All Questions

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 ...

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 ...

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 ...

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 \...

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, ...

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,...