A Liouville Type Theorem For Solutions To Parabolic K-hessian Equations

The theory of fully nonlinear partial differential equations originated from the study of Weyl problem and Minkowshi problem in classical differential geometry and Calabi conjecture in K¨ahler geometry.After the breakthrough in 1970 s,fully nonlinear partial differential equations have developed into an important branch of mathematics.The k-Hessian equation is a kind of fully nonlinear partial differential equation.It appears in a series of important mathematical problems such as the prescribing Weingarten curvature problem and Minkowski problem.In this paper,we consider the Liouville theorem for the entire solutions to the parabolic k-Hessian equation.By establishing the Pogorelov estimate of solutions to the parabolic k-Hessian equation,we obtain the Liouville theorem for(k+1)-convex-monotone solutions satisfying quadratic growth.The main content of this paper includes the following aspects: The first chapter introduces the research background of k-Hessian equation and the main content of this paper.In the second chapter,some mathematical notations and lemmas are given.In the third chapter,we establish the Pogorelov estimate of solutions to the parabolick-Hessian equation,and on this basis,we prove Liouville theorem.Finally,we put forward some follow-up questions that are worth exploring.