Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [mean-curvature-flows]

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

The mean curvature flow is a family of immersions $M_t$ so that $$\partial_t M_t = \vec H_t,$$ where $\vec H_t$ is the mean curvature vector. The mean curvature flow is the gradient flow of the area functional, and is related to the study of minimal surfaces. Any questions concerning mean curvature flow should use this tags. These include

  • Level set flow (consider also using the tags viscosity solution, pde)

  • Brakke flow (consider also using the tag geometric measure theory)

  • Huisken's monotonicity formula, type I/II singularities, self-shrinkers.

  • Self-expanders and translating soliton in mean curvature flow.

  • $F$-stability, entropy stability of self-shrinkers.

  • Normalized mean curvature flow

To improve visibility of the question, consider also using the tags differential-geometry or riemannian-geometry.

155 questions
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Difference between embedded surfaces and immersed surface under mean curvature flow

In Huisken, Gerhard, Asymptotic behavior for singularities of the mean curvature flow, there is a conjecture that the blow up rate of singularities of arbitrary embedded surfaces is same with convex surfaces. (1) $$ \text{blow up rate: }~~~ …
Enhao Lan
  • 6,682
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Gauss-Weingarten relations on an arbitrary Riemannian manifold

I'm trying to derive the Gauss-Weingarten relations on a hypersurface immersed on an arbitrary Riemannian manifold (see page $468$ of this paper for the context): $\begin{cases} \frac{\partial^2 F^{\alpha}}{\partial x_ix_j} - \Gamma^k_{ij}…
George
  • 3,957
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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 $$ \frac{\partial F}{\partial t} = \vec{H}, \: \: \: \: \: (1) $$ where $\vec{H}$ is the mean curvature…
abcdef
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Geometric meaning of $H=\langle x, \nu \rangle$

$M$ is a $n$-dimensional smooth manifold without boundary . $F: M \rightarrow \mathbb R^{n+1}$ is a smooth embedding. $A$ is the second foundamental form , and $H$ is mean curvature. $\nu$ is the normal vector. $x$ is position vector. If…
Enhao Lan
  • 6,682
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3 answers

Which is the best book for studying geometric flows?

I have some knowledge about the basics in Riemannian Geometry (I used Do Carmo's and Petersen's books). Now I would like to focus my attention on geometric flows (mostly mean curvature flow and Ricci flow). Where should I start? Which is the best…
Onil90
  • 2,811
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Tensor calculation on mean curvature flow

I have two questions about tensor calculation. First question : In the book, Lectures on mean curvature flows written by Xi-Ping Zhu, there exists the equaility $g^{mn} \nabla_m \nabla_n h_{ij} = g^{mn} \nabla_m \nabla_i h_{jn}$. I do not…
HK Lee
  • 20,532
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Mean curvature flow - initial condition - mean-convex

The mean curvature flow of a surface given by a graph $X : B \subset \Bbb{R}^n \to [0,\infty)$ is given by $$ X_t (x,t) = H(x,t) \vec n(x,t) $$ where $H$ is the mean curvature and $\vec n$ is the normal vector. A result of Ecker and Huisken says…
curvature
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Mean curvature flow - implementation fails for some meshes

I am working on piece of software to deal with 3D meshes and I need to smooth some meshes. I have implemented MCF by using this formula $\vec{H} = {{t}\over{2}} \sum_{q \in\ link\ p} \vec{Ne} |e| \sin({\theta\over{2}})$ where $\vec{Ne}$ = ${\vec{N1}…
marekszpak
  • 365
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Simons' identity

I'm trying understand the proof of the Simons' identity: $\begin{align*} \nabla_k \nabla_l h_{ij} &= \nabla_i \nabla_j h_{kl} + h_{kl}h_{ip}g^{pq}h_{qj} - h_{il} h_{kp} g^{pq} h_{qj}\\ &+ h_{kj} h_{ip} g^{pq} h_{ql} - h_{ij} h_{kp} g^{pq} h_{qj} +…
George
  • 3,957
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1 answer

Formal Definition of Renormalization Group Flow

I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what this flow actually is, but could not find…
Tom
  • 3,135
5
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1 answer

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 $\phi(p,t)=(p,f(p,t))$. In general $\phi$ is said to flow…
Or Kedar
  • 933
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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 difficulty to apply a PDE's theorem. Let be $X(x,t)$…
George
  • 3,957
5
votes
1 answer

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 $p$ and ending at $q$. When does the curve…
rpf
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1 answer

Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces

Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we take the convention that the $i$-th row of $J_f(x)$…
Nap D. Lover
  • 1,292
4
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1 answer

Inequality in proof of Huisken's theorem

Suppose $M$ is a uniformly convex hypersurface in $\mathbb{R}^{n+1}$ ($A\geq \alpha Hg$ for $\alpha>0$) undergoing mean curvature flow. $A$ denotes the second fundamental form and $H$ denotes the mean curvature. The proof of Huisken's theorem uses a…
Andre of Astora
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