All Questions
Tagged with geometric-measure-theory dg.differential-geometry
120 questions
1
vote
1
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334
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Relationship between volume density and area density
Let $\mu(x)dx$ be a measure in $\mathbb{R}^{2n-2}$, where $\mu$ (a $C^\infty$ and positive function) is the density of the volume in the sense that $\DeclareMathOperator{\Vol}{\mathrm{Vol}} \Vol_\mu(...
0
votes
0
answers
131
views
Barycenters on Hadamard Manifolds
Let $(M,g,m_0)$ be a pointed-Hadamard manifold with Riemmanian distance function $d_g$, $(X,\Sigma,\mu)$ be a finite measure space. We use $L^2(\mu;M,m_0)$ to denote the metric space consisting of ...
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
2
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0
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150
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Extensions of minimal hypersurfaces
Let $B \subset \mathbf{R}^{n+1}$ be the unit ball, and $M \subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We ...
- 5,038
6
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1
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388
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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
1
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1
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146
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Is a locally invertible weak limit of injective maps injective almost everywhere?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries.
Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be Lipschitz injective maps ...
- 6,741
3
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0
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96
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Normal and locally normal currents
I am reading the Geometric Measure Theory book by H. Federer and I have some questions about currents:
Assuming $T \in \mathscr{D}_{m}(U),$ we call $T$ locally normal if and only if $T$ is ...
- 161
9
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2
answers
695
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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
3
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1
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97
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Behaviour of mass for currents with disjoint supports
I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral ...
- 2,095
1
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0
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91
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Why $h_t$ maps into $\mathbb{R}^{\nu}$?
I am studying geometric measure theory (Herbert Federer - Geometric measure theory) and I have a question about class $r$ homotopies. Here's the definition, from p. 363, Section 4.1.9:
Suppose $U$ is ...
- 161
1
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0
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122
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Differentiation under the integral sign for a $L^1$-valued function (shape derivative)
Let
$d\in\mathbb N$;
$U\subseteq\mathbb R^d$ be open and $$\mathcal A:=\{\Omega\subseteq U:\Omega\text{ is bounded and open and }\partial\Omega\text{ is of class }C^{0,\:1}\};$$
$E:=\bigcup_{\Omega\...
- 167
1
vote
0
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81
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zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
- 295
2
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0
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94
views
Closure theorem for weak limits of "foliation currents"
A "foliation current" in the sense of Ruelle-Sullivan (https://www.math.stonybrook.edu/~ebedford/PapersForM655/RS.pdf) is essentially a closed subset of a manifold foliated by equidimensional oriented ...
- 1,601
6
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1
answer
1k
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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
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0
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96
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If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
2
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3
answers
803
views
A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
- 622
2
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0
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90
views
Obstacle problems for minimal hypersurfaces
Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems,
1) fix a homology class $...
- 123
2
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0
answers
92
views
lower volume bound of submanifolds with small mean curvature integral data
$(M^n,g)$ is a smooth submanifold in $\mathbb{R}^p$ ,and $B_1$ is the unit ball centered in the origin 0.
Is there a $\epsilon >0$，
when assuming $\int_{M\cap B_1} |H|^n \leq \epsilon$, and the ...
- 101
1
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0
answers
183
views
Total Mean Curvature as a integral on the whole space
It is well known from De Giorgi that we may express the surface area of a domain $\Omega\subset\mathbb{R}^N$ as:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(\phi(x))||\ dx=\int_{\...
- 1,759
3
votes
2
answers
167
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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 ...
- 409
5
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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
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1
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201
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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
6
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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
10
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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
5
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0
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143
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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
14
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3
answers
1k
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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
6
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0
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388
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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
5
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0
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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
$$
\limsup_{\varepsilon\rightarrow0^{+}}\intop_{\partial\Omega_{...
- 53
10
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0
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265
views
Plank invariant measures on convex bodies
Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
- 7,149
38
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0
answers
1k
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Converse of the Archimedean property of the sphere
In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
- 7,149
7
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1
answer
438
views
An isoperimetric type of inequality in terms of Wasserstein distance/Optimal transport
Let $A \subset \mathbb{R}^n$ be a region having the same volume as an $n$ dimensional ball $B^n_R$ with radius $R$ centring at the origin.
Isoperimetric inequality says:
$ Vol_{n-1} \partial A \geq ...
- 365
4
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0
answers
113
views
Are there any nontrivial examples of $C^1$ hypersurfaces with bounded (integrable) generalized mean curvature?
The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:
Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional ...
- 1,350
9
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2
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299
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Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
- 5,474
1
vote
0
answers
263
views
Does a growing manifold fixed at a point converge to its tangent plane?
Let $M$ be a smooth compact $(n-1)$ dimensional submanifold in $\mathbb{R}^n$. Let $H$ be the $(n-1)$ dimensional Hausdorff measure. Let $f(x,y,t)$ is a function for $x\in\mathbb{R}^n$, $y\in\mathbb{R}...
- 65
2
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1
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316
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Is the $L^p$ space of tensors complete?
On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm
$$||...
- 738
4
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1
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161
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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
3
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0
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91
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Asymptotical control of the measure of tubes covering subsets of fixed Hausdorff dimension
(A version of this question was posted on math stack exchange)
Let $M$ be a $C^1$ submanifold of dimension $n$ of $\mathbb{R}^N$, and denote $\mu$ the standard surface measure on $M$.
Consider a ...
- 1,035
3
votes
1
answer
938
views
Stokes theorem for manifolds with boundary as disjoint union of submanifolds
Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
- 549
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
7
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1
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472
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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
2
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1
answer
163
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harmonic differential form integer class
Let $(M,g)$ be a compact Riemannian three-fold such that $H_2(M,\mathbb{Z}) = \mathbb{Z}$ and $S$ any surface representing 1. By Hodge theory there exist a harmonic differential one-form $\eta$ dual ...
user50806
5
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1
answer
243
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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
3
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0
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172
views
question about currents
I have a question in the field of currents:
Let M be an n-dimensional smooth manifold, and let T be a k-current (induced by a k dimensional sub-manifold), I would like to approximate it by a series of ...
- 31
6
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0
answers
113
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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
3
votes
0
answers
247
views
The projection of density $1$ point on a rectifiable set
I posted this question on MSE but I received no reply. So I repost this here for better luck. Thank you!
Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal ...
- 679
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
7
votes
1
answer
299
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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
3
votes
0
answers
109
views
What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?
Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set $\...
- 1,771
4
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0
answers
326
views
Besicovitch's covering theorem for ellipsoids and shadows
The usual Besicovitch's covering theorem concerns closed balls in $\mathbb{R}^d$. It relies on a property called "directionally limited metric space": the principal ingredient is to say that there can'...
- 41
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