All Questions
Tagged with geometric-measure-theory dg.differential-geometry
120 questions
38
votes
0
answers
1k
views
Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
- 7,149
20
votes
1
answer
3k
views
Hausdorff measure and the volume form
There are two tools, generalizing a concept of a volume to the case of submanifolds in $\mathbb{R}^n$, namely the Hausdorff measure $H^k$ and the volume form. The question is how to show that if $M$ ...
- 1,329
14
votes
3
answers
1k
views
Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?
Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set
$$
\operatorname{perim}(S)=\| \nabla 1_S\|_{TV}
$$
...
- 1,247
12
votes
1
answer
2k
views
Besicovitch Covering Lemma on Manifolds
The classical Besicovitch covering lemma (BCL) asserts that for any $d \geq 1$, there is a constant $N(d)$ with the following property. If $A \subset \mathbb{R}^d$ is any subset and $r : A \to (0,R]$ ...
- 485
11
votes
1
answer
451
views
Does every smooth map of rank at most d factor through a d-manifold?
Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...
- 37.8k
11
votes
1
answer
440
views
Stokes theorem for Lipschitz forms
Assume that $M$ is a smooth oriented compact manifold with boundary and assume that $\omega$ is a Lipschitz $(n-1)$-form on $M$.
Question Is there a published simple proof of the Stokes theorem
$$
\...
- 28k
10
votes
1
answer
978
views
On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?
I could not answer or find references of this question, even for the following special case:
On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function ...
- 1,804
10
votes
1
answer
872
views
Current vs Varifold
I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
- 1,630
10
votes
1
answer
232
views
Is there an inscribed cube for an arbitrary compact closed surface?
Given a compact closed surface $M$ (2-dim topological manifold) isometrically embedded in $\mathbb{R}^3$, are there 8 points $x_i\in M(i=1,\dots,8)$ such that they are the vertices of a cube $C\subset\...
user178596
10
votes
1
answer
696
views
How to shrink a square with minimal distortion?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\euc}{\mathfrak{e}}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\al}{\alpha}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\Lip}{\text{Lip}_{\text{inj}}...
- 6,741
10
votes
0
answers
265
views
Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
- 7,149
9
votes
1
answer
733
views
Calderon-Zygmund decomposition on manifolds?
The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...
- 637
9
votes
2
answers
695
views
Non-calibrated area-minimising surface
Let $(M^{n+k},g)$ be a Riemannian manifold. Call a surface $\Sigma^n \subset M$ calibrated if there is a closed $n$-form $\omega$ defined on a neighbourhood $U \subset M$ of $\Sigma$ so that $\omega \...
- 5,038
9
votes
2
answers
299
views
Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
- 5,474
9
votes
0
answers
202
views
approximation of currents
Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...
- 91
8
votes
3
answers
804
views
How to interpret this quote of Lin?
I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42].
It is a well-known fact that a weakly converging sequence of stationary integral currents may have a ...
- 5,038
7
votes
1
answer
472
views
Geometric Construct for Integrating Symmetric Tensors?
I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes ...
- 73
7
votes
1
answer
299
views
Convexity of Isoperimetric Domains
I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...
- 343
7
votes
1
answer
2k
views
Algebraic geometric measure theory
Suppose I have $V\subset \mathbb{C}^n$ be the zero set of a polynomial $P(z_1, \dotsc, z_n),$ with bounded height of coefficients (where height is, to fix something, $|\log|a||$) and degree $d.$ ...
- 96.4k
7
votes
1
answer
438
views
An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
- 365
6
votes
2
answers
722
views
Stability of minimal surfaces
Let $\Gamma$ be a prescribed $n-2$ dimensional set and assume $S \subset R^n$ is a minimal hyper-surface with respect to some smooth metric $g$ on $R^n$, and $\partial S= \Gamma$. Is $S$ is stable ...
6
votes
1
answer
388
views
A better version of Weyl's Law or uniform estimates of Laplacian higher eigenvalues
Let $(M^n,g)$ be a closed $n$ dimensional Riemannian manifold with $\mathrm{Ric}_g\ge -K$, $(K\ge 0)$. Weyl's law(along with Karamata Tauberian Theorem) asserts that the eigenvalue $\lambda_i$ of $-\...
- 193
6
votes
2
answers
390
views
Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
6
votes
1
answer
634
views
What is the current status on bad tangent cones at isolated singularities?
Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
Question.
...
- 5,038
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
6
votes
1
answer
172
views
Mass minimizing current in real homology class
It is a well-known results by Federer and Fleming that there exists at least one mass-minimizing normal current in every real homology class of a closed $n$-dimensional Riemannian manifold $M$. Their ...
- 61
6
votes
1
answer
400
views
Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
- 831
6
votes
1
answer
159
views
Indecomposable integral currents
Let $\mathbf{I}_k(\mathbb{R}^n)$ denote the space of $k$-dimensional integral currents in $\mathbb{R}^n$ with finite mass. It is said that $T\in \mathbf{I}_k(\mathbb{R}^n)$ is indecomposable if there ...
