| This paper mainly studies the global existence and curvature blowup of Yamabeharmonic flow under compact manifold without boundary,rigidity of compact gradient Yamabe-harmonic soliton and monotonicity of the first eigenvalue of geometrical operators under Rescaled Yamebe flow.In this paper,the specific problems are as follows.On one hand,the issues about Yamabe-harmonic flow:(1)we give higher derivative interior estimaties of Riemannian curvature and Lapse function of Yamabe-harmonic flow under locally conformally flat manifold.(2)we prove the global existence of Yamabe-harmonic flow under locally conformally flat manifold.(3)we prove finite time singularity theorem of Yamabe-harmonic flow,namely Riemann curvature uniform boundedness yielding non-existence of finite time singularities.(4)we introducte of Harnack quantity of compact gradient Yamabe-harmonic soliton,and prove that n(≥ 3)-dimensional cloesd gradient Yamabe-harmonic soliton has constant scalar curvature which equals to Lapse function.(5)we introduce the absolute compound curvature of Yamabe-harmonic flow,and classify the finite time singularities of Yamabe-harmonic flow.(6)we prove monotonicity formula of Yamabe-harmonic flow.On the other hand,the first eigenvalue of geometric operator along curvature flow:(1)we calculate the evolution equation of the first eigenvalue of the operator-Δ +a R along the Rescaled Yamabe flow.(2)we prove monotonicity formula of the first eigenvalue of-Δ + a R along the Rescaled Yamabe flow.(3)we prove monotonicity formula of the first eigenvalue of-Δ + a R along Rescaled Yamabe flow. |
