Heiko Kroener, Matteo Novaga, Paola Pozzi
We consider motion by anisotropic curvature of a network of three curves immersed in the plane meeting at a triple junction and with the other ends fixed. We show existence, uniqueness and regularity of a maximal geometric solution and we prove that, if the maximal time is finite, then either the length of one of the curves goes to zero or the $L^2$ norm of the anisotropic curvature blows up.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove short-time existence of ϕ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.
We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.
We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of $(\mathbb S^2, g)$ in $3$-dimensional warped product space $(N^3, \bar g)$. We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter-Schwarzschild manifolds and the Reissner-Nordström manifolds.
A fourth-order regularization of the curvature flow of immersed plane curves with Dirichlet boundary conditions
本文提出了一种四阶正则化的浸入平面曲线曲率流，通过弹性体类型功能证明其在趋近于0时收敛到具有Dirichlet边界条件的曲率流。
We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bo…
We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $γ$ of $π_2(M^m)$ the limiting lipschitz map…
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R^3, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, …
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, ins…
In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic me…
Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $Σ$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $Φ(H,K_e,K)=0$. In this paper we prove that $Σ$ is a sphere of revolution, provided that the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times \mathbb{R}$ i…
In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic area integrand.
Constant mean curvature disks with circular boundary
该论文证明了三维空间中边界为圆周的紧致浸入常平均曲率圆盘只能是平面圆盘或球冠。
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.
Let $\mathcal{A}$ be a class of immersed surfaces in a three-manifold $M$, and assume that $\mathcal{A}$ is modeled by an elliptic PDE over each tangent plane. In this paper we solve the so-called Hopf uniqueness problem for the class $\mathcal{A}$ under the only mild assumption of the existence of a transitive family of candidate surfaces $\mathcal{S}\subset \mathcal{A}$. Specifically, we prove that any compact immersed surface of genus zero in the class $\mathcal{A}$ is a candidate sphere. This theorem unifies and extends many previous uniqueness results of different contexts. As an applicat…
We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of other lower order anisotropic curvature measures with nonnegative coefficients is a scaled Wulff shape. This generalizes the classical results by Schneider [Comment. Math. Helv. \textbf{54} (1979), 42--60] and by Kohlmann [Arch. Math. (Basel) \textbf{70} (1998), 250--256] to the anisotropic setting. The main ingredients in the proof are the generalized anis…
We consider closed immersed surfaces in R^3 evolving by the geometric triharmonic heat flow. Using local energy estimates, we prove interior estimates and a positive absolute lower bound on the lifespan of solutions depending solely on the local concentration of curvature of the initial immersion in L^2. We further use an ε-regularity type result to prove a gap lemma for stationary solutions. Using a monotonicity argument, we then prove that a blowup of the flow approaching a singular time is asymptotic to a non-umbilic embedded stationary surface. This allows us to conclude that any solution …
We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $ψ$ given a strictly convex initial hypersurface in Euclidean space suitably pinched. We also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere. In dimension two, it is shown that, with a volume preserving rescaling, the limit profile satisfies a soliton equation.
Half-space theorems for translating solitons of the r-mean curvature flow
本文研究了r-均曲率流中translating soliton的半空间定理，证明了在特定生长条件下，此类soliton无法完全包含在某些半空间或锥体中。
We study the geometry of complete immersed surfaces in $\mathbb{R}^3$ 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 $\mathbb{S}^2$; (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 $\mathbb{R}^3$; and (3) if the Wulff shape $W$ of the anisotropic functional is invariant wit…
Some aspects of anisotropic curvature flow of planar partitions
该研究聚焦平面网络的各向异性曲率流几何演化问题，分析了多光滑各向异性下经典解的局部存在性，并探讨了多晶情形的相关特性。
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As an application, we give the existence of smooth solutions to the Orlicz-Christoffel-Minkowski problem.
We study the singular limit of a spatially inhomogeneous and anisotropic reaction-diffusion equation. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. The limit problem involves motion by anisotropic mean curvature. We also prove an optimal estimate of the thickness of the transition layer.
