Related papers: When can minimal hypersurfaces be connected by mea…

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 obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in $\mathbb R^3$, which converges to a plane with multiplicity $2$ as time approaches infinity.

We establish a convergence result for the mean curvature flow starting from a totally real submanifold which is "almost minimal" in a precise, quantitative sense. This extends, and makes effective, a result of H. Li for the Lagrangian mean…

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically…

A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is…

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…

For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily…

We consider mean curvature flow of n-dimensional surface clusters. At (n-1)-dimensional triple junctions an angle condition is required which in the symmetric case reduces to the well-known 120 degree angle condition. Using a novel…

We construct a topological invariant for a Morse-Smale flow on a 3-manifold and prove that the flows are topologically equivalent iff their invariants are same.

We consider intersecting hypersurfaces in curved spacetime with gravity governed by a class of actions which are topological invariants in lower dimensionality. Along with the Chern-Simons boundary terms there is a sequence of intersection…

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after…

In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which…

We propose a construction of mean curvature flows by approximation for very general initial data, in the spirit of the works of Brakke and of Kim & Tonegawa based on the theory of varifolds. Given a general varifold, we construct by…

In this paper, we investigate the preservability of the curvature-adaptedness along the mean curvature flow starting from a compact curvature-adapted hypersurface in locally symmetric spaces, where the curvature-adaptedness means that the…

We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we…

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$…

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower…

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean…

A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of…