# Questions tagged [geometric-measure-theory]

Questions about geometric properties of sets using measure theoretic techniques; rectifiability of sets and measures, currents, Plateau problem, isoperimetric inequality and related topics.

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### Symmetry of the isoperimetric profile

Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as $$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
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### Purely non-atomic measure on the Gromov boundary of a finitely generated free group

In the set-up of my previous post, let $\theta$ be a purely non-atomic probability regular measure defined on the Borel $\sigma$-algebra of the metric space $(\partial F, d)$. We say $\theta$ admits a ...
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### Plateau problem for fluxes of curves

Consider $\mathbb{R}_t \times \mathbb{R}_x ^n$ , let $b_1(t,x)$ and $b_2 (t,x)$ be two velocity fields with all the regularity you want and consider the flow of the point $(0,x_0)$ for a time $T$. ...
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### Use of Stein's maximal principle in Bourgain's paper on Besicovitch sets

I'm trying to understand Bourgain's paper "Besicovitch type maximal operators and applications to Fourier analysis". Let $\xi\in S^2\subset\mathbb{R}^3$ be a unit vector and $\delta>0$, ...
1 vote
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### Singularities of mean-convex MCF in the sphere?

Let $\Sigma^n \subset S^{n+1}$ be a codimension one, embedded minimal hypersurface in the sphere. As the sphere has positive Ricci curvature, this must be unstable. In particular, perturbing $\Sigma$ ...
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### Sweeping out the disk: what comes out?

In 2008, Larry Guth gave a new proof of a theorem of Gromov about the min-max widths of the unit $n$-ball. This states that the $p$-parameter width $\omega_p(k,n)$ (of sweepouts with $k$-dimensional ...
1 vote
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### Plateau problem in the disk: a question about geodesic nets

Consider given a finite collection of points along the boundary of the unit disk $D \subset \mathbf{R}^2$: \begin{equation} p_1,\dots,p_{2n} \in \partial D. \end{equation} We assume that these are all ...
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### Vector measures as metric currents

Currents in metric spaces were introduced by Ambrosio and Kirchheim in 2000 as a generalization of currents in euclidean spaces. Very roughly, a principle idea is to replace smooth test functions (and ...
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### What prevents spontaneous oscillations in minimal surfaces?

Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be an unstable minimal cone with an isolated singularity at the origin. Let $\Sigma \subset \partial B$ be its link, and $(\varphi_i)$ be the eigenfunctions ...
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### Defining minimality 'through deformations'

Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if ...
1 vote
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### Singular asymptotic limits of mean-convex MCF

Let $(M_t \mid t \geq 0)$ be a mean-convex mean curvature flow of hypersurfaces in ambient Riemannian manifold $(N^{n+1},g)$. Brian White proved that this flow (defined 'weakly' as a level set flow ...
1 vote
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### Is an inner product $\langle X, \epsilon\rangle$ between log-concave $X$ and $\epsilon\gets \{0,1\}^n$ log concave?

Let $X$ be a random variable with a density $p(x)$ with respect to the Lebesgue measure. We say that $X$ is log concave if $p(x) = \exp(-V(x))dx$ for $V(x)$ a convex function. Let $X$ be log-concave ...
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### Are $(\Lambda,r_0)$-perimeter minimising sets $C^{1,1}$?

I've tried to find counterexamples or results in this direction, but I haven't found what I'm after (except for the $\mathbb{R}^2$ case). Allard's regularity theorem guarantees that $(\Lambda,r_0)$-...
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### A geometric criterion for uniqueness in the Plateau problem?

Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi_t = \{ z = t\}$ at ...
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### A strong maximum principle for varifolds of arbitrary codimension

Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...
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### Concentration of volume towards the boundary

Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let $$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$ be the set of all ...
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### Regularity of the spherical mean of a compactly-supported function

The problem Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$. Then, consider ...
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### What is the area-decreasing 'convex hull'?

Let $K \subset \mathbf{R}^3$ be a compact set. What is the smallest set $C$ containing $K$, with the property that in a neighbourhood of $C$, the closest-point projection of surfaces onto $C$ ...
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### Finding set of best approximations from a point in $c_0$ to its subspace

Given $X$=$c_0$, null sequence space with sup norm. Consider a subspace $Y$ of $c_0$ consisting of elements of $c_0$ as, $Y=\{x\in c_0 : x_{2i}=i.x_{2i-1}, i \geq 1\}$. I need to find the set of best ...
Let $T$ be a $k$-dimensional varifold in a Riemannian manifold $M$. Assume that $f$ is a smooth function on $M$ which is weakly (sub-)harmonic on $T$; that means that  \int \langle \nabla_\omega f, ...
Let $M$ be an analytic manifold with boundary $\partial M$, equipped with a Riemannian metric $g$, which is also analytic up to and including the boundary. Are there Lojasiewicz–Simon estimates ...