arXiv subjects

Saul Ancari

Publications and source records attributed to Saul Ancari.

4 recordsLinked to original sources

arXiv2022

Some rigidity properties for $λ$-self-expanders

$λ$-self-expanders $Σ$ in $\mathbb{R}^{n+1}$ are the solutions of the isoperimetric problem with respect to the same weighted area form as in the study of the self-expanders. In this paper, we mainly extend the results on self-expanders which we obtained in \cite{ancari2020volum} to $λ$-self-expanders. We prove some results that characterize the hyperplanes, spheres and cylinders as $λ$-self-expanders. We also discuss the area growths and the finiteness of the weighted areas under the control of the growth of the mean curvature.

math.DG↗

arXiv2020

Volume properties and rigidity on self-expanders of mean curvature flow

In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems that characterize the hyperplanes through the origin as self-expanders. We estimate upper bound of the bottom of the spectrum of the drifted Laplacian. We also give the upper and lower bounds for the bottom of the spectrum of the $L$-stability operator and discuss the $L$-stability of some special self-expanders. Besides, we prove that the surfaces $Γ\times\mathbb{R}$ with the product metric are the only complete self-expander surfaces immersed in $\mathbb{R}^3$ with constant scalar curvature, where $Γ$ is a complete self-expander curve (properly) immersed in $\mathbb{R}^2$.

math.DG↗

arXiv2020

Rigidity theorems for constant weighted mean curvature hypersurfaces

In this article, we study hypersurfaces $Σ\subset \mathbb{R}^{n+1}$ with constant weighted mean curvature. Recently, Wei-Peng proved a rigidity theorem for CWMC hypersurfaces that generalizes Le-Sesum classification theorem for self-shrinker. More specifically, they showed that a complete CWMC hypersurface with polynomial volume growth, bounded norm of the second fundamental form and that satisfies $|A|^2H(H-λ)\leq H^2/2$ must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions if the reverse inequality holds then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete CWMC hypersurface with $H\geq 0$ must either be a hyperplane or a generalized cylinder.

math.DG↗

arXiv2019

Volume estimates and classification theorem for constant weighted mean curvature hypersurfaces

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $Σ\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.

math.DG↗