Gap type results for submanifolds with weighted mean curvature in Gaussian space

Abstract

We study complete n -dimensional submanifolds immersed in the Gaussian space $$\mathbb {G}^{n+p},$$ that is, the Euclidean space $$\mathbb {R}^{n+p}$$ endowed with its standard metric and the Gaussian weight $$e^{-|x|^2/4}$$ . More precisely, in codimension $$p\ge 2,$$ we consider submanifolds with parallel f -mean curvature vector, while in codimension one we deal with hypersurfaces having constant f -mean curvature. By combining Stokes’ Theorem with Kato’s inequality, and assuming suitable weighted integrability conditions involving the norm of the covariant derivative of the second fundamental form, we obtain gap type rigidity results for the norm of the second fundamental form. In particular, under appropriate upper bounds depending on the f -mean curvature, we prove that either | A | vanishes identically, in which case the submanifold is a hyperplane, or | A | is identically equal to the corresponding critical value. As consequences, we derive gap type results for complete self-shrinkers in Euclidean space.