Inverse Mean Curvature Research Articles

Isoperimetry and the properness of weak inverse mean curvature flow

Isoperimetry and the properness of weak inverse mean curvature flow

Weak solutions of anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials

We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in RN with no ellipticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.

Inverse mean curvature flow and Ricci-pinched three-manifolds

Abstract Let ( M , g ) (M,g) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε ⁢ tr ⁡ ( Ric ) ⁢ g \mathrm{Ric}\geq\varepsilon\operatorname{tr}(\mathrm{Ric})g for some ε > 0 \varepsilon>0 . In this note, we give a new proof based on inverse mean curvature flow that ( M , g ) (M,g) is either flat or has non-Euclidean volume growth. In conjunction with the work of J. Lott [On 3-manifolds with pointwise pinched nonnegative Ricci curvature, Math. Ann. 388 (2024), 3, 2787–2806] and of M.-C. Lee and P. Topping [Three-manifolds with non-negatively pinched Ricci curvature, preprint (2022), https://arxiv.org/abs/2204.00504], this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon [Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds, preprint (2022), https://arxiv.org/abs/2203.15313] using Ricci flow.

A Minkowski-type inequality for capillary hypersurfaces in a half-space

In this article, we investigate a flow of inverse mean curvature type for capillary hypersurfaces in the half-space. We establish the global existence of solutions for this flow and demonstrate that it converges smoothly to a spherical cap as time tends to infinity. As a result, we derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces for the whole range of contact angle θ∈(0,π).

Inverse mean curvature flow with a free boundary in hyperbolic space

Inverse mean curvature flow with a free boundary in hyperbolic space

The first nonzero eigenvalue of the (p,q)-Laplace system along the inverse mean curvature flow with forced term

In this paper, we study the first nonzero eigenvalue of the (p,q)-Laplace system along the inverse mean curvature flow with forced term in Euclidean space. Some monotonicity formulas for the first nonzero eigenvalue are obtained.

Another proof of the existence of homothetic solitons of the inverse mean curvature flow

Abstract We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in ℝ n × ℝ {\mathbb{R}^{n}\times\mathbb{R}} , n ≥ 2 {n\geq 2} , of the form ( r , y ⁢ ( r ) ) {(r,y(r))} or ( r ⁢ ( y ) , y ) {(r(y),y)} , where r = | x | {r=|x|} , x ∈ ℝ n {x\in\mathbb{R}^{n}} , is the radially symmetric coordinate and y ∈ ℝ {y\in\mathbb{R}} . More precisely for any 1 n < λ < 1 n - 1 {\frac{1}{n}<\lambda<\frac{1}{n-1}} and μ < 0 {\mu<0} , we will give a new proof of the existence of a unique solution r ⁢ ( y ) ∈ C 2 ⁢ ( μ , ∞ ) ∩ C ⁢ ( [ μ , ∞ ) ) {r(y)\in C^{2}(\mu,\infty)\cap C([\mu,\infty))} of the equation r y ⁢ y ⁢ ( y ) 1 + r y ⁢ ( y ) 2 = n - 1 r ⁢ ( y ) - 1 + r y ⁢ ( y ) 2 λ ⁢ ( r ⁢ ( y ) - y ⁢ r y ⁢ ( y ) ) , r ⁢ ( y ) > 0 , \frac{r_{yy}(y)}{1+r_{y}(y)^{2}}=\frac{n-1}{r(y)}-\frac{1+r_{y}(y)^{2}}{% \lambda(r(y)-yr_{y}(y))},\quad r(y)>0, in ( μ , ∞ ) {(\mu,\infty)} which satisfies r ⁢ ( μ ) = 0 {r(\mu)=0} and r y ⁢ ( μ ) = lim y ↘ μ ⁡ r y ⁢ ( y ) = + ∞ {r_{y}(\mu)=\lim_{y\searrow\mu}r_{y}(y)=+\infty} . We prove that there exist constants y 2 > y 1 > 0 y_{2}>y_{1}>0 such that r y ⁢ ( y ) > 0 r_{y}(y)>0 for any μ < y < y 1 \mu<y<y_{1} , r y ⁢ ( y 1 ) = 0 r_{y}(y_{1})=0 , r y ⁢ ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y ⁢ y ⁢ ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 \mu<y<y_{2} , r y ⁢ y ⁢ ( y 2 ) = 0 r_{yy}(y_{2})=0 and r y ⁢ y ⁢ ( y ) > 0 {r_{yy}(y)>0} for any y > y 2 {y>y_{2}} . Moreover, lim y → + ∞ ⁡ r ⁢ ( y ) = 0 {\lim_{y\to+\infty}r(y)=0} and lim y → + ∞ ⁡ y ⁢ r y ⁢ ( y ) = 0 {\lim_{y\to+\infty}yr_{y}(y)=0} .

