GAP AND RIGIDITY THEOREMS OF λ-HYPERSURFACES.

We study λ-hypersurfaces that are critical points of a Gaussian weighted area functional ... dA for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete λ-hypersurfaces in terms of the norm of the second fundamental form |A|. Second, we show that in one dimension, the only smooth complete and embedded λ-hypersurfaces in ℝ² with λ ≥ 0 are lines and round circles. Moreover, we establish a Bernstein-type theorem for λ-hypersurfaces which states that smooth λ-hypersurfaces that are entire graphs with polynomial volume growth are hyperplanes. All the results can be viewed as generalizations of results for self-shrinkers. [ABSTRACT FROM AUTHOR]