# Questions tagged [mean-curvature-flows]

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

99 questions
Filter by
Sorted by
Tagged with
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-...
20 views

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 ...
63 views

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), ...
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, ...
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 ...
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.  A circle should have the ...
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 ...
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 ...
45 views

59 views

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 ...
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 $$\omega(x)=\frac{x-\langle x, \theta\rangle \theta}{|x-\langle x, \theta\rangle \theta|}$$ ...
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 $n$-...
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 ...
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 ...