L2L2 norm preserving flow in matrix geometry

In this paper, we study a new L2L2 norm preserving heat flow in matrix geometry. We show that if the initial data has trace zero and has unit L2L2 norm, this flow has a global solution and enjoys the entropy stability in any finite time. We show that as the time is approaching infinity, the flow has its limit as an eigen-matrix of the Laplacian operator. Interesting operator convex property of heat equation is also derived.