Questions tagged [mean-curvature-flows]

For questions about different versions of mean curvature flow, including the level set flow and Brakke flow.

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77 views

Why we need the ancient solution?

Recently, I read the Angenent, Sigurd B., Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, Proc. 3rd Conf., Gregynog/UK 1989, Prog. Nonlinear Differ. Equ. Appl. 7, 21-...
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20 views

Mean curvature of a fiber in a product manifold

I'm reading this paper and I'm stuck in the lemma $2.1$. There are three points that I can't understand. My first doubt is why the following equality is valid? $$\frac{\partial^2 p^{\alpha}}{\...
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1answer
46 views

Mean Curvature Flow of graphs

Let $\Omega\subset \mathbb{R}^n$ be open and let $f:\Omega\times [0,T)\to \mathbb{R}$ be a smooth function. Consider the graph function $\phi:\Omega\times [0,T)\to \mathbb{R}^{n+1}$ given by $\phi(p,t)...
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33 views

Computation of the Laplacian form of $g_{\sigma, \eta}$ in Huisken and Sinestrari's paper

I'm trying understand how to compute evolution equation for $g_{\sigma, \eta}$ defined on page $3$ of this article and I'm reading the Lectures on Mean Curvature Flows by Xi-Ping Zhu, which gives some ...
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1answer
83 views

Lemma $3.2$ - Mean curvature flow singularities for mean convex surfaces

This is a lemma of the paper "Mean curvature flow singularities for mean convex surfaces" by Gerhard Huisken and Carlo Sinestrari (the paper is available here): $\textbf{Lemma 3.2.}$ Suppose $(1 + \...
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27 views

How to view this algorithm as a tridiagonal system? (Curve shortening flow)

I'm trying to create an implementation of a numerical method to approximate curve shortening flow, following "Computation of geometric partial differential equations and mean curvature flow" by ...
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42 views

Research trends on mean curvature flow

It's been a few months since I've been curious about mean curvature flow and now I'm reading Robert Haslhofer's lecture notes. I like the subject and would like to do research on it, but I know ...
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1answer
87 views

Evolution equation of Christoffel symbols under Mean Curvature Flow

I'm reading a master thesis about Mean Curvature Flow and I'm trying understand how was developed the following equation: $$\frac{\partial \Gamma^i_{jk}}{\partial t} = \nabla A \ast A.$$ It's how ...
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45 views

If a hypersurface has a self-intersection point, then the second fundamental form it is negative in a neighborhood of this point

I'm studying by myself Mean Curvature Flow and I'm reading Lecture Notes on Mean Curvature Flow by Xi-Ping Zhu. The doubt of the title of this topic arised when I read the proof of the following ...
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81 views

Proposition $2.1$ - Lectures on Mean Curvature Flow by Xi-Ping Zhu

I'm studying by myself Mean Curvature Flow by Zhu's book and I never did a PDEs course, so I'm trying to learn a little of PDEs as I progress in the study of Mean Curvature Flow and I found some ...
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26 views

Curve shortening flow on riemann surface with singular metric

Are there any results on curve shortening flow for curves in a Riemann surface with singular metric? I haven't been able to find any papers on this.
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66 views

Proof of Huisken's monotonicity formula

Firstly, I would like to say that I'm studying by myself Riemannian Geometry in order to be able to understand Mean Curvature Flow, so there are some computations that I don't understand well (as I ...
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1answer
72 views

Application of the Maximum Principle for the squared norm of the second fundamental form

Firstly, I would like to say that I didn't do a PDE's course and I'm trying study by myself Mean Curvature Flow and consult references for PDEs theory when needed, so I would appreciate if someone ...
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1answer
80 views

Curve shortening flow with boundary

Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(\mathbb{R}^2,\delta_{ij})$) and $p,q\in M$ two points with $p\neq q$. Let $\gamma:I\to M$ be a smooth embedded curve starting at $...
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1answer
26 views

Elementary Computation of the Flow of the Metric Tensor under Mean Curvature Flow

Context: I have just been introduced to mean curvature and would like to verify an elementary computation to ensure that I am not kidding myself when learning this material. Let us recall that a ...
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1answer
118 views

Lemma 2.3 - Lectures on Mean Curvature Flow

I'm self-study Mean Curvature Flow and I'm stuck on item $(ii)$ of the lemma below My doubts are referent the equalities marked with a red rectangle Why $|A|^2 = \langle h_{ij}, h_{ij} \rangle$ ? I ...
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1answer
212 views

Coordinates of the tensor of the second fundamental form for a submanifold

I'm self-study Riemannian Geometry in order to be able to understand this lecture notes about Mean Curvature Flow. I'm reading the first chapter, which is a review of Riemannian Geometry, and I'm ...
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1answer
108 views

