arXiv research

Unified estimates for mean curvature in Lorentz-Minkowski space.

Survey on rigidity results for graphs with prescribed mean curvature.

Radial graphs with constant mean curvature found in Euclidean space.

Strict convexity of graphs with constant mean curvature is proven under certain conditions.

Study on $t$-graphs with prescribed mean curvature in Heisenberg groups.

Proves existence of graph on torus with prescribed curvature.

Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.

New examples of mixed-type zero-curvature graphs found.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

Smoothness of graphs evolving by fractional mean curvature is proven.

Develops a duality for graphs in Riemannian and Lorentzian spaces with prescribed mean curvature.

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains.

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier conditi…

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

Estimates mean curvature, scalar curvature, shape operator in warped products.

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.

This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an impor…

Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

In this paper, we obtain an Ecker-Huisken type result for entire graphs with parallel mean curvature.

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Symmetric graphs flow without singularities on their axis.

Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which contacts the cylinder over the domain orthogonally along its boundary.

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.

Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…

The aim of this paper is to give two uniqueness results for the Dirichlet problem associated to the constant mean curvature equation. We study constant mean curvature graphs over strips of R^2. The proofs are based on height estimates and the study of the asymptotic behaviour of solutions to the Dirichlet problem.

Study on mean curvature flow of graphs in higher dimensions.

We prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.

This paper classifies complete translating solitons in 3D space.

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…

The paper classifies and constructs translating graphs in 3D and higher dimensions.

Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.

Estimates prove existence of curvature flow in curved spaces.

The abstract discusses graphs with prescribed mean curvature on Riemannian manifolds, including existence and translating graphs.

Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.

This paper studies mean curvature flows near cylindrical singularities.

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(κ,τ)$-spaces with $κ\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(κ,τ)$. We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(κ…

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the…