Related papers: A note on the Gauss curvature flow

We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general…

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface…

Starting with a model conical K\"ahler metric, we prove a uniform scalar curvature bound for solutions to the conical K\"ahler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also…

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified…

In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature $R$ of the standard solution satisfies…

We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with…

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and…

This paper aims to investigate a Harnack inequality for non-negative solutions of the normalized infinity Laplacian with nonlinear absorption and gradient terms. More specifically, we establish a Harnack inequality for non-negative…

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…

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…

We study a singular limit problem of the Allen-Cahn equation with the homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the…

We show the existence of continuous viscosity solutions to the equation describing the flow of a graph in the Carnot group G x R according to its horizontal Gauss curvature. In doing so, we prove a comparison principle for degenerate…

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and…

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$…

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 develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving…

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…

We study solutions of the mean curvature flow which are defined for all negative curvature times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of…

We study the obstacle problem associated to mean curvature flow. We add to the geometric vanishing-viscosity approximation of Evans and Spruck a singular perturbation that penalizes the violation of the constraint, and pass to the limit.…

We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This…