Related papers: Positively curved surfaces in the three-sphere
In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^\alpha f^{-\beta}$, where $u$ is the support function of the hypersurface, $f$ is a…
In this paper we investigate the flow of surfaces by a class of symmetric functions of the principal curvatures with a mixed volume constraint. We consider compact surfaces without boundary that can be written as a graph over a sphere. The…
In this work we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow…
We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…
We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely,…
Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic…
Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still…
The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a…
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.
The study of quadric surfaces of revolution is a cornerstone of classical Euclidean geometry, but its extension to the three-dimensional sphere $\mathbb{S}^3$ has not been sufficiently explored. This article addresses this important gap by…
Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, 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 $\mathbb{R}^{n+1}$…
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If…
We study the curve shortening flow on Riemann surfaces with finitely many conformal conical singularities. If the initial curve is passing through the singular points, then the evolution is governed by a degenerate quasilinear parabolic…
We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For…
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed…
In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We…
We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…
We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous…
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…