相关论文: One dimensional solutions of the $\lambda$-self sh…
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…
The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is…
In this article we study smooth asymptotically conical self shrinkers in $\mathbb{R}^4$ with Colding-Minicozzi entropy bounded above by $\Lambda_{1}$.
In this article we show the existence of closed embedded self-shrinkers in $\Bbb{R}^{n+1}$ that are topologically of type $S^1\times M$, where $M\subset S^n$ is any isoparametric hypersurface in $S^n$ for which the multiplicities of the…
In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space.…
We provide a new class of exact solutions for the interior in (2 + 1) dimensional spacetime. The solutions obtained for the perfect fluid model both with and without cosmological constant ($\Lambda$) are found to be regular and singularity…
In this paper, we study scalar curvature of $n$-dimensional self-shrinkers in the Euclidean space $\mathbb R^{n+1}$. If the scalar curvature of an $n$-dimensional self-shrinker is a positive constant, then we prove that the scalar curvature…
In this paper we show the existence of a closed, embedded $\lambda$-hypersurfaces $\Sigma \subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n)…
We solve the Euclidean Einstein equations with non-Abelian gauge fields of sufficiently large symmetry in various dimensions. In higher-dimensional spaces, we find the solutions which are similar to so-called scalar wormholes. In…
Given a smooth, symmetric, homogeneous of degree one function $f\left(\lambda_{1},\cdots,\,\lambda_{n}\right)$ satisfying $\partial_{i}f>0$ for all $i=1,\cdots,\,n$, and a rotationally symmetric cone $\mathcal{C}$ in $\mathbb{R}^{n+1}$, we…
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and…
Higher dimensional, static, cylindrically symmetric vacuum solutions with and without a cosmological constant in the Brans-Dicke theory are presented. We show that, for a negative cosmological constant and for specific values of the…
$\lambda$-self-expanders $\Sigma$ 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…
Lie point symmetries of the 2+1-dimensional cubic Schr\"odinger equation to obtain new analytic solutions in a systematic manner. We present an analysis of the reduced ODEs, and in particular show that although the original equation is not…
Given $m \in \mathbb{N} \setminus \{0\}$ and $\rho > 0$, we find solutions $(\lambda,u)$ to the problem \begin{equation*} \begin{cases} \bigl(-\frac{\mathrm{d}^2}{\mathrm{d} x^2}\bigr)^m u + \lambda G'(u) = F'(u)\\ \int_{\mathbb{R}} K(u) \,…
We derive spherically symmetric solutions of the classical \lambda-R model, a minimal, anisotropic modification of general relativity with a preferred foliation and two local degrees of freedom. Starting from a 3 + 1 decomposition of the…
In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma$ contained in one of the closed…
We show that the method used in the Schwarzschild black hole for finding the elementary solution of the electrostatic equation in closed form cannot extend in higher dimensions. By contrast, we prove the existence of static, spherically…
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…