The tangential gradient of a function $f$ defined on a manifold

I'm self study Riemannian Geometry to be able to understand this lecture notes about Mean Curvature Flow. The first chapter is a review of Riemannian Geometry and I'm stuck in the following part: "...
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1answer
96 views

Integrating an inner product by parts

I am trying to integrate the following: $$ \int \langle F, -(\partial_u \kappa) T\rangle du $$ where $F: S^1 \times [0,T) \to \mathbb{R}^2$ is a simple closed plane curve given by spatial parameter $...
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1answer
159 views

Huisken's monotonicity formula

In mean curvature flow there is an important tool, namely the Huisken's monotonicity formula: For a solution of the mean curvature flow $F: M^n \times [0,T) \rightarrow \mathbb{R}^m$ we have $$ \...
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1answer
43 views

length of a curve evolving under a vector field

Suppose we have a family of closed curves $\gamma_t: [0,1] \rightarrow \mathbb{R}^n$ where $\gamma_0$ is given and $\gamma_t$ is defined to be the curve formed by allowing $\gamma_0$ to evolve under a ...
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63 views

Dirichlet Problem for Quasi-linear Parabolic Equation that Degenerates at Endpoints

The equation: Let $\Omega = (0,1) \times (0, T) \subset \mathbb R \times \mathbb R.\\$ I would like to find a solution $u(t,x)$ to the following Dirichlet problem: $$ u_t = h(x, u) \frac{u_{xx}}{...
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47 views

How to understand the flow of time-depended vector field

Maybe, this question shows my stupidity. When I read Amitai Yuval's answer, I can't understand it. In fact, I only know what is the flow of a fixed vector field, fixed means it does not depend of ...
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24 views

Equivalence class of the normal flow

Consider a compact manifold $M^n$ embedding in $\mathbb R^{n+1}$. We can deform it by some geometric flow. A kind of all, we only move it in the normal direct, for example, mean curvature $$ \...
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62 views

Good references to start studying the curve shortening flow

I want to study the curve shortening flow on curves in $\mathbb{R^2}$. I have knowledge of the local theory of plane and space curves (obviously that comes with linear algebra and calculus as well), ...
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42 views

How to get $\partial_t |A| \le \Delta |A| +2 |A|^3 + |A|^2$ from $\partial_t |A|^2 = \Delta |A|^2 -2|\nabla A|^2 +2|A|^4 -2\operatorname{ tr}(A^3)$?

How to get $$ \partial_t |A| \le \Delta |A| +2 |A|^3 + |A|^2 $$ from $$ \partial_t |A|^2 = \Delta |A|^2 -2|\nabla A|^2 +2|A|^4 -2 \operatorname{tr}(A^3) $$ where $A$ is second fundamental form, ...
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1answer
184 views

What is little Holder space?

In The volume preserving mean curvature flow near spheres, the little Holder space $h^s(U)$ is defined as the closure of $C^\infty(U)$ in the usual Holder norm of $C^s(U)$, where $s >0$. But as I ...
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2answers
134 views

Curvature of Circles in different Radius [closed]

I am not a professional in the curvature. So I only think of it intuitively at the moment. So please understand, and please let me know how to think of this correctly. [1] A circle should have the ...
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60 views

Blow up of mean curvature in mean curvature flow

When I read the proof of Theorem 1.1 of Blow-up of the mean curvature at the first singular time of the mean curvature flow I can't understand why there are only two things are becoming strictly mean ...
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1answer
57 views

An application of Topping's diameter estimates

We recall Topping's diameter estimates (Theorem 1.1 here): (Topping): Let $M$ be an n-dimensional closed, connected manifold smoothly immersed in $\mathbb R^N$, where $N\ge n+1$. Then the ...
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1answer
45 views

A inequality in stability of mean curvature flow

$C_1,C_2$ are positive constants, and $\alpha\in(0,1)$ is a constant too. If for any $\epsilon>0$, we have $$ \sum_{i<j}(k_i-k_j)^2 \le C_1 \epsilon^{2\alpha} \\ \sum k_i \ge C_2 \epsilon^\...
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64 views

Different equations for mean curvature flow

Let $M$ be a manifold. Then a smooth family of immersions $F:M \times [0,T) \rightarrow \mathbb{R}^m$ is said to be a mean curvature flow if $$ \frac{\partial F}{\partial t} = \vec{H}, \: \: \: \: \: ...
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1answer
59 views

An inequality obtained in a minimum point

I'm trying understand the article "The Heat Equation Shrinking Convex Planes" by Gage and Hamilton and I'm get stuck in the proof of Theorem $3.2.1$, specifically, in the proof of this inequality: $...
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1answer
131 views

Roadmap to the geometry of lie groups and Duistermaat ideas.

I'm currently an advanced undergrad studentwith interest in differential geometry and topology, but this term I took a lie groups class and it was fascinating, nevertheless, there seems to be a break ...
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26 views

How to show $|f''|=|1+f'^2|(tf'-f)$ has unique solution?

