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I have a question about how to think about the Ryu-Takayanagi entanglement entropy mathematically. For simplicity, let's work in the simplified setting of a time-symmetric slice of $AdS_4$ space -- i. …

You can't make such a relaxation, at least in the world of weak set flows. See Example 7.3 (specifically comment iv) of Ilmanen's paper Generalized Flow of Sets by Mean Curvature on a Manifold I'd be …

As @alesia points out $2\Gamma$ means to take the curve ``with multiplicity two" (this is usually understood to be in the context of currents or flat chains -- for an oriented curve you can think abou …

This is a little long for a comment. I also haven't double checked things so may have made mistakes. I'm going to treat the case of a convex curve in the plane (I imagine something similar works for …

The issue here is that (I suspect) your intuition is failing you as your first bullet point is incorrect. A good first visualization is the Clifford torus in $\mathbf{S}^3$ all the parallel surfaces a …

I think (with one caveat) that your question has a negative answer for families of very squashed spheres. Let $$ D_a^\pm=\{(x,y,\pm a^{-1}): x^2+y^2\leq a\} $$ be the disk of radius $a$ at height $\pm …

The answer is no. Take two parallel circles of unit radius in $z=\pm \epsilon$ with $\epsilon$ small. Tilt the two circles very slightly toward one another. This satisfies your hypotheses. There are …

Your question is actually related to a somewhat non-trivial property of the minimal graph equation (and related non-linear elliptic equations) that goes back to work of Jenkins and Serrin. The basic i …

I think there are two issues at play. MCF of $n$ dimensional hypersurfaces is analogous to $2n$ dimensional Ricci flow (at least for $n=1$ and $n=2$). The Riemann curvature tensor is more algebraica …

A fairly elementary proof is as follows (I assume you are using the convention that $f$ convex means $\Omega$ is convex and $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$). For $x,y\in \Omega$ let $[x,y]\subset …

EDIT As DimitriPanov points out Claim 1' is false and this leads to an unfixable gap. I'll leave the answer in case someone finds it useful. Ironically, it looks like I can salvage the proof only f …

This is a straightforward comparison argument. Let $\omega_n$ be the volume of $\partial B_1\subset \mathbb{R}^{n+1}$. For generic $R$, one has $\partial B_R \cap T=\tau$ a smooth submanifold. By Al …

Properness is expected to hold for finite genus embedded minimal surfaces while it seems likely that there are infinite genus counterexamples. Both of these claims are completely open (and are extrem …

Here's a half-baked idea involving minimal surfaces (when I have time I will see if I can make this a little more meaningful). Consider the compact three manifold $M=\Sigma \times \mathbb{S}^1$ with …

This computation is only used when computing the evolution of the function $\phi$ defined on the previous page and $\phi$ is defined so it is supported in a small neighborhood of the smooth solution.