The Bernstein problem for timelike surfaces

The aim of this thesis is to study Bernstein's Theorem for timelike surfaces. In the first chapter, there is information about the history of Bernstein's Theorem, the transfer of the Bernstein's Theorem to higher dimensions, the Lorentz spaces and the adaptation of the Bernstein's Theorem to the Lorentz spaces. In the second part, there are some definitions, theorems and proofs about the geometry of semi-Riemannian manifolds and Lorentz spaces. In the third chapter, the graphs on the timelike and spacelike planes are found by obtaining the shape operators and the equations that they need to provide to be minimal surface. In the fourth chapter, minimal surface finding problem is reduced classification problem of isometric immersions from E² to H₁³ . Then, the first and second fundamental forms of isometric immersions from E² to S³ are studied according to the global asymptotic coordinates {u,v} in E² and according to the Euclidean coordinate system {x,y} respectively. Then the isometric immersions are classified from E² to H₁³. After that, theorems are given to produce the entire surface that has zero mean curvature. Finally, sectional curvatures of such graphs with zero mean curvature are calculated in several ways.