Minimizing movements for forced anisotropic curvature flow of droplets
本文研究非均匀水平超平面上液滴的受迫各向异性曲率流，建立了光滑流和1/2-赫尔德连续极小化移动解的存在性，并探讨了其比较原理、一致有界性及与光滑流的一致性。
We fix a Finsler norm $F$ and, using the anisotropic curvature flow, we prove that the anisotropic maximum curvature $k^F_{\max}$ of a smooth Jordan curve is such that $ k^F_{\max}(γ)\geq \sqrt{κ/A}$ , where $A$ is the area enclosed by $γ$ and $κ$ the area of the unitary Wulff shape associated to the anisotropy $F$.
We prove that two spheres of the same constant mean curvature in an arbitrary homogeneous three-manifold only differ by an ambient isometry, and we determine the values of the mean curvature for which such spheres exist. This gives a complete classification of immersed constant mean curvature spheres in three-dimensional homogeneous manifolds.
In this paper we consider the weak formulation of the inverse anisotropic mean curvature flow, in the spirit of Huisken-Ilmanen. By using approximation method involving Finsler-p-Laplacian, we prove the existence and uniqueness of weak solutions.
In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with $λ_1\ge0$ and small $L^2$ norm of tracefree curvature. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.
Stability of the Wulff shape with respect to anisotropic curvature functionals
本文基于函数的无迹F-海森堡L^p范数给出了超曲面到Wulff形状的距离估计，并以此证明了多种各向异性问题的定量稳定性结果。
Short-time existence of Lagrangian mean curvature flow
本文利用直接的P.D.E.方法和带corners的流形概念，证明了Lagrangian平均曲率流在更强意义下收敛的短时间存在性解。
We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.
Rods in flows: the PDE theory of immersed elastic filaments
该研究针对低雷诺数流体中浸没Kirchhoff杆的运动，推导其能量在各向异性度量下的梯度流演化方程，分别分析无标架、平面及三维情形的解的适定性与收敛性，完善浸没弹性细丝的PDE理论。
We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean …
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.
Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions
本文在任意维数有界严格凸域上，针对带接触角和Neumann边界条件的各向异性平均曲率流建立先验梯度估计，并证明解收敛到时间平移不变的解。
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.
A variational approach to the volume-preserving anisotropic mean curvature flow in 2D
本文提出一种基于最小化移动的变分算法，用于二维体积保持各向异性平均曲率流，证明经典解存在性及算法收敛于全局解。
Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows
本文研究带外力和迁移率的各向异性非均匀平均曲率流，证明最小运动方案的收敛性，去掉经典理论的平移不变性假设，推广了平均曲率演化的相关研究。
We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the i…
Some results on anisotropic fractional mean curvature flows
本文研究了带有时间依赖强迫项的各向异性分数均值曲率流阈值动力学算法的一致性，证明了凸集在演化中的保持性及有界凸集演化的唯一性。
Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature
本文研究了各向异性非局部设置中具有恒定各向异性非局部平均曲率的周期回转面的存在性和对称性，并推广了Delaunay经典结果。
Area-Preserving Anisotropic Mean Curvature Flow in Two Dimensions
该研究针对平面体积约束下的各向异性曲率驱动集合运动问题，采用Cahn-Hoffman映射参数化边界，证明保面积各向异性平流指数收敛到等面积Wulff形不交并，还得到收敛轮廓与初始数据距离的一致界。
Two dimensional anisotropic mean curvature flow with contact angle condition
研究二维严格凸域上具接触角边界条件的各向异性平均曲率流，建立先验梯度估计，并证明解收敛到时间平移不变解，方法亦适用于高维Dirichlet情形。
Short time existence for a surface diffusion evolution equation with curvature regularization is proved in the context of epitaxially strained three-dimensional films. This is achieved by implementing a minimizing movement scheme, which is hinged on the $H^{-1}$-gradient flow structure underpinning the evolution law. Long-time behavior and Liapunov stability in the case of initial data close to a flat configuration are also addressed.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。