,and then we proved the uniqueness of the solution of the linearized equation by using separation of variables method and the properties of the fundamental solutions in the partial differential equation.A new proof is given for the infinitesimal rigidity of this kind of special surface.Finally,the corresponding Reilly formula and Pohozave formula are obtained for S² with planar boundary in Minkowski problem.It is found that the infinitesimal rigidity can be proved as long as a suitable weight function V is found.Therefore,it can be extended to the general case,namely the general convex surface in Minkowski problem,which can also be proved in this way.It can refer to the idea of Xia^[21].