Questions tagged [geometric-measure-theory]
The study of the geometric structure of measures, as well as the study of geometry from a measure-theoretic viewpoint, geometric measure theory has applications in partial differential equations, harmonic analysis, differential geometry, Riemannian geomerty, sub-Riemannian geometry, as well as calculus of variations. Statements such as the isoperimetric inequality and the coarea formula, and subjects such as the Plateau problem belong under this tag.
1,011 questions
4
votes
1
answer
144
views
Riesz Representation of Sections of a Vector Bundle
Let $M$ be a smooth manifold and $E \rightarrow M$ a Riemannian vector bundle over $M$. Let $\Gamma_c^0(E)$ denote the space of continuous compactly supported sections of $E$ given the $||\cdot||_\...
- 189
3
votes
1
answer
121
views
Can the surface of revolution of an increasing $C^1$ function from $(0,0)$ to $(1,1)$ have an area of exactly $3\pi$?
Recently, we had a mathematical competition with the following question:
Suppose $f\in C^1([0,1])$ is a non-decreasing function with $f(0)=0, f(1)=1$. Let $S(f)$ denote the surface area of the solid ...
4
votes
1
answer
94
views
Validity of a propostion on Sobolev composition with locally Lipschitz maps
I have a question regarding the correctness of the following proposition regarding Sobolev spaces and the validity of its proof that I have been using in some personal work. Additionally, if correct, ...
2
votes
0
answers
31
views
Help with a proof on characterization of curvature in metric measure spaces
I'm currently studying the paper "On the geometry of metric measure spaces I." from Sturm. Specifically I'm trying to understand the curvature bounds for metric measure spaces. I'll set up ...
- 21
0
votes
0
answers
34
views
Finiteness almost everywhere of upper density
I've tried proving the following claim:
$$
\Theta_k^*(\mu,x)<\infty\text{ for $\mathcal{H}^k$-a.e. point $x\in\Omega\subseteq\mathbb{R}^n$},
$$
where $\Omega$ is an open subset and $\mu$ is a ...
0
votes
0
answers
38
views
Existence of Distance Non-Increasing Isotopies of Simple Closed Curves in $\mathbb{R}^2$
Suppose $C, D$ are simple closed curves in $\mathbb{R}^2$. It's known that there exists an isotopy $F: C \times I \rightarrow \mathbb{R}^2$ with $F(C, 0) \equiv \text{id}_C$ and $F(C, 1) = D$.
Call ...
- 972
2
votes
1
answer
94
views
Minkowski-Steiner formula
If one looks at Wikipedia's entry for the Minkowski-Steiner formula:
$$\lambda (\partial A) := \liminf_{\delta \to 0} \frac{\mu \left( A + \overline{B_{\delta}} \right) - \mu (A)}{\delta}$$
It is ...
- 1,601
4
votes
0
answers
71
views
Area formula for Riemannian manifolds and Lipschitz submanifolds
Let $N^n,M^m$ be smooth Riemannian manifolds with $m\geq n$ and $P\subseteq N^n$ be a $k-$dimensional Lipschitz submanifold. Suppose that $f:P\to M^m$ is Lipschitz. I suppose that the following area ...
- 73
10
votes
0
answers
171
views
Fourier Transform of mutually singular measures
The Fourier Transform $\int_{\mathbb R^d}e^{-2\pi is\cdot x}\mathrm{d}\mu$ is an injective linear map from the space of bounded-variation Borel measures to the space of continuous functions; so in ...
- 1,664
2
votes
0
answers
72
views
Second Variation and Index of a Varifold
I have not seen the concepts of second variation and index of a varifold defined in some common references as opposed to the first variation (for example: Leon Simon's Introduction to Geometric ...
- 451
2
votes
1
answer
52
views
Do Steiner symmetrizations with respect to orthogonal directions commute?
Denote by $S_a$ the Steiner symmetrization in $\mathbb R^n$ with respect to the hyperplane perpendicular to $a\in \mathbb R^n$ with $\|a\|_2=1$.
Let now $a,b$ be two unit vectors that are orthogonal. ...
- 56k
1
vote
1
answer
120
views
Bounds on Ahlfors regularity constants for volume measure on compact Riemannian manifolds
Let $X$ be a metric space of finite diameter $D$, and let $\mu$ be a Borel measure on $X$. Let $B(x,r)$ denote the open ball of radius $r>0$ centered at $x\in X$. If there exist positive constants $...
- 3,080
3
votes
1
answer
174
views
Proof of Gauss-Green Theorem (Thm 6.2.6) in Krantz-Parks (Geometric Integration Theory)
In Krantz and Parks ``Geometric Integration theory'' they prove the Gauss-Green theorem for rectangles (i.e., axis-aligned cubes/cuboids) first:
Corollary 6.2.5 Let $V$ be a $C^1$ vector field and $R$...
- 56k
1
vote
0
answers
47
views
Sufficient conditions for restriction of Lebesgue measure on $E\subseteq \mathbb R^n$ to be Ahlfors regular
Let $X$ be a metric space and $\mu$ a Borel regular measure on $X$. Let $B(x,r)$ denote the open ball at $x\in X$ of radius $r>0$.
If there exist constants $c, q > 0$ such that for all $x\in X$ ...
- 3,080
1
vote
1
answer
160
views
Smooth functions with arbitrarily large regular level sets
Let $f:M\to\mathbb{R}^k$ be smooth, where $M$ is a closed manifold of dimension $n>k$. I am thinking about the continuity properties of the map $c\mapsto\mathcal{H}^{n-k}(f^{-1}(c))$. Clearly this ...
- 103