Related papers: On the second boundary value problem for mean curv…

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a…

Dense granular flows exhibit both surface deformation and secondary flows due to the presence of normal stress differences. Yet, a complete mathematical modelling of these two features is still lacking. This paper focuses on a steady…

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving…

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a…

We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space $\mathbb{R}^{n,m}$, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet…

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower…

We study a class of boundary value problems with $\varphi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\varphi(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u')\right)'=\lambda f(u)\; \text{ on }(-L,…

In this paper, we consider noncompact ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) which are strictly convex, uniformly two-convex, and noncollapsed. We prove that such an ancient solution is a…

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In…

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed…

We prove the longtime existence for the mean curvature flow problem with a perpendicular Neumann boundary condition in a Generalized Robertson Walker (GRW) spacetime that obeys the null convergence condition. In addition, we prove that the…

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the…

Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow…

In this paper, we can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold $M^{n}\times\mathbb{R}$, where $M^{n}$ is an $n$-dimensional…

In this paper, for the Lorentz manifold $M^{2}\times\mathbb{R}$, with $M^{2}$ a $2$-dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^{2}$,…

A manifestly covariant equation is derived to describe the second order perturbations in topological defects and membranes on arbitrary curved background spacetimes. This, on one hand, generalizes work on macroscopic strings in Minkowski…

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio…

In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified…

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second…

In this study, we deal with non-degenerate translators of the mean curvature flow in the well-known hyperbolic Einstein's static universe. We classify translators foliated by horospheres and rotationally invariant ones, both space-like and…