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Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$. Is $\sqrt g$ is infinitely (right) differentiable at $x=0$? …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ This is a cross-post from MSE. Let $\M,\N$ be Riemannian m …

Let $M$ be a smooth manifold with non-empty boundary. Let $g$ be a smooth Riemannian metric on $M$. Is the following true? For every $\epsilon >0$ there exist a Riemannian metric $g_{\epsilon}$ w …

Indeed, the result can be found in the book Harmonic morphisms between Riemannian manifolds, by Paul Baird, John C. Wood. The relevant statement is Corollary 3.5.2.

$\newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\VolM}{\text{Vol}_{\M}} \newcommand{\VolN}{\text{Vol}_{\N}}$ This question is mainly a reference request. (It is a cross-post fro …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\TM}{T\mathcal{M}}$ $\newcommand{\Ric}{\operatorname{Ric}}$ $\newcommand{\Volg}{\operatorname{Vol}_g}$ I would like to find a reference for the following c …

This is a cross-post from MSE (where I got no answer). It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic. I discovered lately that in dimension $d>2$, co …

This is a cross-post. It seems to be folklore knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps. Specifically, I have found this nice proof for the following …

Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group. A proo …

Let $f:M \to N$ be a smooth weakly conformal map between connected $d$-dimensional Riemannian manifolds, i.e. $f$ satisfies $df^Tdf =(\det df)^{\frac{2}{d}} \, \text{Id}_{TM}$. Assume $d \ge 3$ a …

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 $H^k_{g_0}=\text{ker} \Delta_{g_0}$. …

$\newcommand{\M}{M}$ 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. …

I am looking for a reference for the following claim: Let $\phi:\mathbb (a,b) \to \mathbb R$ be a continuous function, and let $c \in (a,b)$ be fixed. Suppose that "$\phi$ is convex at $c$". i.e. for …

This is just an elaboration on Robert's great answer: The key idea is to use the fact that the induced metric on the Hom -space of two inner product spaces is "linear" in the "metric" on the target. …

Well, for conformal maps equation $(1)$ is merely $d$-harmonicity in disguise:) The equation is $$ \delta\big((\det df)^{1-\frac{2}{d}} df\big)=0. \tag{1} $$ Since for conformal maps, $\det df=\|df\ …