In mathematical physics, Cauchy surfaces play a crucial role in defining initial data for the evolution of physical systems. The characteristic nature of Cauchy surface means that information about the future evolution of the system can be obtained from the data on the surface itself. However, not all Cauchy surfaces are characteristic. This article aims to investigate the noncharacteristic nature of Cauchy surfaces and their applications in various physical scenarios. We demonstrate that the noncharacteristicity of a Cauchy surface has important implications for the behavior of physical systems, including the existence of singularities and the formulation of initial value problems. We explore the applications of noncharacteristic Cauchy surfaces in classical and quantum field theory, general relativity, and quantum gravity. Our analysis provides a deeper understanding of the mathematical and physical properties of Cauchy surfaces and their significance in the study of fundamental physical phenomena. In this work we show that under the noncharacteristic condition one can prove a local existence result for the Cauchy problem similar to the semi-linear and quasilinear cases. For this, we consider the Eikonal equation and Hamilton-Jacobi equations. Therefore, we solve simultaneously for the characteristics and the solution value, resulting in curves that live in the solution graph.