# 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 ...
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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 ...
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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 ...
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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 ...
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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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### 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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### 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, ...
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 ...
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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^{...