Mean curvature type flow and sharp Micheal-Simon inequalities

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface Σ0 is of smoothly closed starshaped, then the solution Σt of the flow (1.5) exists for all time and converges to a sphere in C∞-topology. Following this flow argument, we obtain a new proof of the celebrated sharp Michael-Simon inequality (1.2) for mean curvatures on smooth, compact, and starshaped hypersurface M0 (possibly with boundary). Specifically, if M0 is closed, we find a necessary and sufficient condition for the equality of (1.2 ′) holds.In the second part of this paper, by exploiting the properties of the inverse mean curvature type flow (1.6) obtained from the expanding geometric flow in [7] by rescaling, we develop and present a new sharp Michael-Simon inequality (1.7) for the k-th mean curvatures. When M0 is closed, starshaped, (k−1)-convex and g is of constant, the inequality (1.7) is equivalent to the Alexandrov-Fenchel type inequality (1.8).

First eigenvalues of free boundary hypersurfaces in the unit ball along the inverse mean curvature flow

In this note, we consider the first nonzero eigenvalue λp,1 of the p-Laplacian on free boundary proper hypersurfaces in the unit ball evolving along the inverse mean curvature flow. We show that λp,1 is monotone decreasing along the flow. Using the convergence of free boundary disks in the unit ball, we give a lower bound of λp,1 of a free boundary disk type hypersurface in the unit ball.

Evolution of the first eigenvalue along the inverse mean curvature flow in space forms

In this paper, we show that the first nonzero eigenvalues λ1 of the Laplacian and the p-Laplacian are decreasing along the inverse mean curvature flow in the hyperbolic space and in the sphere. For a convex closed surface M⊂S3, we show that the geometric quantity λ1⋅Area(M) is monotonically decreasing along the inverse mean curvature flow in S3.

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space ${\mathbb R}^{n+1}_1$

Classification of Ruled Surfaces as Homothetic Self-Similar Solutions of the Inverse Mean Curvature Flow in the Lorentz–Minkowski 3-Space

Classification of Ruled Surfaces as Homothetic Self-Similar Solutions of the Inverse Mean Curvature Flow in the Lorentz–Minkowski 3-Space

A Study on Normal Motion of the Torus of Revolution in ℝ3

Abstract In the present research paper, we investigate the motion of surfaces in ℝ3 according to their curvatures. We study the motion of the torus of revolution via the normal velocity. We consider two cases: when the normal velocity is a function of both the time and the coordinates of the torus, and when it is a function of time only. We also study how the torus moves under different types of curvature flows, such as inverse mean curvature flow, inverse Gaussian curvature flow, and harmonic mean curvature flow. Moreover, we present some new applications of these flows.

Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in the Lorentz manifold bmM^n×mathbbR

令$M^{n}\times\mathbb{R}$是$(n+1)$-维Lorentz流形, 其中$M^{n}$为Ricci曲率非负、曲率张量及其一阶协变导数有界的、且含有一个极点的$n$-维($n\geq 2$)完备黎曼流形，$\mathbb{R}$是1维欧氏空间，$\Omega$是$M^{n}\times\mathbb{R}$中的有界类空凸超曲面.本文考虑了$M^{n}\times\mathbb{R}$中定义在$\Omega$上的类空图超曲面，沿着具有退化Neumann边值条件的一类各向异性逆平均曲率流的演化过程，可证明该流的长时间存在性.此外，通过适当的伸缩变换后我们可以得到，当$t\rightarrow\infty$时，演化类空图超曲面光滑地收敛为定义在$\Omega$上的一个常值函数的图曲面.

Inverse mean curvature flow of rotationally symmetric hypersurfaces

Inverse mean curvature flow of rotationally symmetric hypersurfaces

Hyperbolic inverse mean curvature flow with forced term: Evolution of plane curves

The motion of plane curves under hyperbolic inverse mean curvature with a constant force field is considered. The preserving convexity of the evolving curve under this flow is proved. Furthermore, we give an example to understand how the forced term affects the evolution of plane curves. Particularly, the asymptotic behavior of the flow is discussed.

Motion of level sets by inverse anisotropic mean curvature

Motion of level sets by inverse anisotropic mean curvature

Inverse mean curvature flow and the stability of the positive mass theorem

Inverse mean curvature flow and the stability of the positive mass theorem

The asymptotic behaviour of p-capacitary potentials in asymptotically conical manifolds

We study the asymptotic behaviour of the p-capacitary potential and of the weak inverse mean curvature flow of a bounded set along the ends of an asymptotically conical Riemannian manifolds with asymptotically nonnegative Ricci curvature.