Mean Curvature Flow Research Articles

Abstract Given $$\lambda \in \mathbb {R}$$ and $${\textbf {v}}\in \mathbb {L}^3$$ , a $$\lambda $$ -translator with velocity $${\textbf {v}}$$ is an immersed surface in $$\mathbb {L}^3$$ whose mean curvature satisfies $$H=\langle N,{\textbf {v}}\rangle +\lambda $$ , where N is a unit normal vector field. When $$\lambda =0$$ , we fall into the class of translating solitons of the mean curvature flow. In this paper we study $$\lambda $$ -translators in $$\mathbb {L}^3$$ that are invariant under a 1-parameter group of translations and rotations. The former are cylindrical surfaces and explicit parametrizations are found, distinguishing on the causality of both the ruling direction and the $$\lambda $$ -translators. In the case of rotational $$\lambda $$ -translators we distinguish between spacelike and timelike rotations and exhibit the qualitative properties of rotational $$\lambda $$ -translators by analyzing the non-linear autonomous system fulfilled by the coordinate functions of the generating curves.

We prove convergence of the nonlocal Allen–Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a W 1 , 1 -kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an L 2 -estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm Abels, H., & Hurm, C. (2024). Journal of Differential Equations , 402: 593–624. To this end, we prove a nonlocal Ehrling-type inequality to show uniform H 3 -estimates for the nonlocal solutions.

Abstract In this paper, we prove that every strictly convex 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, min-max theory and degree theory. We also establish the existence of smooth free boundary mean-convex foliations. In stark contrast to our prior work in the closed setting, the present result is sharp for generic metrics.

Abstract Bamler–Kleiner recently proved a multiplicity-one theorem for mean curvature flow in R 3 \mathbb{R}^{3} and combined it with the authors’ work on generic mean curvature flows to fully resolve Huisken’s genericity conjecture. In this paper, we show that a short density-drop theorem plus the Bamler–Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken’s conjecture – without relying on the strict genus-drop theorem for one-sided ancient flows previously established by the authors.

Abstract Bounds of total curvature and entropy are two common conditions placed on mean curvature flows. We show that these two hypotheses are equivalent for the class of ancient complete embedded smooth planar curve shortening flows, which are one-dimensional mean curvature flows. As an application, we give a short proof of the uniqueness and classification of tangent flow at infinity of an ancient smooth complete non-compact curve shortening flow with finite entropy embedded in $$\mathbb {R}^2$$ R 2 .

A priori estimates for Singularities of the Lagrangian Mean Curvature Flow with supercritical phase

Stability of the area preserving mean curvature flow in asymptotic Schwarzschild space

Abstract A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.

Mean curvature flow with generic low-entropy initial data II

Type II singularities of Lagrangian mean curvature flow with zero Maslov class

Abstract Given a smooth closed embedded self-shrinker S with index I in $$\mathbb {R}^{n}$$ R n , we construct an I-dimensional family of complete translators polynomially asymptotic to $$S\times \mathbb {R}$$ S × R at infinity, which answers a long-standing question by Ilmanen. We further prove that $$\mathbb {R}^{n+1}$$ R n + 1 can be decomposed in many ways into a one-parameter family of closed sets $$\coprod _{a\in \mathbb {R}} T_a$$ ∐ a ∈ R T a , and each closed set $$T_a$$ T a contains a complete translator asymptotic to $$S\times \mathbb {R}$$ S × R at infinity. If the closed set $$T_a$$ T a fattens, namely it has nonempty interior, then there are at least two translators asymptotic to each other at an exponential rate, which can be viewed as a kind of nonuniqueness. We show that this fattening phenomenon is non-generic but indeed happens.

Abstract In this paper we prove that in $$\mathbb {R}^3$$ the minimizing movement solutions for mean curvature motion of droplets, obtained in Bellettini (JMPA 117:1–58, 2018), coincide with the smooth mean curvature flow of droplets with a prescribed (possibly nonconstant) contact angle.

