Related papers: Curvature flows and CMC hypersurfaces

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…

In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the…

In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…

We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and…

A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the…

In this paper, we introduce a new constrained mean curvature type flow for capillary boundary hypersurfaces in space forms. We show the flow exists for all time and converges globally to a spherical cap. Moreover, the flow preserves the…

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.

The most used form of the Shallow Water Equations doesn't take into account the variation of the curvature of the base flow surface. In this paper, we compare the theoretical and numerical solutions of the standard model and with the…

We study the regularity of the $p$-Gauss curvature flow with flat side. In our previous paper(arxiv:2403.12292), we obtained the regularity of the interface, namely the boundary of the flat part. In this paper, we study the regularity of…

In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $\Omega\times \R$, where, $A$ is a constant, $\Omega$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $\Omega$, $V$ and $H$ denote…

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of…

In this paper, we study curve shortening flows on rotational surfaces in $\mathbb{R}^3$. We assume that the surfaces have negative Gauss curvatures and that some condition related to the Gauss curvature and the curvature of embedded curve…

In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex…

Given an initial $C^1$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean…

Using the Bryant representation, we define a new flux on homology classes of CMC-1 surfaces in hyperbolic 3-space, satisfying a balancing formula which is useful to show nonexistencd of certain kinds of complete CMC-1 surfaces.

In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show existence and uniqueness of weak solutions. We also prove existence of weak…

In [12], the existence of ideal circle patterns in Euclidean or hyperbolic background geometry under the combinatorial conditions was proved using flow approaches. It remains as an open problem for the spherical case. In this paper, we…

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

Studying various functionals and associated gradient ows are known problems in differential geometry. The perpose of this article is to provide a general overview of curvature functionals in Finsler geometry and use their information for…