# 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 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 ...
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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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1 vote
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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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