Questions tagged [mean-curvature-flows]
For questions about different versions of mean curvature flow, including the level set flow and Brakke flow.
144
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mean curvature flow for non-mean-convex surfaces
There are many results for mean curvature flow starting from mean-convex surfaces: Huisken and coauthor even proved long-time behavior in this case https://ems.press/journals/jems/articles/15560
I was ...
- 93
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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 ...
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Viscosity solutions of (graph) mean curvature flow
Let $w=w(x,t)$, $x\in\mathbb{R},\ t\in[0,T)$ be a classical solution of the graph curve shortening flow (call (A))
$$
\begin{cases}
w_t=\frac
{w_{xx}}{1+w_x^2}\quad &\text{ in } \mathbb{R}\times(0,...
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Mean curvature is extrinsic measure
Mean curvature is 1/2 of the trace of the shape operator, the shape operator can be found through a multiplication of the matrices $\begin{pmatrix}
E& F\\
F&G
\end{pmatrix}^{-1}\begin{...
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An 'obvious' inequality in Mean curvature flow
My question is from the mean curvature flow paper by Gerhard Huisken. First $H:=g^{ij}h_{ij}=tr(h)$ where $h$ is the second fundamental form of the manifold. Then $A:=\left\{h_{ij}\right\}$.
It states ...
- 306
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1
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75
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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 ...
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mean curvature flow vs. ricci flow
Both of these seem to describe very similar flows: curve-shortening flows of a high-dimensional hypersurface, which may form singularities, where the flow velocity at any point is normal to the ...
- 1,716
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Calculation of curvature in the level set method [duplicate]
I understand that the gradient of the unit normal vector is the curvature, but how was equation (4) in the link below derived?
https://math.mit.edu/classes/18.086/2007/levelsetnotes.pdf
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Definition of the mean curvature of the boundary of a smooth domain $\Omega \in \mathbb{R}^n$
In some papers and notes on differential geometry, I found the statement like
Let $\Omega \subset \mathbb{R}^n$ be open, smooth, convex such that its boundary $\partial \Omega$ has non-negative mean ...
- 2,372
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29
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PDE for the area-preserving non-parametric curve shortening flow
In dimension $1$, it is well-known that the motion of a (unparametrized) plane curve $\{(x,f(x,t);\,x \in I \subset \mathbb R)\}$ under the curve shortening flow can be encoded by the PDE $$\partial_t ...
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Is it possible to apply mean curvature flow (i.e., nonlinear diffusion) to Dirac mass?
It is well-known that the mean curvature flow applied to a time-dependent graph $\gamma := (x,\rho(x,t))$ leads to the PDE $$\partial_t \rho = \frac{\partial_{xx} \rho}{1+|\partial_x \rho|^2} = \...
- 2,372
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Self similar solutions of mean curvature flow and Gaussian curvature flow
I want to know if the mean curvature flow has such a relationship ,and if so, what is like?
The solution $F_t$ to the Gaussian curvature flow $\partial_t{F}=-K\nu$ is the self-similar if for some ...
- 31
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PDE for the Mean Curvature Flow applied to a graph of a function
Given a graph of a function $\gamma = (z,\rho(z))$, we know that the curvature $\kappa$ at the point $(z,\rho(z))$ is given by $$\kappa(z) = \frac{\rho''(z)}{\left(1+|\rho'(z)|^2\right)^{\frac 32}}.$$ ...
- 2,372
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Is there any kind of "flow" that describe balloon-like expansion for 2d surfaces constructed via a 1d paths?
I've looked into Ricci Flow and mean curvature flow/curve shortening flow, but I dont think any of these describe the math behind balloon-like expansion (or even simple diffusion) well enough given ...
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Merriman-Bence-Osher scheme for mean curvature flow on graphs for image segmentation
I have been following some papers on the application of the Merriman-Bence-Osher (MBO) scheme for mean curvature flow and its application to image segmentation. I have some of the citations below by ...
- 1,873
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2
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Mean curvature flow vs. diffusion
Mean curvature flow (MCF) and diffusion-type flows both have smoothing effects on a curve.
I can tell there is a deep connection between the two, as seen in the Merriman-Bence-Osher (MBO) numerical ...
- 1,716
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0
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36
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Why are there different defining equations for curvature flow problems? What are the key differences?
Given an initial hypersurface in $\mathbb{R^{n+1}}$ with position vector $V_0$ there are two ways I have seen to define the evolution of $V_0$ due to a curvature flow;
$$(1) \hspace{1cm} \frac{\...
- 513
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Book on the geometry of rotationally symmetric riemannian manifolds
I would like to find some references where there are specific computations and properties of rotationally symmetric riemannian manifolds, e.g. spectrum of the laplacian, schrödinger operators, ...
- 33
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97
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Avoidance principle of mean curvature flow at singularities
The avoidance principle for mean curvature flow states as follows.
$\textbf{Theorem.}$ Let $M_0$ and $N_0$ be two smooth closed surfaces and let $M_t$ and $N_t$ be their evolutions under mean ...
- 1,637
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62
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Allen-Cahn equation
Why is it that the Allen-Cahn equation is in some lecture notes written as $u_t = \Delta u - \epsilon^{-2} f(u)$ and in others $u_t = \epsilon \Delta u - \epsilon^{-1} f(u)$? The Allen-Cahn equation ...
