Anisotropic Mean Curvature Research Articles

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.

Diffuse-interface approximation and weak–strong uniqueness of anisotropic mean curvature flow

Abstract The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

A note on the anisotropic Bernstein problem in ℝ³

It was proved by Jenkins [Arch. Rational Mech. Anal. 8 (1961), 181–206] that a smooth entire graph in R 3 {\mathbb {R}}^3 with vanishing anisotropic mean curvature must be a plane. By using a calibration argument and a stability inequality we show here a different self-contained proof of this result, which is still valid when the anisotropic mean curvature is constant.

Boundary regularity for anisotropic minimal Lipschitz graphs

We prove that m-dimensional Lipschitz graphs in any codimension with C 1 , α boundary and anisotropic mean curvature bounded in Lp , p > m, are regular at every boundary point with density bounded above by 1 / 2 + σ , provided the anisotropic energy satisfies the uniform scalar atomic condition.

The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

AbstractWe prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.

Minimizing movements for anisotropic and inhomogeneous mean curvature flows

AbstractIn this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions à la Luckhaus–Sturzenhecker to such flows, the latter result holding in low dimension and conditionally to the convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows one to simplify many arguments.

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 ${\mathbb R}^{n+1}_1$

Complete surfaces of constant anisotropic mean curvature

We study the geometry of complete immersed surfaces in R3 with constant anisotropic mean curvature (CAMC). Assuming that the anisotropic functional is uniformly elliptic, we prove that: (1) planes and CAMC cylinders are the only complete surfaces with CAMC whose Gauss map image is contained in a closed hemisphere of S2; (2) Any complete surface with non-zero CAMC and whose Gaussian curvature does not change sign is either a CAMC cylinder or the Wulff shape, up to a homothety of R3; and (3) if the Wulff shape W of the anisotropic functional is invariant with respect to three linearly independent reflections in R3, then any properly embedded surface of non-zero CAMC, finite topology and at most one end is homothetic to W.

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.

Alexandrov’s Theorem for Anisotropic Capillary Hypersurfaces in the Half-Space

In this paper, we show that any embedded capillary hypersurface in the half-space with anisotropic constant mean curvature is a truncated Wulff shape. This extends Wente’s result (Pac J Math 88:387–397, 1980. https://doi.org/10.2140/pjm.1980.88.387) to the anisotropic case and He–Li–Ma–Ge’s result (Indiana Univ Math J 58(2):853–868, 2009. https://doi.org/10.1512/iumj.2009.58.3515) to the capillary boundary case. The main ingredients in the proof are a new Heintze-Karcher inequality and a new Minkowski formula, which have their own interest.

Some new characterizations of the Wulff shape

We consider characterizations of a closed orientable connected hypersurface $X:M\to \mathbb R^{n+1}$ in terms of anisotropic higher order mean curvature, which is an extension of the usual mean curvature. On the one hand, we improve the result of Leyla On

Motion of level sets by inverse anisotropic mean curvature

Motion of level sets by inverse anisotropic mean curvature

Image denoising based on a new anisotropic mean curvature model

A number of variational models for image denoising have been proposed in the last few years in order to advance the denoising performance. To improve the denoising quality, it is very significant to describe the local structure of image in the proposed models. To this end, this paper proposes a novel denoising model which combines the gradient operator $ \nabla $ with the adaptive weighted matrix $ W $ in the mean curvature regularized term such that the proposed model can describe the local features in image efficiently. Since the proposed model is a high-order nonlinear and nonconvex optimization problem, we need to use the operator splitting method to tansform it into a multi-variable optimization problem and then the alternating direction method of multipliers (ADMM) can be applied to solve it. Numerical experiments demonstrate that the proposed model yields good performance compared with other well-known gradient-based models.

Contour parametrization via anisotropic mean curvature flow

We present a new implementation of anisotropic mean curvature flow for automatic contour parametrization. Our procedure couples the mean curvature flow of planar closed smooth curves with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as background. A parallelized algorithm that realizes our scheme is explained, and we have made its open-source code available. We also include a stability criterion for our numerical approximation.

Regularity for graphs with bounded anisotropic mean curvature

We prove that \(m\)-dimensional Lipschitz graphs with anisotropic mean curvature bounded in \(L^p\), \(p>m\), are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a \(C^{1,1}\) neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a \(C^{1,1}\) (resp. \(C^1\)) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

Graphical translators for anisotropic and crystalline mean curvature flow

In this paper we discuss existence, uniqueness and some properties of a class of solitons to the anisotropic mean curvature flow, i.e., graphical translators, either in the plane or under an assumption of cylindrical symmetry on the anisotropy and the mobility. In these cases, the equation becomes an ordinary differential equation, and this allows to find explicitly the translators and describe their main features.

Sharp bounds for the anisotropic p-capacity of Euclidean compact sets

We prove various sharp bounds for the anisotropic p-capacity CapF,p(K) (1<p<n) of compact sets K in the Euclidean space Rn (n≥2). Our results are mainly the anisotropic generalizations of some isotropic ones in [Ludwig, Xiao, Zhang, Math. Ann. 2011], [Xiao, Ann. Henri Poincaré 2016], [Xiao, Adv. Math. 2017] and [Xiao, Adv. Geom. 2017]. Key ingredients in the proofs include the inverse anisotropic mean curvature flow (IAMCF), the anisotropic Hawking mass and its monotonicity property along IAMCF for certain surfaces, and the anisotropic isocapacitary inequality.

A fully nonlinear locally constrained anisotropic curvature flow

Given a smooth positive function F∈C∞(Sn) such that the square of its positive 1-homogeneous extension on Rn+1∖{0} is uniformly convex, the Wulff shape WF is a smooth uniformly convex body in the Euclidean space Rn+1 with F being the support function of the boundary ∂WF. In this paper, we introduce the fully nonlinear locally constrained anisotropic curvature flow ∂∂tX=(1−Ek1/kσF)νF,k=2,…,nin the Euclidean space, where Ek denotes the normalized kth anisotropic mean curvature with respect to the Wulff shape WF, σF the anisotropic support function and νF the outward anisotropic unit normal of the evolving hypersurface. We show that starting from a smooth, closed and strictly convex hypersurface in Rn+1 (n≥2), the smooth solution of the flow exists for all positive time and converges smoothly and exponentially to a scaled Wulff shape. A nice feature of this flow is that it improves a certain isoperimetric ratio. Therefore by the smooth convergence of the above flow, we provide a new proof of a class of the Alexandrov–Fenchel inequalities for anisotropic mixed volumes of smooth convex domains in the Euclidean space.

The anisotropic p-capacity and the anisotropic Minkowski inequality

In this paper, we prove a sharp anisotropic Lp Minkowski inequality involving the total Lp anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in ℝn. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward F-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).

Surfaces of constant anisotropic mean curvature with free boundary in revolution surfaces

Surfaces of constant anisotropic mean curvature with free boundary in revolution surfaces