Related papers: A new way to Dirichlet problems for minimal surfac…
By making use of the approximation method, we obtain the existence and regularity of the viscosity solutions for the generalized mean curvature flow. The asymptotic behavior of the flow is also considered. In particular, the Dirichlet…
In this paper, we study a class of special Lagrangian curvature potential equations and obtain the existence of smooth solutions for Dirichlet problem. The existence result is based on a priori estimates of global $C^{0}$, $C^{1}$ and…
We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-\Delta u=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a…
The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of…
We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…
In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. First, we prove that if the ratio $\lambda$ of the maximum and the minimum of the holomorphic sectional curvatures…
We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and…
We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of…
We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three…
In this paper, we prove convergence of the high codimension mean curvature flow in the sphere to either a round point or a totally geodesic sphere assuming a pinching condition between the norm squared of the second fundamental form and the…
We study the mean curvature flow of smooth $m$-dimensional compact submanifolds with quadratic pinching in the Riemannian manifold $\mathbb{C}P^n$. Our main focus is on the case of high codimension, $k\geq 2$. We establish a codimension…
We prove a Minkowski type inequality for weakly mean convex and star-shaped hypersurfaces in warped cylinders which are asymptotically flat or hyperbolic. In particular, we show that this sharp inequality holds for outward minimizing…
We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form,…
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains…
In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…
We consider properly immersed finite topology minimal surfaces S in complete finite volume hyperbolic 3-manifolds N, and in M x S(1), where M is a complete hyperbolic surface of finite area. We prove S has finite total curvature equal to…
In this article we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with…
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…
We study asymptotic behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures.…
We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains have and are more or less determined implicitly by a mean value property for harmonic…