Related papers: Notes on translating solitons for Mean Curvature F…

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property…

A $\lambda$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +\lambda$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $\lambda \in \mathbb{R}$. In…

The Hodge star mean curvature flow on a 3-dimensional Riemannian or pseudo-Riemannian manifold is a natural nonlinear dispersive curve flow in geometric analysis. A curve flow is integrable if the local differential invariants of a solution…

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 produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in $\mathbb{C}^{m}$…

Mainstream flow matching methods typically focus on learning the local velocity field, which inherently requires multiple integration steps during generation. In contrast, Mean Velocity Flow models establish a relationship between the local…

Transformers flexibly operate over sets of real-valued vectors representing task-specific entities and their attributes, where each vector might encode one word-piece token and its position in a sequence, or some piece of information that…

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.

We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed…

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in ${\mathbb{R}^3}$. As application we study when the euclidean Gauss map has a harmonic argument and solve a general…

A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^\alpha$-translator of the flow by the powers of Gauss curvature if satisfies $K^\alpha= \langle N,\vec{v}\rangle$, $\alpha \neq 0$, where $K$ is the Gauss curvature, $N$…

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.

We complete the theoretical framework required for the construction of a Morse homology theory for certain types of forced mean curvature flows. The main result of this paper describes the asymptotic behaviour of these flows as the forcing…

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

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed…

We introduce and prove basic results about several graph-theoretic notions relevant to the multiresolution analysis of flow graphs that represent the transfer of control in computer programs. We take a category-theoretical viewpoint to…

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

Building on recently established enumerative connections between lambda calculus and the theory of embedded graphs (or "maps"), this paper develops an analogy between typing (of lambda terms) and coloring (of maps). Our starting point is…

A $K^\alpha$-translator is a surface in Euclidean space $\r^3$ that moves by translations in a spatial direction and under the $K^\alpha$-flow, where $K$ is the Gauss curvature and $\alpha$ is a constant. We classify all…