All Questions
9 questions
2
votes
0
answers
134
views
Critical points of a strictly subharmonic function
Let $M$ be a smooth, compact manifold with boundary. Let $u: M \to \mathbf{R}$ be a smooth function that has its Riemannian Laplacian equal to a positive constant:
\begin{equation}
\Delta u = A > 0....
- 5,038
8
votes
2
answers
773
views
Points where harmonic functions fail to give a coordinates system
Let $\Omega$ be a bounded domain in $\mathbb R^n$ with a smooth boundary and let $g$ be a smooth Riemannian metric on $\Omega$. Let $f_1,f_2,\ldots,f_n$ be non-constant smooth functions on $\partial \...
- 4,135
5
votes
1
answer
162
views
Is the evaluation map from harmonic forms on the torus surjective on flat neighbourhoods?
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. Fix ...
- 6,741
1
vote
0
answers
105
views
Is every "higher-order" harmonic morphism conformal?
$\newcommand{\TM}{\operatorname{TM}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\operatorname{T\N}}$
$\newcommand{\TstarM}{...
- 6,741
8
votes
2
answers
471
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a ...
- 6,741
1
vote
1
answer
126
views
Invariance of the space of harmonic functions under derivation associated to a non-vanishing vector field
Let $X$ be a non-vanishing real analytic vector field on an open manifold $M$. What kind of obstructions would appear when we search for a Riemannian metric on $M$ such that the space of ...
- 356
14
votes
0
answers
632
views
Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
- 6,741
1
vote
3
answers
258
views
Harmonic Function with special property
I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two ...
- 4,135
2
votes
0
answers
104
views
Existence of harmonic maps between loops
Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
$$E[\...
- 9,853