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Let $\mathbb D^2$ be the closed unit disk in $\mathbb R^2$. Let $f:\mathbb D^2 \to \mathbb{R}^2$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^2 \to \mathbb{R}^2$ be t …

This is a follow-up question of this one. Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a sm …

While analyzing a variational problem, I came to the following question: Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ …

It seems that the answer is negative for dimension $n=2$ . I am not sure if higher dimensions can be reduced to the $2D$ case. Here is the argument for $n=2$: Suppose that there exist $f_k \in C^{\i …

Let $\mathbb D^n$ be the closed unit ball in $\mathbb R^n$. Let $f:\mathbb D^n \to \mathbb{R}^n$ be a real-analytic orientation preserving immersion, and let $\omega:\mathbb D^n \to \mathbb{R}^n$ be t …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\ep}{\epsilon}$ I will sketch here a different approach than the one given by Robert's answer. (It is loosely based on Alex's answer). We want to prove that …

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open neighbourhood …

In a nutshell: Given a metric on the torus $\mathbb{T}^n$, can we extend any element $\sigma \in \bigwedge^k T_p^*\mathbb{T}^n$ to a global harmonic form? Let $\mathbb{T}^n$ be the $n$-Torus. F …

$\newcommand{\M}{M}$ This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$. Explicitly, let $\M$ be a smooth oriented Riemannian manifold. …

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{g_0}$. …

I think the answer is positive. Let $D$ be the subspace of smooth closed $k$-forms on $M$. Equip $D$ with the supremum- $C^1$ norm: $$ \| \omega \|_{C^1,sup}:=\max\{ \|\omega\|_{sup}, \|T\omega\|_{su …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly …