Self-expanders of mean curvature flow and constant weighted mean curvature hypersurfaces - Villca, Saul Ancari (2023)
In this thesis, we study self-expanders of the mean curvature flow and special constant weighted mean curvature hypersurfaces in Euclidean space. In the first part of this thesis, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growth and the finiteness of the weighted volumes. We get some properties that characterize the hyperplanes through the origin as self-expanders. We give sufficient conditions for the self-expander hypersurfaces to be products of self-expander curves and flat subspaces. We also study the spectrums of the weighted Laplacian and the L-stability operator. The upper bound of the bottom of the spectrum of the weighted Laplacian, and upper and lower bounds for the bottom of the spectrum of the L-stability operator are given. In the second part, we study two kinds of constant weighted mean curvature hypersurfaces in Euclidean space: λ-hypersurfaces and λ-self-expanders, that are the hypersurfaces Σ whose mean curvature H satisfies H = λ + hx,ni 2 and H = λ − hx,ni 2 , respectively, where λ ∈ R is constant, x is the position vector in R n+1 and n is the outward unit normal field on Σ. They are solutions of the Gaussian isoperimetric problem and the isoperimetric problem with the same weighted volume form as self-expanders’, respectively. We obtain various results that characterize the hyperplanes, spheres and cylinders as λ-hypersurfaces and λ-self-expanders, respectively. Besides, in the case of properly immersed λ-self-expanders, we get the discreteness of the spectrum of the weighted Laplacian, give the upper and lower bounds for the bottom of the spectrum of the weighted Laplacian and prove an inequality between the bottom of the spectrum of the weighted Laplacian and the bottom of the spectrum of the L-stability operator. The results in the thesis have been partially included in our articles [AC20], [AM21a] and [AM21b].