All Questions
Tagged with dg.differential-geometry geometric-measure-theory
120 questions
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 ...
- 708
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
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
CommunityBot
- 1
3
votes
1
answer
189
views
Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
CommunityBot
- 1
11
votes
1
answer
440
views
Stokes theorem for Lipschitz forms
Assume that $M$ is a smooth oriented compact manifold with boundary and assume that $\omega$ is a Lipschitz $(n-1)$-form on $M$.
Question Is there a published simple proof of the Stokes theorem
$$
\...
- 6,367
1
vote
1
answer
113
views
Are smooth surfaces embedded in R3 , with finite area, always the boundary of a finite perimeter set?
Are smooth compact surfaces embedded in R3 (with no boundary) , with bounded area, always the boundary of a finite perimeter set?
Take a smooth surface S in R3 embedded , with finite area. Can we say ...
- 708
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 \...
CommunityBot
- 1
2
votes
2
answers
154
views
Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
- 128k
2
votes
0
answers
75
views
Connectedness of Space of Caccioppoli Sets?
Let $(M^n, g)$ a closed, connected riemannian manifold. Is the space of all caccioppoli sets on $M$ connected with respect to the flat norm? How about with respect to the F norm (Flat metric on the ...
- 337
1
vote
1
answer
83
views
When is a $1$-varifold $V$ the associated varifold of the reduced boundary of some Caccioppoli set?
Let $v_1$, $v_2$, $\cdots$, $v_l\in\mathbb{R}^n$ be unit vectors, $\mathbb{R}_v^+:=\{\lambda v:\lambda>0\}\subset\mathbb{R}^n$ be the ray in $v$'s direction; $n_1$, $n_2$, $\cdots$, $n_l>0$ be ...
- 2,504
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 $M$ ...
- 45k
1
vote
1
answer
181
views
For proper group action on closed Riemannian manifold, must the union of orbits with non-unique closest points to a given point be of 0 volume measure
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}d\operatorname{vol}_g \forall E \in \mathcal{B}(M),$ the ...
- 10.6k
1
vote
0
answers
63
views
Boundary behavior for submanifolds with bounded second fundamental form
I am interested in a boundary version of this question About hypersurfaces in R^n+1 with bounded 2nd fundamental form.
The question is as follows. Let $\Sigma^k\subset \Bbb R^n$ be a submanifold with ...
- 151
1
vote
1
answer
260
views
What does $g_x^{-1}$ mean where $g$ is a Riemannian metric?
I am reading this paper about the Wasserstein-Fisher-Rao distance and they define the Wasserstein-Fisher-Rao distance on a manifold as follows (page 9):
Here, $M$ is a compact Riemannian manifold, $\...
- 5,380
6
votes
1
answer
401
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 ...
- 2,504
1
vote
1
answer
160
views
Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
- 125
2
votes
0
answers
91
views
Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
- 125
2
votes
0
answers
91
views
Equivalence class of parametrized surfaces which induce the same current
Suppose $M$ is a smooth manifold of dimension $n \geq 2$. A $k$-current is a linear functional on compactly supported smooth forms on $M$, denoted $T: \Omega^k_c(M) \to \mathbb{R}$.
Let $X: [0,1]^2 \...
- 21
1
vote
1
answer
318
views
What is the limit of a helix as the frequency tends to infinity?
Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$
My initial ...
CommunityBot
- 1
0
votes
0
answers
94
views
Bounding the area of the image of a set by product of maximum of lengths
Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.
My question feels ...
- 502
1
vote
0
answers
67
views
Limits of branched minimal immersions into the sphere
Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds?
The case ...
- 5,038
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 ...
- 12.9k
1
vote
1
answer
123
views
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$ ...
- 2,478
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 ...
- 587
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 \...
- 587
2
votes
0
answers
134
views
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 ...
- 5,038
2
votes
1
answer
281
views
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 ...
- 2,478
3
votes
0
answers
123
views
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 ...
- 415
2
votes
0
answers
141
views
Harmonic functions on varifolds
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, ...
- 415
3
votes
0
answers
100
views
Are there Lojasiewicz-Simon estimates with boundary?
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 ...
- 5,038
2
votes
0
answers
119
views
How do you construct barriers for minimal surfaces?
There is no comparison principle for minimal surfaces: two minimal surfaces $M_1, M_2 \subset B$ in the unit ball of $\mathbf{R}^3$, with the boundary $\partial M_1 \subset \partial B$ lying 'above' $\...
- 5,038
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 ...
- 2,504
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
$$
I_M(v)=...
- 45k
4
votes
0
answers
192
views
What are the next-simplest area-minimizing cones?
The simplest area-minimizing, codimension one cones $\mathbf{C} \subset \mathbf{R}^{n+1}$ are the Simons cones. I am trying to understand the behavior of area-minimizing cones a bit better, but these ...
- 5,038
1
vote
1
answer
267
views
A paradox based on Simons cones
Let $\mathbf{C}_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\...
- 5,038
2
votes
0
answers
90
views
Why are $S_1,S_2$ oriented boundaries of least area?
I am trying to understand the paper by Bombieri and Giusti on Harnack inequality on minimal surfaces: https://link.springer.com/article/10.1007/BF01418640.
In particular, I am trying to understand the ...
- 63.9k
6
votes
1
answer
635
views
What is the current status on bad tangent cones at isolated singularities?
Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface.
Question.
...
- 5,038
11
votes
1
answer
452
views
Does every smooth map of rank at most d factor through a d-manifold?
Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...
- 108k
14
votes
3
answers
1k
views
Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli?
Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be Caccioppoli if the indicator function $1_S$ is BV, and we set
$$
\operatorname{perim}(S)=\| \nabla 1_S\|_{TV}
$$
...
- 512
0
votes
0
answers
83
views
Distortion estimates to control Hausdorff measure of a curve
I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...
- 869
9
votes
0
answers
202
views
approximation of currents
Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...
- 91
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
9
votes
1
answer
734
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 $f$ ...
- 63.9k
0
votes
0
answers
57
views
Can iterative application of ham sandwich cuts form streamlines of an ODE?
It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $...
- 23
0
votes
0
answers
425
views
Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 63.9k
3
votes
2
answers
167
views
Least area bounded by multiple of curves
In this paper SOME EXTREMAL QUESTIONS FOR SIMPLICIAL COMPLEXES, the author discussed about minimal area of bounded by multiples of a curve. Say we have a (well-behaved) curve $\Gamma$, the solution to ...
- 2,478
3
votes
0
answers
102
views
When is the least-area surface unique?
Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-...
- 5,038
1
vote
0
answers
134
views
Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
- 63.9k
2
votes
1
answer
742
views
Continuity of the perimeter of level sets w.r.t. level function
Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
- 60.5k
2
votes
0
answers
354
views
Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
- 1,759
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)\...
- 8,992