GAP THEOREMS FOR COMPLETE lambda-HYPERSURFACES

摘要

An n-dimensional lambda-hypersurface X : M -> Rn+1 is the critical point of the weighted area functional integral(M) e(-1/4|X|2) d mu for weighted volume-preserving variations, which is also a generalization of the self-shrinking solution of the mean curvature flow. We first prove that if the L-n-norm of the second fundamental form of the lambda-hypersurface X : M -> Rn+1 with n >= 3 is less than an explicit positive constant K(n, lambda), then M is a hyperplane. Secondly, we show that if the L-n-norm of the trace-free second fundamental form of M with n >= 3 is less than an explicit positive constant D(n, lambda) and the mean curvature is suitably bounded, then M is a hyperplane. We also obtain similar results for lambda-surfaces in R-3 under L-4-curvature pinching conditions.