On length-preserving and area-preserving inverse curvature flows in the hyperbolic plane

In this paper, we study the area-preserving and length-preserving ${\kappa }^{\alpha }$-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane ${\mathbb{H}}^2$ for $\alpha <0$ and prove that convexity is preserved along the flows. Assuming that the flows exist for all time, we show that the evolving curves converge smoothly to geodesic circles. Furthermore, we also derive a sufficient condition for global existence of the flows.