Aerobic exercise is essential to maintain the cardiovascular system's health, yet accurate analysis of its ‎complex motions poses severe challenges. Traditional methods often encounter computational loads, overfitting, ‎and unfavorable generalization in real applications. In trying to resolve these issues, this work introduces the MCT-‎MCT-MCT-MCT-MCT-MCT-MCT-HCNanoNet-AM framework, which combines motion capture technology (MCT) and a Humboldt Squid ‎Optimization Algorithm (HSOA)-optimized Compact-NanoNet Deep Convolutional Neural Network (CNanoNet). ‎This blended approach is going to enhance the accuracy and efficiency of aerobic movement analysis. The MCT-‎MCT-MCT-MCT-MCT-MCT-MCT-HCNanoNet-AM system workflow incorporates several key steps. Initially, high-speed motion capture technology ‎is utilized to capture data. The raw data are processed beforehand using Mean Curvature Flow, which is a technique ‎to remove noise at the first step. Afterwards, feature extraction and movement detection are done using the ‎CNanoNet model. The HSOA is then employed to optimize the CNanoNet for improving model performance. The ‎optimized model is then utilized for real-time aerobic movement recognition with personalized feedback to the ‎users. Experimental results indicate that the MCT-HCNanoNet-AM system noticeably enhances the recognition ‎and classification of aerobic movements with an exceptional set of 99.98% accuracy, 99.99% precision, 99.986% ‎specificity, 99.99% recall, and 99.98% F-Score. The system takes 91 seconds in computation and area under the ‎curve (AUC) of 0.9976. Integration of real-time feedback mechanisms not only helps in refining the techniques of ‎users but also in the prevention of injuries by identifying potential risks. Overall, the MCT-HCNanoNet-AM system ‎is a major advance in aerobic movement recognition technology that enhances performance as well as overall ‎physical well-being through innovative technological advancements‎.

Solitons of the spacelike weighted mean curvature flow in spatially GRW spacetimes and the weighted mean curvature equation

Abstract In this article, we study the convergence rate of the following Yamabe-type flow R ϕ ( t ) m = 0 in M and ∂ ∂ t g ( t ) = 2 ( h ϕ ( t ) m − H ϕ ( t ) m ) g ( t ) ∂ ∂ t ϕ ( t ) = m ( H ϕ ( t ) m − h ϕ ( t ) m ) on ∂ M {R}_{\phi \left(t)}^{m}=0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}M\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}\left\{\begin{array}{l}\frac{\partial }{\partial t}g\left(t)=2\left({h}_{\phi \left(t)}^{m}-{H}_{\phi \left(t)}^{m})g\left(t)\\ \frac{\partial }{\partial t}\phi \left(t)=m\left({H}_{\phi \left(t)}^{m}-{h}_{\phi \left(t)}^{m})\end{array}\right.\hspace{0.33em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial M on a smooth metric measure space with boundary ( M , g ( t ) , e − ϕ ( t ) d V g ( t ) , e − ϕ ( t ) d A g ( t ) , m ) \left(M,g\left(t),{e}^{-\phi \left(t)}{\rm{d}}{V}_{g\left(t)},{e}^{-\phi \left(t)}{\rm{d}}{A}_{g\left(t)},m) , where R ϕ ( t ) m {R}_{\phi \left(t)}^{m} is the weighted scalar curvature, H ϕ ( t ) m {H}_{\phi \left(t)}^{m} is the weighted mean curvature, and h ϕ ( t ) m {h}_{\phi \left(t)}^{m} is the average of the weighted mean curvature.

In this paper we introduce a new two-player zero-sum game whose value function approximates the level set formulation for the geometric evolution by mean curvature of a hypersurface. In our approach the game is played with symmetric rules for the two players and probability theory is involved (the game is not deterministic).arXiv admin note: text overlap with arXiv:2409.06855

Dynamic Ritz Projection of Mean Curvature Flow and Optimal \(\boldsymbol{L^2}\) Convergence of Parametric FEM

In this paper, we study the graphical mean curvature flow in a warped product rG/K×I, where G/K is a symmetric space of compact type, I is an open interval, and r is a smooth positive function on I. If the initial hypersurface is K-equivariant, then the K-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group K acts naturally on both G/K and rG/K×I. If the flow is graphical, then it follows from the K-equivariance of the flow that it can be described by using K-invariant functions on G/K. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that G/K is a rank one symmetric space of compact type and the warping function r satisfies certain additional properties. The proof is carried out by estimating the gradient of the K-invariant functions satisfying the flow equation.

Classification of convex ancient free boundary mean curvature flows in the ball

Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions