Smooth local solutions to weingarten equations and ${\sigma }_k$-equations

Abstract

In this paper, we study the existence of smooth local solutions to Weingarten equations and ${\sigma }_k$-equations. We will prove that, for $2\le k\le n$, the Weingarten equations and the ${\sigma }_k$-equations always have smooth local solutions regardless of the sign of the functions in the right-hand side of the equations. We will demonstrate that the associate linearized equations are uniformly elliptic if we choose the initial approximate solutions appropriately.