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

Examples of indefinite Einstein non-Ricci-flat metric on solvmanifolds

A solvmanifold (resp. nilmanifold) is a compact quotient of a solvable (risp. nilpotent) Lie group G. Usually one consider a co-compact lactice Γ on G and the solvmanifold is G/Γ. Moreover, if there are left-invariant structures on G (e.g. metrics, complex strucutres etc.), those structures define analogous structure on the compact quotient. Moreover, left-invariant structures … 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

Compact images of nowhere dense closed convex sets in a Hilbert space

Let B=−B be a nowhere dense bounded closed convex set in the Hilbert space ℓ2 such that the linear hull of B is dense in ℓ2. Question. Is there a non-compact linear bounded operator T:ℓ2→ℓ2 such that T(B) is compact? Answer Your revised assumption is that the norm (rather than semi-norm after the revision) on … 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