arXiv research

Distance between evolving hypersurfaces is a PDE solution.

The paper extends a Harnack inequality to noncompact evolving hypersurfaces.

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the…

Localizes curvature estimates for evolving hypersurfaces under various flows.

Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of …

Study shows how curved surfaces evolve smoothly to spherical shapes.

Avoids noncompact hypersurfaces from touching in evolving flows.

We consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface. We study this problem by level set method. We give some criteria to judge whether the interface evolution is fattening or non-fattening.

The paper studies how surfaces evolve in a cone under a specific flow.

The paper proves smooth convergence of evolving hypersurfaces to critical points.

The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.

We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0<p\leq 1.$ If $f$ is the mean curvature, we obtain stronger Harnack inequalities.

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, the function which gives the curvature of the largest interior sphere touching the…

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…

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 $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbol…

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case…

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescale…

The paper studies how certain spacelike surfaces evolve over time in a specific space.

The paper studies how certain surfaces evolve over time.

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 convex hypersurface converges to a round sphere. The proof is based on the mono…

Study curves evolving on hypersurfaces with free boundaries, preserving length.

The paper estimates the volume of singular points in evolving surfaces.

We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 nμ^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the relation between $μ$ and the upper or lower bound of the normal curvatures of the ev…

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …

We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews.…

We study spacelike hypersurfaces in anti-De Sitter spacetime that evolve by the Lagrangian angle of their Gauß maps.

We study the topology of the space $\d\K^n$ of complete convex hypersurfaces of $\R^n$ which are homeomorphic to $\R^{n-1}$. In particular, using Minkowski sums, we construct a deformation retraction of $\d\K^n$ onto the Grassmannian space of hyperplanes. So every hypersurface in $\d \K^n$ may be flattened in a canonic…

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that i…

Study constructs solutions for evolving hypersurfaces using inverse spacetime mean curvature.

Analyzed geometric and diffusion properties of a coupled system.

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

The paper studies how convex hypersurfaces evolve under curvature flows in space forms.

Huisken and Sinestrari have recently defined a surgery process for mean curvature flow when the initial data is a two-convex hypersurface. The process depends on a parameter H. Its role is to initiate a surgery when the maximum of the mean curvature of the evolving hypersurface becomes H, and to control the scale at wh…

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as for example on the curvature, we prove that it converges to a hemisphere, when t…

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

By studying the monotonicity of the first nonzero eigenvalues of Laplace and p-Laplace operators on a closed convex hypersurface $M^n$ which evolves under inverse mean curvature flow in $\mathbb{R}^{n+1}$, the isoperimetric lower bounds for both eigenvalues were founded.

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

Study shows how certain hypersurfaces evolve under mean curvature flow.

We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to…

The paper studies how certain surfaces evolve in space-time.

We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…

In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts which define free/almost free divisors near the focal point.

The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study …

In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second fundamental form of these hypersurfaces in some space-time region $B_R(y)\times (0, t_0)$ …

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We p…