Related papers: Mean Curvature Type Flows of Graphs in Product Man…

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally…

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve…

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…

We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for…

We obtain height, gradient, and curvature a priori estimates for a modified mean curvature flow in Riemannian manifolds endowed with a Killing vector field. As a consequence, we prove the existence of smooth, entire, longtime solutions for…

In this paper, we study the mean curvature flow of graphs with Neumann boundary condition. The main aim is to use the maximum principle to get the boundary gradient estimate for solutions. In particular, we obtain the corresponding…

We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial…

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various…

We consider the warped product manifold, $\mathbb{R}_+ \times_{\bf{Id}} M^n$, with Riemannian metric $\gamma\equiv \mathrm{d} r^2 \oplus r^2 \sigma$, where $(M^n, \sigma)$ is a smooth closed Riemannian $n$-manifold. We investigate what…

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…

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…

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we…

This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian…

In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial…

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in…

We study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature flow. As an application, the existence of the mean curvature flow of graphs…

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

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to…

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…

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow…