The Positive Mass Theorem On Non-Spin Manifolds With Distributional Curvature And Applications Of Ricci Flow

In this thesis,we prove the positive mass theorem on non-spin manifolds with dis-tributional curvature by using the mollifiers and Ricci flow technique.In our case,the regularity of the metric is low so that the curvature can only be defined in the distribu-tional sense.With this low regular metric,we prove that the positive mass theorem still holds under some extra geometric conditions.The positive mass theorem was first proved by Schoen and Yau in 1979 for mani-folds whose dimension is less than eight and then by Witten in 1981 for spin manifolds with any dimension.In physical terms,the positive mass theorem says that an isolated gravitational system with nonnegative local energy density must have nonnegative to-tal energy.Mathematically speaking,the positive mass theorem means a Riemannian manifold with nonnegative scalar curvature must have nonnegative mass.Moreover,the manifold is isometric to the standard Euclidean space when the mass is zero.We consider whether this problem can be true under low regular metric and the necessary conditions for the positive mass theorem.Our theorem is the following.Given a smooth Rimannian n-manifold(M,h),where h is a smooth background metric,we consider another asymptotically flat metric g ? C0 ? W-q1,p,p>n,q>n/2 on M,where the function space W-q1,p is the weighted Sobolev space that indicates the decay rate of the matric effectively.Without the assumption of spin structure,we show that the generalized ADM mass mADM(M,g)is non-negative under the following conditions.1.g has bounded curvature Rm(g)C'?Rm(g)?C in the sense of Aleksandrov,and the set ?={x ? M:Rm(g)(x)?0} is compact;2.g has non-negative scalar curvature distribution Rg?Rg,u??0 for any smooth compactly supported non-negative function u on M;3.the Ricci curvature distribution?Rij,u?is in L-q-2p;4.(?)where l(x)=dist(x,?),for any smooth background metric h,·and(?)are inner product and Levi-Civita connection with respect to h,V is as in the definition of scalar curvature distribution;5.Rg is a finite,signed measure outside a compact set;6.q=n-2.Our approach is to construct smoothings g? of the metric such that g? converge to g,and use Ricci flow to get a family of Ricci flows g?(t)in a time interval.Condition 1 guarantees that the smoothings g? have bounded curvature and thus have the short time existence of Ricci flow on complete noncompact manifolds;condition 2 requires the scalar curvature distribution is nonnegative,together with condition 4,we can get the scalar curvature of g? is almost nonnegative and after using the Ricci flow,we obtain the nonnegativity of the scalar curvature of g?(t);using the decay rate of Ricci curvature,condition 3 ensures the decay rate of the smoothings g?,this results from the property of Laplace operator on the weighted Sobolev spaces;condition 5 and 6 is the reason for the mass being finite and for the mass of g? converging to that of g as ? tends to zero.Finally,we use classical positive mass theorem on g?(t)to get the mass of g? is non-negative.We also study the isoperimetric inequality on the steady gradient Ricci soliton.We prove that the isoperimetric inequality holds in the cigar steady soliton and in the Bryant steady soliton.Since both of them are Riemannian manifolds with warped product metric,we utilize the result of Guan-Li-Wang[13]to get our conclusion.For the sake of the structure of Ricci soliton,the conditions for the isoperimetric inequality is naturally satisfied.