All Questions
8 questions
2
votes
0
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91
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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
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
$$
\int \langle \nabla_\omega f, ...
- 415
5
votes
0
answers
261
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Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary
In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
- 409
0
votes
0
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425
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Compact connected Riemannian manifolds are Ahlfors regular metric space
Let $(M,g)$ be a compact connected $n$-dimensional Riemannian manifold; let $(X,d)$ denote its associated metric (length) space. A comment on the original formulation of this post mentioned that $(X,...
- 5,405
1
vote
1
answer
258
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
- 459
6
votes
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
6
votes
0
answers
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
3
votes
0
answers
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