Hyperbolic Inverse Curvature Flows In Warped Products

Hyperbolic inverse curvature flow in recent years is one of the popular research fields in differential geometry.The research theory is closely to Riemannian geometry,partial differential equations,etc.Let M0 be a compact,mean convex,star—shaped smooth hypersurface of the n+1 dimensional warped product space Wnn+1(n≥2),which is given an embedding X0:Nn→Wn+1.We study the following initial value problem:#12where F defined on an open cone Γ is a function of principle curvature,v is the unit outward normal vector on X(x,t).X0 be a smooth function and X1 be a smooth vector-valued function on Nn.We prove the short time existence of the solution of this equation,and then calculate the evolution equations of some geometric quantities on the smooth hypersurfaces.