Research On Two Problems Of Submanifolds

In this thesis,we mainly study the following two problems on submanifolds.First,we study the problem on logarithmic Sobolev inequality on submanifolds in Riemannian manifolds.Logarithmic Sobolev inequality is a very common type of inequality,it played a key role in Perelman’s proof of the Poincare conjecture using Ricci flows,and has many applications in fields such as mean curvature flow.In 2019,Brendle extended the logarithmic Sobolev inequality on Gross’ s Euclidean space to the submanifold version of Euclidean space.Based on a better property,found by us,of the contact-like set which was defined by Brendle using the ABP method and a representation of the asymptotic volume ratio of a complete non compact Riemannian manifold with non negative Ricci curvature which was found by us,on the one hand,we proved a logarithmic Sobolev inequality on a closed submanifold in a complete noncompact Riemannian manifold with nonnegative sectional curvature and Euclidean volume growth.This extends the results of Gross and Brendle.On the other hand,according to that the asymptotic volume ratio of a complete noncompact Riemannian manifold with nonnegative Ricci curvature is invariant to metric dilation,as an application of this inequality,we also obtained the nonexistence of any codimensional closed minimal submanifolds in a complete noncompact Riemannian manifold with nonnegative sectional curvature and Euclidean volume growth.On the one hand,this result can be seen as a supplementary generalization to Agostiniani,Fogagnolo,and Mazzieri’s results on the nonexistence of closed minimal hypersurfaces in complete noncompact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth in the case of high codimensional minimal submanifolds.On the other hand,this result can also be used as a corollary of Brendle’s isoperimetric inequality.Next,we studied the rigidity problem of a 4 dimensional complete self-shrinker in5-dimensional Euclidean space with constant square norm of the second fundamental form.Self-shrinkers are closely related to the self shrinking solution of the mean curvature flow,they play an important role in the singularity analysis of mean curvature flows.Therefore,the study of these singular models is of paramount importance.For the classification problem of n dimensional complete self-shrinker in(n + 1)-dimensional Euclidean space with a constant second fundamental form,a complete classification result was obtained by Cheng Qingming and Ogata only when n = 2,partial results were obtained by Cheng,Li and Wei.This problem still opens in high-dimensional cases.For this,first,we obtain a rigid result for a 4dimensional complete self-shrinker in 5-dimensional Euclidean space with constant square norm of the second fundamental form.With the help of research techniques of Peng and Terng in the research of Chern conjecture for minimal hypersurfaces in a unit sphere,we obtain a representation of the invariant A-2Bon self-shrinkers.Previous studies of this type of problem have focused on discussing one of the maximum,minimum,or null point of a particular function.Differently,we first discuss at the null point of corresponded function,after obtaining some global information,turn to the maximum and minimum points of the function for further discussion.Finally,as an application,we obtained a new classification result for 4 dimensional complete self-shrinkers under the above conditions.