- 181
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What is meant by divergence of a matrix
I have the following issue regarding motion by mean curvature, but I suspect it's more an issue of understanding the different notions of the gradient and the divergence of a vectorfield.
We are ...
- 181
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35
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Decomposition of tensors in mean curvature flow
I have come across a lot of decompositions of tensors when I am reading mean curvature flow by Huisken https://projecteuclid.org/download/pdf_1/euclid.jdg/1214438998. For example, on page 4 in order ...
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75
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Inequality used to bound curvature terms
I've been poring over the article: Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves
(here). In Lemma 4.4.2 , it's supposed to find bounds for the higher derivatives of k.In the part ...
- 502
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1
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73
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A small calculating problem in the article by M. Gage and R. S. Hamilton
In the article " The heat equation shrinking convex plane curves " by M. Gage and R. S. Hamilton, I didn't finish the calculation in 4.3.4:$$\frac{\partial }{{\partial t}}\int\limits_0^{2\pi ...
- 502
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0
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31
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Contraction of tensor by metric in mean curvature flow
Picture below is from the 255th page of
Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001.
In my view,
$$
A*\nabla A = g^{ia}...
- 5,487
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1
answer
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An inequality about second fundamental form under volume preserving mean curvature flow.
If $M_t$ evolve under volume preserving mean curvature flow, assume
$$
A=(h_{ij})
$$
is the second fundamental form of $M_t$, and
$$
|A|^2=h_i^jh_j^i
$$
then, we have
$$
\partial_t|A|^2= \Delta |A|^2- ...
- 5,487
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1
answer
43
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How to understand the sum in calculation of $\partial_t \sqrt g$?
Assume $g_{ij}$ is Riemannian metric. Under the mean curvature flow, we have
$$
\partial_t g_{ij}= -2H h_{ij}
$$
where $H$ is mean curvature , and $h_{ij}$ is second fundamental form. (Forgive me ...
- 5,487
3
votes
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answers
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Curvature Flow and Green's Functions
I'm trying to find out if there is a research area with associated literature linking the solution of Laplace's and Helmholtz's equations in 3D to curvature flow. If you look at the solution of ...
- 1,310
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Laplacian of the squared distance function
I'm having trouble to understand some properties of the squared distance function assigned to a curve which moves by mean curvature.
We assume that a smooth mean curvature flow of an embedded curve $\...
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answers
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Computing $L^2$ Gradient Flow for a "compactness energy" over a weighted graph.
I'm reading the following paper on applying mean curvature flow to gerrymandering:
Matt Jacobs, Olivia Walch, A partial differential equations approach to defeating partisan gerrymandering.
The ...
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1
answer
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A frame in **Flow by mean curvature of convex surfaces into spheres**
Picture below is from 242th page of
Huisken, Gerhard, Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom. 20, 237-266 (1984). ZBL0556.53001.
$H$ is the mean curvature. I ...
- 5,487
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0
answers
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Avoidance principle for mean curvature flow
I am reading Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré, Carlo Sinestrari, Vicente Miquel and Joan Porti and I am trying understand the use of an hypothesis. The Avoidance ...
- 3,619
2
votes
0
answers
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Building a numerical simulation of curvature flows. Where to start?
I would like to simulate the evolution of surfaces under various different curvature flows e.g. mean and gauss among others.
There are many different computer-graphics softwares which support ...
- 513
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0
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56
views
Is there a theorem in Differential Geometry for surfaces can be deformed without stretching?
I'm an amateur mathematician who is keen on self-studying DG. I want to know if there is a theorem in DG that proves that surfaces can be "continuously deformed" without stretching, from one surface ...
- 477
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26
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Proposition $3.4$ of Mean curvature flow with surgeries of two–convex hypersurfaces
I'm studying by myself "Mean curvature flow with surgeries of two–convex hypersurfaces" and I'm trying understand the last three lines of the proof of the proposition $3.4$ on page $149$:
Then it ...
- 3,619
2
votes
0
answers
77
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Lemma $10.6$ - Lectures on Mean Curvature Flow by Xi-Ping Zhu
I'm studying by myself Mean Curvature Flow and I'm stuck in the proof of the lemma $10.6$. I will put some notations before the lemma:
My doubts are how the terms highlighteds are derived. I couldn't ...
- 3,619
5
votes
1
answer
467
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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}\\
&+ ...
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4
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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 ...
- 1,168
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1
answer
51
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Area derivative under curve shortening flow of surfaces
For a compact Riemannian surface $\Sigma$. For an initial embedded closed curve $\gamma_0$ in $\Sigma$, a family $\gamma_t$ $(0\leq t<T)$ is parametrized by
\begin{equation}
F : S^{1} \times[0, T) \...
- 23
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votes
1
answer
120
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Proposition $3.2$ of Mean curvature flow with surgeries of two–convex hypersurfaces
I'm reading "Mean curvature flow with surgeries of two–convex hypersurfaces" by Gerhard Huisken and Carlo Sinestrari and I didn't understand how to prove the proposition $3.2$ on page $148$. The ...
- 3,619
2
votes
1
answer
130
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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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0
answers
32
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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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4
votes
1
answer
386
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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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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 ...
- 3,619
2
votes
1
answer
117
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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):
$\textbf{Lemma 3.2.}$ ...
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37
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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 ...
- 81
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0
answers
78
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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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votes
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answer
365
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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 ...
- 3,619
3
votes
0
answers
78
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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 ...
- 3,619
5
votes
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answers
188
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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 ...
- 3,619