Related papers: Geometry of Two-dimensional Self-shrinkers
Given a smooth, symmetric, homogeneous of degree one function $f=f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\, n$, and an oriented, properly embedded smooth cone $\mathcal{C}^n$ in…
We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\RR^3$. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all…
In this note we will prove that an $n$ dimensional graphic self-shrinker in $R^{n+m}$ with flat normal bundle is a linear subspace. This result is a generalization of the corresponding result of Lu Wang in codimension one case.
In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n \geq 2$ in the…
We establish a uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and a uniform curvature bound. Additionally, we extend the non-collapsing result to a broader class of smooth metric measure…
We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted…
Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show…
We study constant mean curvature 1/2 surfaces in H2xR that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H2 admits a structure of infinite dimensional manifold with…
A model describing cell membranes as optimal shapes with regard to the $L^2$-deficit of their mean curvature to a given constant called spontaneous curvature is considered. It is shown that the corresponding energy functional is lower…
In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in $\mathbb{R}^{n+1}$. First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the…
We investigate the singularities of two-ruled hypersurfaces in the Euclidean four-space. By considering the points that minimize the distance between adjacent rulings, we obtain a characterization the striction curve. We introduce the…
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these…
In this paper, we study complete self-shrinkers in Euclidean space and prove that an $n$-dimensional complete self-shrinker in Euclidean space $\mathbb{R}^{n+1}$ is isometric to either $\mathbb{R}^{n}$, $S^{n}(\sqrt{n})$, or $S^k…
We study the geometry of a general class of vacuum asymptotically Anti-de Sitter spacetimes near the conformal boundary. In particular, the spacetime is only assumed to have finite regularity, and it is allowed to have arbitrary boundary…
We introduced in a previous paper a general notion of asymptotic morphism of a given local net of observables, which allows to describe the sectors of a corresponding scaling limit net. Here, as an application, we illustrate the general…
We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures…
In this paper, we prove a pinching theorem for $n-$dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: $ |…
In this paper, we study $\lambda$-submanifolds of arbitrary codimensions in Gauss spaces. These submanifolds can be seen as natural generalizations of self-shrinker and $\lambda$-hypersurfaces. Using a divergence type theorem and some…
We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian…
In this paper, we classify $3$-dimensional complete self-shrinkers in Euclidean space $\mathbb R^{4}$ with constant squared norm of the second fundamental form $S$ and constant $f_{4}$.