- 415
6
votes
1
answer
802
views
Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 143
6
votes
0
answers
156
views
Does an 8 dimensional compact Riemannian manifold contain an embedded minimal hypersurface?
It is well know that if $(M^{n+1},g)$ is a compact Riemannian manifold and $n \leq 6$ then there exists a smooth, embedded minimal hypersurface $\Sigma^n$ in $M$ (infinitely many such $\Sigma$, even)
...
- 1,179
6
votes
0
answers
388
views
What’s the limit of a vector bundle?
In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current.
In the geometric ...
- 8,723
6
votes
0
answers
113
views
Does the Hodge *-operator act on the tangent space at 0 to the space of integral (n-1)-cycles in a conformal manifold of dimension d=2n?
Suppose $M$ is a compact, oriented conformal manifold of even dimension $d=2n$.
Write ${\cal D}^{\mathit{int}}_{k}(M)$ for the space of integral
$k$-currents in $M$
and write ${\cal D}^{\mathit{int}}...
- 351
5
votes
1
answer
412
views
Continuous deformation of soap films
Let $S$ be a soap film bounded by an unknotted wireframe cycle (in $R^3$). Why is it the case that as we deform the wireframe in $R^3$, $S$ deforms continuously?
- 51
5
votes
2
answers
1k
views
Continuity of Hausdorff measure on level sets
Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that:
$\bullet$ $\phi^{-1}(0)\neq\emptyset$;
$\bullet$ $\nabla\phi(x)\...
- 1,759
5
votes
1
answer
278
views
Approximating Jordan curves
I'd like to capture the intuitive notion that a Jordan curve $\gamma_2$ “follows” or “approximates” another Jordan curve $\gamma_1$, i.e. goes somehow “parallel” ...
- 9,720
5
votes
1
answer
411
views
Averaging maps of Riemannian manifolds
Let $M$ be a compact Riemannian manifold. We know how to average functions $f\colon M\to {\mathbb R}$; the integral $\frac{\int_M f}{\int_M 1}$ returns a value in ${\mathbb R}$. If intead $f\colon M\...
- 666
5
votes
1
answer
201
views
The Hausdorff dimension of the union of singular orbits and exceptional orbits
Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional ...
user78128
5
votes
1
answer
243
views
sequence of graphs converge in the sense of varifold to multiplicity 2 plane
Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
- 49
5
votes
1
answer
670
views
Signed distance function and level set
For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...
- 1,759
5
votes
1
answer
196
views
isoperimetric problems on Alexandrov spaces
For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by
$$
I_M(v)=...
- 51
5
votes
1
answer
504
views
Tangent cones at zero and infinity to minimal surfaces
Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth:
$\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \...
- 5,038
5
votes
0
answers
261
views
Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary
In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
- 409
5
votes
0
answers
273
views
Is there any geometrical/homological intuition behind symmetrized gradient?
The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
- 980
5
votes
0
answers
143
views
Smoothing properties of convolutions of $P^1(\mathbb{R})$ by $SL(2,\mathbb{R})$
Consider the action of $SL_2(\mathbb R)$ on real projective space $P^1(\mathbb R)$; given $A \in SL_2(\mathbb R)$ and $\alpha \in P^1(\mathbb R)$ we write $A . \alpha \in P^1(\mathbb R)$ for this ...
- 485
5
votes
0
answers
240
views
The boundary integral of a harmonic function
Let $\Omega\subset\mathbb{R}^{n}$ be a bounded
domain with smooth boundary and $f$ be a harmonic function on $\Omega.$
It is known that
$$
\limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...
- 53
4
votes
2
answers
286
views
Area-minimising hypersurface with unbounded area growth
Let $T$ be an $n$-dimensional area-minimising hypersurface in $\mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $\mathcal{H}^n(T \cap B_R) \...
- 5,038
4
votes
2
answers
718
views
What is the constant in the rate of exponential convergence for mean curvature flow?
Given a domain $\Omega \subset \mathbb{R}^d$ which is convex and smooth and $|
\Omega|=1$, it is well known that the metric converges exponentially fast to that of the sphere under volume preserving ...
- 2,641
4
votes
1
answer
161
views
Is a minimal surface $S$ that is bounded by an analytic closed curve $C$, analytic?
Let $C$ be an analytic closed curve (in the form of an unknot) in $\mathbb{R}^3$ and let $S$ be a minimal surface (a disk) bound by $C$. Is $S$ always analytic? Can you point out some references?
- 415
4
votes
1
answer
259
views
normal form of currents?
(this question did not get any answers on math.SE, so I am reposting it here)
Let $M$ be an $n$-dimensional manifold. Then the space of currents $\mathcal D^k(M)$ of degree $k$ on $M$ is the space ...
- 143
4
votes
0
answers
192
views
What are the next-simplest area-minimizing cones?
The simplest area-minimizing, codimension one cones $\mathbf{C} \subset \mathbf{R}^{n+1}$ are the Simons cones. I am trying to understand the behavior of area-minimizing cones a bit better, but these ...
- 5,038