Approximation argument in geometric flows

I’m studying by myself Mean Curvature Flow and I’m reading the paper “Interior estimates for hypersurfaces moving by mean curvature” by Klaus Ecker and Gerhard Huisken and I’m stuck in the following theorem: Theorem 5.1: M0≡F0(Rn) be a locally Lipschitz continuous entire graph over Rn, then the initial value problem (1) has a smooth solution … Read more

Double commutant of compact operators

So my question is straightforward. Let $\mathfrak{X}$ be a (complex, if necessary) Banach space and $K\colon\mathfrak{X}\to\mathfrak{X}$ a nonzero compact operator. Denote by $\mathcal{C}(K)$ the commutant of $K$—i.e. the algebra of operators that commute with $K$, and $\mathcal{C}^2(K):=\mathcal{C}(\mathcal{C}(K))$. For every $T\in \mathcal{C}^2(K)$ that is not a multiple of the identity, is it the case that $\mathcal{C}^2(T)$ … Read more

Weak compactness of order intervals in L1L^1

Let (Ω,μ) be a measure space, say σ-finite for the sake of simplicity, and let L1:=L1(Ω,μ) denote the real-valued L1-space over (Ω,μ). For all f,h∈L1 we call the set [f,h]:={g∈L1:f≤g≤h} the order interval between f and h. Order intervals have the following property: Proposition 1. Every order interval in L1 is weakly compact. However, the … Read more

Can the degree of kk-nilpotence of a simple simply connected compact Lie group be in (0,1)(0,1)?

Let G be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of k-nilpotence of G to be the Haar measure of the set {(x1,…,xk+1):[x1,…,xk+1]=1}. ([x,y]=x−1y−1xy and [x1,…,xk+1]:=[[x1,…,xk],xk+1].) The following question is raised in Martino, Tointon, Valiunas, and Ventura – Probabilistic nilpotence in infinite groups: Does a … Read more

When is a Nemytskii map between Sobolev spaces compact?

Let f:R→R be a smooth function with bounded derivative. Define the Nemytskii map F:H1(Ω)→H1(Ω) by F(u)(x):=f(u(x)). Here Ω is a bounded smooth domain. There exists work where we can deduce continuity and differentiability of F under some assumptions on f. What conditions on f will ensure that F satisfies the following compactness criterion: if un⇀u … Read more

How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$ a nice description of the complex $E_6$ … Read more

Pullback of Morse form satisfies Palais Smale

Let $(\alpha,g)$ be a Morse-Smale pair on a closed smooth manifold $M$, i.e. $\alpha$ is a Morse form and $g$ a Riemannian metric on $M$ such that stable and unstable manifolds of the gradient vector field $X$ intersect transversally. Let $\pi \colon \tilde{M} \to M$ be the associated cover to $\ker [\alpha]$. Denote by $\tilde{f} … Read more

Is this property of continuous maps equivalent to some more familiar condition?

Let f:X→Y be a continuous map. Suppose that, for each collection of open sets {Vi}i∈I⊂X, ⋃U⊂Y open, f−1(U)⊂⋃i∈IViU=⋃i∈I⋃U⊂Y open, f−1(U)⊂ViU. I wonder whether this property is equivalent to being proper for locally compact hausdorff spaces. Below I have a proof of one direction. Write f∗(V)=⋃f−1(U)⊂VU. (1) says that f∗ preserves joins, so that it has a right adjoint f!. … Read more

Reeb stability counterexample: foliation in Sn−2×S1×S1S^{n-2}\times S^1\times S^1 with non-diffeomorphic leaves

Reeb’s global stability theorem requires the foliation to be of codimension 1. As a counterexample, in “Geometric theory of foliations”, Camacho and Lins Neto present the following. Consider the manifold Sn−2×S1×S1 with coordinates (x1,…,xn−1,φ,θ) such that ∑n−1i=1x2i=1 , and two differential one-forms η1=dθ and η2=((1−sin(θ))2+x21)dφ+sin(θ)dx1. Frobenius’ integrability condition is satisfied as can be easily checked, so there’s … Read more

Compactness of symmetric power of a compact space

Suppose I have a compact metric space (X,d) and let X=XK be the product space. Consider the equivalence relation ∼ on X given as: for α,β∈X, α∼β iff there exists a permutation τ of {1,…,K} such that αi=βτ(i) for all i. Consider the quotient space X/∼ and define a metric ρ on this quotient space … Read more