Finiteness Of Switches In Discrete Yamabe Flow

Conformal mapping keeps the angle invariant, and the geometry studying con-formal mapping is called the conformal geometry. Conformal geometry is closely re-lated with Riemann Surface, harmonic analysis, differential geometry and so on, and has rich mathematical structure. Conformal geometry is also closely related with hy-dromechanics, electromagnetism and other physical subjects. Now discrete conformal geometry has more and more broad application in such as computer imaging, face recognition, medical imaging. Meanwhile, discrete geometry itself produced many interesting structures and problems.In2004, Luo gave a definition of conformal equivalence in the triangulated sur-face by defining the discrete conformal factor on the vertexes. Using the corresponding discrete Yamabe flow one gave a new method to compute the conformal mapping on the surfaces. This method has some advantages compared to old method. In2013, Luo and others introduced "switch" to the discrete Yamabe flow and get the discrete uni-formization theorem. Given a "proper" discrete curvature, the corresponding Yamabe flow would converge the the surface with the given curvature. One important question is wether the Yamabe flow need only finitely many switches. We proved this.We have already known that the Yamabe flow is on Rn=∪i=1mDi, which is an analytic cell decomposition, and the discrete Yamabe flow is continuous globally and analytic on each Di. Our problem equal to that Yamabe flow intersect the faces finitely many times. Using these facts, we may prove (1)locally Yamabe flow intersect the faces finitely many times,(2)near the limit point, Yamabe flow intersect the faces finitely many times. These2properties imply the global finiteness.In part (1) we use the Taylor expansion, carefully study the direction of flow in any subspace; In part (2) we characterized the expansion of the Yamabe flow on the limit point, and then use the argument in part (1).