BV weak solutions to Gauss–Codazzi system for isometric immersions

The isometric immersion problem for surfaces embedded into R3 is studied via the fluid dynamic framework introduced in Chen et al. (2010) [6] as a system of balance laws of mixed-type. The techniques developed in the theory of weak solutions of bounded variation in continuum physics are employed to deal with the isometric immersions in the setting of differential geometry. The so-called BV framework is formed that establishes convergence of approximate solutions of bounded variation to the Gauss–Codazzi system and yields the C1,1 isometric realization of two-dimensional surfaces into R3. Local and global existence results are established for weak solutions of small bounded variation to the Gauss–Codazzi system for negatively curved surfaces that admit equilibrium configurations. As an application, the case of catenoidal shell of revolution is provided.