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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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Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
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Huisken's monotonicity formula (mean curvature flow)

Let $x_0\in \mathbb{R}^2$ and define the backward heat kernel relative to $(x_0, T )$ as $\rho_{x_0}(x,t)=\frac{e^{-\frac{|x-x_0|^2}{4(T-t)}}}{\sqrt{4\pi(T-t)}}$. The article that I'm reading says ...
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Second fundamental form of a general immersion $F:M^n\to(N^{n+1},\bar{g})$

I'm working on evolution equations of mean curvature flow, and to take an initial step, I guess I would have to know the second fundamental form anyway. Thank you. In literature, the mean curvature ...
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A distance Comparison principle for evolving curves (Huisken)

I am reading "A distance Comparison principle for evolving curves", an article where Huisken gives an alternative proof of Grayson's theorem. I can't understand the proof of Theorem 2.1. Why ...
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Heat equation of the distance function and Ricci flow

I would like to get more acquainted to with Ricci flow and evolving metrics. Suppose we are given a familly $\{M, g(t)\}_t$ of Riemannian manifolds with time-dependent metrics. We would like to ...
Rundasice's user avatar
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A consequence of Huisken's rescaled monotonicity formula : Stone's Lemma

The context is that of the mean curvature flow, more precisely, concerning Type I singularities and the rescaling procedure. The text I am following is by Mantegazza: "Lecture notes on Mean ...
Seurat's user avatar
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How do normal coordinates take effect in the proof of evolution equations?

Though the title includes "(geometric) evolution equations", my question is really more of how to use normal coordinates to help us prove an identity involving components of a tensor. My ...
Boar's user avatar
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What does it mean to induce a Riemannian metric on an evolving hypersurface immersed in a Riemannian manifold?

Oftentimes in some journal articles, I encounter a statement like "Let $F:M^n\times[0,T]\to N^{n+1}$ be a one-parameter family of immersions in a Riemannian manifold $(N,g)$ and let $g_t$ be the ...
Boar's user avatar
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Maximum Principle Theorem

I am reading the mean curvature flow from Mantegazza. The maximum principle is stated as follows: Two questions: Why does $ u_{\text{max}}:=\max u(p,t) $ exist? Why is there $T'$ instead of $T$ in ...
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Mean Curvature flow: singularities and rescaling procedure

In the context of Mean Curvature Flow (MCF), it's explained that in order to study the development of singularities, two tools are necessary: the monotonicity formula of Huisken and the rescaling ...
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Translating parabolic PDEs (e.g. heat equation) to geometric flows (e.g. mean curvature flow)

Suppose $X:\mathcal S\to\mathbb R^3$ is an embedding of a surface $\mathcal S\subset\mathbb R^3$. Then, the mean curvature normal of $\mathcal S$ is the Laplacian applied to the coordinates $\Delta ...
Justin Solomon's user avatar
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Alternative definitions of mean curvature flow

I am reading the mean curvature flow from a thesis. Here, the mean curvature flow is defined as follows: Let $(M,\varphi_0)$ be an hypersurface. Given $T>0$, the geometric mean curvature flow of $(...
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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 ...
James Chiu's user avatar
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1 answer
115 views

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 ...
Andre of Astora's user avatar
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319 views

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 ...
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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
Daisuke Sugawara's user avatar
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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 ...
Fei Cao's user avatar
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1 vote
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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 ...
Fei Cao's user avatar
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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} = \...
Fei Cao's user avatar
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1 vote
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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 ...
Study 's user avatar
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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 $$\kappa(z) = \frac{\rho''(z)}{\left(1+|\rho'(z)|^2\right)^{\frac 32}}.$$ ...
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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 ...
PythonMan1029's user avatar
3 votes
1 answer
354 views

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 ...
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4 votes
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547 views

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 ...
900edges's user avatar
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1 vote
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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{\...
valcofadden's user avatar
3 votes
0 answers
298 views

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, ...
user2002's user avatar
1 vote
1 answer
147 views

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 ...
User's user avatar
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2 votes
0 answers
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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 ...
mathstu's user avatar
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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 ...
mathstu's user avatar
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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 ...
STUDENT's user avatar
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1 vote
1 answer
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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 ...
sraung Jo's user avatar
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1 answer
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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 ...
sraung Jo's user avatar
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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}...
Enhao Lan's user avatar
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1 vote
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- ...
Enhao Lan's user avatar
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2 votes
1 answer
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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 ...
Enhao Lan's user avatar
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3 votes
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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 ...
Jap88's user avatar
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1 vote
1 answer
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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 $\...
mathstu's user avatar
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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 ...
YLP's user avatar
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1 vote
1 answer
60 views

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 ...
Enhao Lan's user avatar
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3 votes
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182 views

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 ...
George's user avatar
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2 votes
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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 ...
valcofadden's user avatar
1 vote
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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 ...
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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 ...
George's user avatar
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2 votes
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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 ...
George's user avatar
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5 votes
1 answer
612 views

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}\\ &+ ...
George's user avatar
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5 votes
1 answer
197 views

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 ...
Tom's user avatar
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8 votes
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399 views

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^{...
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