Existence And Uniqueness Of Ricci Flow On Surfaces With Initial Curvature Unbounded Below

In 1982, Richard Hamilton in introduced Ricci flow equaiton in[H1], and started the research of Ricci flow theory. As an important analyzing tool, Riccci flow theory is widely applied in Geometrical Analysis and Physics, such as the proof of Poincare conjecture,the research of black hole in astrophysics etc. On the other hand, as a parabolic equation, the research of existence and uniqueness of Ricci flow under various of initial conditions is also very important to the development of equation field. For this reason, since Hamilton proved the short-time existence and uniqueness of Ricci flow on closed manifolds in[H1], the research of existence and uniqueness of Ricci flow is still a front topic today.In[H1] Hamilton proved the following theorem:Theorem 0.0.4.If(Mn, g0) is a closed Riemannian manifold, with go smooth, then there exists a unique Ricci flow g(t),t∈[0,δ),δ> 0, with g(0)= g0.The theorem is very great, though the original proof is very complicated. Not long after, DeTurck improved Hamilton's proof by using the Ricci-DeTurck flow, and this method has become a standard method in Ricci flow research.In 1989, Wanxiong Shi extended Hamilton work to noncompact case, and proved the following theorem:Theorem 0.0.5.([S1])Let (Mn, g0) be a complete noncompact Riemannian manifold, with Riemannian curvature Rm bounded, then there exists a complete Ricci flow g(t), t∈[0, t), such that g(0)= g0,and supM×[0,T)|Rm|=∞or T=∞。On the research of uniqueness of Ricci flow on complete noncompact manifold, there have been plenty of results these years too. In[LT], Peng Lu and Gang Tian used DeTurck's trick to prove the uniqueness of the standard solution of Ricci flow on Rn, n≥3, which is radially symmetric about the origin. Hsu extended Lu and Tian's result in [Hs], and proved the uniqueness of the solution of the radically symmetric solution of the Ricci harmonic flow associated with the standard solution of Ricci flow. Nevertheless, the first general uniqueness result is proved by Bing-Long Chen and Xi-Ping Zhu in 2006:Theorem 0.0.6.([CZ]) Let (M, g) be a complete noncompact smooth Riemannian manifold, with Rm(g) bounded. Suppose there exist two Ricci flow g1(t) and g2(t), with t∈[0,T], and g1 (0)= g2(0)= g. If both Rm(g1(t)) and Rm(g2(t)) are bounded for t∈[0,T], then g1 (t)＝g2(t).Due to the importance of Ricci flow in both mathematical and physical field, it's necessary to study the existence and uniqueness of Ricci flow under weaker initial conditions.In 2007, Peter Topping firstly tried to prove the existence of Ricci flow on surfaces with unbounded initial curvature in [T1]. This is a very interesting try, though there were a few defaults inside. Topping's theorem is as follows:Theorem A Let M be an open surface equipped with a smooth metric g, and the Gaussian curvature is bounded above only. Then, there exists a constant T> 0 depending only on the supremum of K(g), such that a smooth Ricci flow g(t) exists on M for t∈[0, T], and the Gaussian curvature is bounded (?)t∈[t0, T], (?)t0> 0.In Chapter 4 of this paper, we give a complete proof, see also in[CY].Besides, as Ricci flow in dimension 2 remains the conformal class of initial met-ric, we can simplify the Ricci flow equation and prove the uniqueness under certain assumption.The uniqueness theorems will be proved are:Theorem B Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K< 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T<∞, and satisfy for some positive constant C, here po(p,x) means the distance between x and a fixed point p with respect to initial metric g(0),i=1,2, then g1(x,t)＝g2(x,t),(?) (x,t)∈M×[0,T].Besides, we still haveTheorem C Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K> 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T< K-1/2, and satisfy then g1(x,t)=g2(x,t),(?)(x,t)∈M×[0,T].When K> 0, we have the following theorem:Theorem D Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) bounded above by K, K> 0. If now g1(t) and g2(t) are both Ricci flow, for (x, t)∈M×[0, T], with g(0),0< T< K/2,and they both satisfy3. there exists some domain E and a small time interval [0, t0](?) [0, T], such that Ki(x, t) is nonpositive for (x, t)∈(M\Σ)×[0, t0],i=1,2, then g1(x, t)= g2(x,t), (?)(x, t)∈M×[0, T].All the uniqueness theorems mentioned above will be proved in chapter 5.