Related papers: On the Lagrangian minimal surface equation and rel…
By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in…
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence…
We obtain some equations for Hamiltonian-minimal Lagrangian surfaces in CP^2 and give their particular solutions in the case of tori.
For each surface besides the sphere, projective plane, and Klein bottle, we construct a face-simple minimal quadrangulation, i.e., a simple quadrangulation on the fewest number of vertices possible, whose dual is also a simple graph. Our…
We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a…
The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on 3-dimensional Euclidean space is described in terms of the moduli parameter k. We find that the distribution of Gaussian curvature on the surface reflects the…
There exist rotationally symmetric translating solutions to mean curvature flow that can be written as a graph over Euclidean space. This result is well-known. Its proof uses the symmetry and techniques from partial differential equations.…
The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the…
For the minimal graph with strict convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal…
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow…
In this short note, we present new observations and examples concerning the existence and rigidity of solutions to the Allen-Cahn equation with degenerate minimal hypersurfaces as their limit interfaces.
A priori estimates for the mean curvature evolution of Killing graphs in Cartan-Hadamard manifolds with asymptotic Dirichlet conditions are established. As an application, the existence of the corresponding parabolic flow is proved,…
In this overview report we generalize Erhard Heinz' curvature estimate for minimal graphs in R^3 to graphs in R^n of prescribed mean curvature. Secondly, we analyse these problems in the frame of the outer differential geometry which leads…
In this paper, we solve the longstanding Gaussian curvature conjecture of a minimal graph $S$ over the unit disk. The conjecture asserts that for any minimal graph above the unit disk, the Gaussian curvature at the point directly above the…
In this paper, we employ the loop group method to study the construction of minimal Lagrangian surfaces in the complex projective plane for which the surface is contractible. We present several new classes of minimal Lagrangian surfaces in…
The General Lagrangian Mean (GLM) theory uses a set of averaged equations of fluid dynamics to describe interactions between mean flows and waves. These equations are formulated in coordinates that follow the fluid's average velocity and…
Considering the second boundary value problem of the Lagrangian mean curvature equation, we obtain the existence and uniqueness of the smooth uniformly convex solution, which generalizes the Brendle-Warren's theorem about minimal Lagrangian…
In this paper, we firstly prove that every hyper-Lagrangian submanifold $L^{2n} (n > 1)$ in a hyperk\"ahler $4n$-manifold is a complex Lagrangian submanifold. Secondly, we demonstrate an optimal rigidity theorem with the condition on the…
We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in…
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean…