The Convexity Of The Level Set Of Extremely Strict Space-like Hypersurfaces In The Form Of Two-dimensional Space

For an elliptic partial differential equation,it is interesting and important to consider some geometrical properties of the solution.Particularly,the level sets of the solution to elliptic partial differential equations.The constant rank theorem combing with deformation process is applied in this paper to discussed some geometrical properties of the level sets of maximal strictly space-like hypersurfaces defined on 2-dimensional space forms.Finally,we derive some geometrical properties,including the regularity and the strict convexity and the curvature description of the level sets.The paper is organized with five section:In section 1,we introduced the history of this research and the relevant conclusions.Then we give two theorems in this paper,Theorem 1.4 and theorem 1.6.In section 2,we list the notations and the preliminaries being used during the process of the proof.Especially,we deduce the equation of the maximal space-like graph on Riemannian manifolds and the curvature of the level sets.In section 3,we will derive a constant rank type theorem to describe a property about the convex level sets.In section 4,we set out to prove Theorem1.4 and in section 5 we come to finish the proof of Theorem 1.6 by a lengthly calculation.The two main theorems of this paper are as follow:Theorem 1.4 Let?M²,g?be a space form with constant sectional curvature?,and?₀and?₁be bounded smooth strictly convex domains in M²,???₁????₀.Assume that the following maximal strictly space-like graph equation defined on?=?₀\???₁????1.3?has a smooth strictly space-like solution u on???.Then?u?0 is valid everywhere on???and all the level sets of u are strictly convex with respect to?u.Theorem 1.6 Let u be the solution to equation?1.3?on a domain in Riemannian manifold M²with constant Gaussian curvature denoted by?.Let k_gbe the geodesic curvature of the level sets of u.For?=???,the following relation is satisfied:???Where a_ij=?1-|?u|²??_ij+u_iu_jis the maximal operator.