Related papers: An immersed $S^2$ self-shrinker

We construct new embedded self-shrinkers of genus 3, 5, 7, 11 and 19 using variational methods. Our self-shrinkers resemble doublings of the Platonic solids and were discovered numerically by D. Chopp in 1994.

We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form $\mathbb S^k\times\R^{n-k}\subset \R^{n+1}$. We use a…

Rationally convex topological embeddings of compact surfaces (closed or with boundary) into $\mathbb{C}^2$ are constructed.

In the paper, we construct, for $\lambda>0$, complete embedded and non-convex $\lambda$-hypersurfaces, which are diffeomorphic to a cylinder. Hence, one can not expect that $\lambda$-hypersurfaces share a common conclusion on the planar…

We construct an sl(2)-action on the equivariant skein lasagna module.

In this paper, we introduce a new family of functions to construct Schr\"odinger operators with embedded eigenvalues. This particularly allows us to construct discrete Schr\"odinger operators with arbitrary prescribed sets of eigenvalues.

In this paper, we generalize some halfspace type theorems for self-shrinkers of codimension 1 to the case of arbitrary codimension.

Let $C$ be an $m$-dimensional cone immersed in $\mathbb{R}^{n+m}$. In this paper, we show that if $F:M^m \rightarrow \mathbb{R}^{n+m}$ is a properly immersed mean curvature flow self-shrinker which is smoothly asymptotic to $C$, then it is…

We describe a construction of complete embedded self-translating surfaces under mean curvature flow by desingularizing the intersection of a finite family of grim reapers in general position.

We show that there exist infinitely many pairwise non-isotopic splitting spheres for two unlinked, unknotted $S^2$'s in $S^4$. This answers a question posed by Hughes, Kim, and Miller.

We show that a recently given nonautonomous nonlinear Schrodinger equation (NLSE) can be transformed into the autonomous NLSE.

For any $H$ in (0,1/2), we construct complete, non-proper, stable, simply-connected surfaces embedded in $H^2xR$ with constant mean curvature $H$.

We show that any open orientable surface S can be properly embedded in H^2xR as an area minimizing surface.

We construct three kinds of complete embedded minimal surfaces in $\Bbb H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These…

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove…

We prove an eigenvalue estimate which holds on every properly embedded self-similar shrinker for mean curvature flow. This generalizes earlier work of Ding and Xin to the noncompact case.

We provide conditions under which an isometric immersion of a (warped) product of manifolds into a space form must be a (warped) product of isometric immersions.

We prove a variant of the so-called bilinear embedding theorem for operators in divergence form with complex coefficients and with nonnegative locally integrable potentials, subject to mixed boundary conditions, and acting on arbitrary open…

A semigroup amalgam (S; T1, T2) is known to be non-embeddable if T1 and T2 are both groups (completely regular semigroups, Clifford semigroups) but S is not such. We prove some non-embeddability conditions for semigroup amalgams (S; T1, T2)…

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers…