Self-shrinkers And Applications Of Curvature Flow

This thesis mainly focuses on the studies of the properties of self-shrinking solutions of mean curvature flow, and the geometry application of inverse mean curvature flow.An immersed submanifold X:Mnâ†'Rn+P in Euclidean space is called a self-shrinker, if its mean curvature vector H and the position vector X satisfywhere⊥denotes the projection onto the normal bundle of M. Self-shrinker not only arises as the self-shrinking solution of mean curvature flow, but also describes singularity model of mean curvature flow, which plays an important role in the study of mean curvature flow. In the chapters3,4,5, we focus on the properties of self-shrinker. We prove that any complete noncompact properly immersed self-shrinker must have at least linearly volume growth estimate; We also study the classification of self-shrinker that satisfies H≠0and the principal normal v=H/|H|is parallel, and we show that Smoczyk’s theorem[1] on such self-shrinker also holds under a weaker condition. As applications of this result, we prove several rigidity theorems for self-shrinkers under some special assumptions. We next study the f-stability of self-shrinker with arbitrary codimension. Based on Colding-Minicozzi’s[2] work about f-stability of hypersurface self-shrinker, we calculate the first and second variation formula of the f-functional for self-shrinker with arbitrary codimension, and prove that an n-dimensional minimal submanifold in sphere Sn+P((?)2n) as a self-shrinker is f-stable if and only if it is the sphere Sn((?)2n). We also prove some f-stability theorems for self-shrinkers with parallel principal normal, and for product self-shrinkers.Apart from the mean curvature flow, the F-flow also attracts many interests, where by F-flow we means that the speed of the flow is a function F=F(k) of the principal curvatures k=(k1,···,Kn) of the flow hypersurface. When F=H, the F-flow is just the mean curvature flow. In chapter6of this thesis, we study the noncollapsing estimate of F-flow. We generalize Ben Andrews’s[3,4] noncollapsing estimates on F-flow in the Euclidean space to F-flow in the sphere and in the hyperbolic space.The geometric curvature flow is important, mainly because of its applications in geometry and topology. In the chapter7of this thesis, we use Gerhardt’s inverse curvature flow[5] to prove a sharp Alexandrov-Fenchel type inequality for star-shaped and2-convex hypersurface Mn(n≥2) in the hyperbolic space Hn+1:where σ2=σ2(k) is the second order elementary polynomial of the principal curvature k of M, ωn is the area of unit sphere Sn c Rn+1and|M|is the area of M. Moreover, equality holds in (4) if and only if M is a geodesic sphere in Hn+1.