Blow-up Analysis And Its Geometric Applications

Blow-up analysis is an important topic in the study of partial differential equations and geometric analysis, which usually include energy identity and neck analysis. In the present paper, we are concerned on the popularization of harmonic maps and harmonic map flow, such as Dirac-harmonic maps, biharmonic maps and biharmonic map flow. I will prove in this paper that there is no neck during the blow-up process of Dirac-harmonic maps and biharmonic maps. On the other hand, I will construct some initial maps and the target manifold with some topology condition to prove the biharmonic map flow must blowup in finite time. In detail, the main results of the paper are as follows.â… .Neck analysis of Dirac-harmonic mapsTheorem0.1. Let (M, g) be a compact Riemannian surface and SM be the spin bun-dle over M.(N, h) be another compact Riemannian manifold. For a sequence of smooth Dirac-harmonic maps {(φκ,ψκ)} with uniform bounded energy E(φκ,ψκ）≤∧<A∞, we assume that{(φκ,ψκ)} weakly converges to a Dirac-harmonic map{(φ,ψ)} in W1,2(M, N)×L4(∑M(?)RK) with a finite blow-up points {pl,..., pI}, then after taking subsequence, still denoted by{(φκ,ψκ)}, we can find a finite set of Dirac-harmonic spheres (σil,fεil):S2â†'N,i=1,....I;l=1,...,Li such that the image φ(D) UiI=1UlLi=1(σil(S2)) is a connected set.â…¡.Neck analysis of biharmonic mapsTheorem0.2. Suppose B4(?) R4is a unite sphere and (N, h) is a compact Riemannian manifold. Let Ui be a sequence of biharmonic maps from B4to N satisfying for some A>O. Assume that there is a sequence positive λiâ†'O such that ui(λix)â†'w on any compact set K K (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then,Remark0.3. For convenient, we make a unnecessary assumption in the above theorem, that is there is only one bubble in the blow-up process. But, according to Ding and Tian[15I’s induction method, we know the above theorem also holds for two or more bubbles.By the proof of above theorem, I can give a new method to prove the energy identity and removable singularity of biharmonic maps. So, we can get the following two corollary.Corollary0.4. Under the same assumption of above theorem, we haveCorollary0.5. Let u be a smooth biharmonic map on B1\{0}. If then u can be extended to a smooth biharmonic map on B1.â…¢.Finite time blow-up of biharmonic map flowTheorem0.6. Let ui be a sequence of approximate biharmonic maps from B4to (N, h) satisfying with for some∧>0and p≥4/3. Assume that there is a positive sequence λiâ†'0such that on any compact set K C (?) R4, that ui converges weakly in W2,2to u∞and that w is the only bubble. Then, Theorem0.7. Suppose that M’is any closed manifold of dimension m>4with nontrivial Ï€4(M’) and let M’##Tm be the connected sum of M’ with the torus of the same dimension. For any Riemannian metric g on M, we can find (infinitely many) initial map u0:S4â†'M such that the biharmonic map flow starting from u0develops finite time singularity.