I calculate the 1-dimensional self-similar solution of mean curvature flow, and found this quesiton is equal to prove $$ |f''|=|1+f'^2|(tf'-f) $$ has unique solution. Where $f=f(t): \mathbb R\...
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61 views

Equivalence between the Curve Shortening Flow and a parabolic PDE with a Boundary Value Problem

I'm reading The heat equation shrinking convex plane curves and I'm trying understand the theorem $4.1.4$ that asserts that the Curve Shortening Flow is equivalent to a parabolic PDE with a Boundary ...
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23 views

A calculation in MCF of cylindrical graphs

Let $M$ be an n-dimensional hypersurface in $\mathbb R^{n+1}$. For any $x\in \mathbb R^{n+1}$, we define $$ \omega(x)=\frac{x-\langle x, \theta\rangle \theta}{|x-\langle x, \theta\rangle \theta|} $$ ...
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140 views

Compactness argument for type I singularities of the mean curvature flow

My question is about the compactness argument to find a limiting self-similar solution to the mean curvature flow near a singularity. Consider a mean curvature flow $(M_t)_{t \in I}$ of $n$-...
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50 views

Application or meaning of $\partial_t F = -(H - \gamma )\nu$

About two month ago, I try to study this flow $$ \partial_t F = -(H - \gamma )\nu $$ where $F$ is position vector , $H$ is mean curvature , $\gamma$ is positive constant, $\nu$ is normal vector. It is ...
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36 views

Normalizing a geometric flow liking mean curvature flow

I try to normalize a flow, which is slight modification of mean curvature flow, $$ \partial_t F = -(H-1)\nu $$ where $F$ is position vector, $H$ is mean curvature, $\nu$ is normal vector. According to ...
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87 views

Motivation for the Huisken monotonicity formula

In mean curvature flow one evolves an immersed manifold $F_0: M^n \rightarrow \mathbb{R}^{n+1}$ along it's mean curvature, i.e. one searches for a solution $F:M \times [0,T) \rightarrow \mathbb{R}^{n+...
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88 views

Why there are two different area preserving mean curvature flow ?

I see two different area preserving mean curvature flow. First, in 260th page of Flow by mean curvature of convex surfaces into spheres, $$ \partial _t F = -H\nu + \frac{1}{n} h_1 F $$ where $F$ ...
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38 views

Where can download Local and global behavior of hypersurfaces moving by mean curvature

The only where I think I can download Local and global behavior of hypersurfaces moving by mean curvature is ResearchGate. But I fail to login. Where I can download this paper, thanks.
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50 views

Moving hypersurface along the normal vector by different speed

Assume $M\subset \mathbb R^n$ is a hypersurface, I move $M$ in normal direct, i.e. $$ \partial_t x = f(x)\nu(x) \tag{1} $$ where $x$ is the position vector, $\nu$ is normal vector, $f(x)$ is speed. ...
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161 views

PDE of the curve shortening flow for convex curves

I'm trying study the curve shortening flow for convex curves. I'm studying for the Lectures on Mean Curvature Flows by Xi-Ping Zhu and I found difficult to understand how the got the equation 1.2 (...
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63 views

Effect of diameter estimate in estimating the norm of traceless second fundamental form

The traceless second fundamental form is $$ \mathring A = A -\frac{H}{n}g $$ where $A$ is second fundamental form, $H$ is mean curvature, $g$ is metric, $n$ is dimension, $M_t\subset \mathbb R^{n+1}$ ...
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72 views

Why small $C^2$-perturbation of the sphere is a convex hypersurface?

Why small $C^2$-perturbation of the sphere is a convex hypersurface? In fact, I don't know what is $C^2$-perturbation. Whether it mean the position vector $F(x,t)$ is $C^2$ about $t$? But in my view,...
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1answer
112 views

Compare of a PDE and ODE

On closed smooth convex manifold $M^n\subset \mathbb R^{n+1}$, $H(x,t)$ is mean curvature , $A$ is second fundamental form, satisfy $$ \partial_t H =\Delta H + |A|^2H - |A|^2 $$ Consider ODE $$ \...
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1answer
190 views

Maximum principle in a proof of mean curvature flow

On closed smooth manifold $M$ $$ \partial_t H \ge \Delta H + \frac{1}{n} H^3 $$ Let $\varphi$ be the solution of $$ \frac{d \varphi}{dt}=\frac{1}{n}\varphi ^3 ~~~~ \\ \varphi(0)=H_\min(0) >0 $$ ...
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64 views

Non-negative solution of $\partial_t H = \Delta H + |A|^2(H-1)$

This question is from a transformation of mean curvature flow. I want to find a precise initial conditions to keep $H \ge 1$ or $H\le 1$ under this flow. Consider PDE on closed smooth manifold $M^n\...