Vanishing theorems, higher order mean curvatures and index estimates for self-shrinkers
In this paper the authors investigate non-compact self-shrinkers of the mean curvature flow both in general codimension and in codimension 1 in particular. First they give a new proof of a classification theorem for self-shrinkers in arbitrary codimension due to [\textit{H.-D. Cao} and \textit{H. Li}, Calc. Var. Partial Differ. Equ. 46, No. 3--4, 879--889 (2013; Zbl 1271.53064); \textit{Q. Ding} and \textit{Y. L. Xin}, Trans. Am. Math. Soc. 366, No. 10, 5067--5085 (2014; Zbl 1316.53076)], and show an analytic result by adapting an argument from \textit{S. Pigola} et al. [Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique. Basel: Birkhäuser (2008; Zbl 1150.53001)]. With these preparations they obtain a gap result for the norm of the second fundamental form for complete self-shrinkers in arbitrary codimension, extending a theorem of \textit{M. Rimoldi} [Proc. Am. Math. Soc. 142, No. 10, 3605--3613 (2014; Zbl 1297.53048)] in the codimension-1 case. They also derive some estimates involving the higher-order mean curvatures of an orientable self-shrinker in the codimension-1 case. Finally they give some results on the index of a self-shrinker in the codimension-1 case.
scientific article; zbMATH DE number 6923334
scientific article