All Questions
Tagged with geometric-measure-theory dg.differential-geometry
120 questions
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 ...
0
votes
0
answers
148
views
existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
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
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
3
votes
0
answers
354
views
Brakke's theorem for gap in entropy between self-shrinkers
In their paper The round sphere minimizes entropy among closed self shrinkers, Colding-Ilmanen-Minicozzi-White state "It follows from Brakke's theorem that $\mathbb{R}^n$ has the least entropy of any ...
- 61
3
votes
1
answer
437
views
How to define a generalized differential form through its values on submanifolds
Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, ...
- 335
2
votes
1
answer
391
views
(n-1)-dimensional normal currents and Smirnov's paper
I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
http://www.unige.ch/~...
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
2
votes
1
answer
168
views
Generalization of an inequality due to Gage for curve shortening Part II
I asked a question recently about generalizing an inequality due to Gage. This inequality asserts that given a convex domain $\Omega$ in $\mathbb{R}^2$ with support function $p(X) = \langle X, \nu \...
- 2,641
2
votes
1
answer
319
views
Generalization of an inequality due to Gage for curve shortening
There is a well known inequality due to Gage which asserts the following.
Let $\Omega$ be a smooth, convex set in $\mathbb{R}^2$ and let $p = \langle X, \nu \rangle$ be the support function of $\Omega$...
- 2,641
2
votes
3
answers
2k
views
basic question about rectifiability
The definition that Leon Simon gives (lect. of Geom. Measure theory) for n-rectifiability implies that every subset of an n-rectifiable set is rectifiable. To quote: a set $M\subset \mathbb{R}^{n+k}$ ...
- 31
3
votes
1
answer
348
views
Classification of limits under volume preserving mean curvature flow?
It is well known that if you start with a domain $\Omega \subset \mathbb{R}^d$ which is uniformly convex, then it converges exponentially fast to the ball when evolved under volume preserving mean ...
- 2,641
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
3
votes
0
answers
301
views
What does convergence in the $L^2$ sense to a constant mean curvature surface imply?
I have been thinking about the following question and have been unable to find any literature on the subject.
Question: Assume I have a sequence of smooth, simply connected, compact domains $\...
- 2,641
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
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
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
2
votes
2
answers
643
views
Estimating the Hausdorff measure of a subset of the sphere
Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these ...
- 2,270
1
vote
2
answers
336
views
What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that `$\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$` has piecewise-smooth boundary?
What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that $\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$ has piecewise-smooth boundary?
Some remarks:
I don't mind if the ...
- 6,694
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