All Questions
Tagged with harmonic-functions dg.differential-geometry
23 questions
2
votes
0
answers
86
views
Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space
Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
- 1,407
2
votes
1
answer
147
views
Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
- 117
4
votes
1
answer
200
views
Bounded covariant derivative of curvature tensor
Let $M$ be a complete Riemannian manifold.
Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...
- 45k
1
vote
0
answers
64
views
Integrability (and hence regularity) of $\alpha$-harmonic maps
To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^...
- 219
0
votes
0
answers
146
views
Generalized harmonic map
Let $M, N$ be closed Riemannian manifolds and $c$ be a constant. For a map $f:M\to N$, define the energy as
$$E(f) = \frac{1}{2} \int_M\Big( \| df(x)\|^2 - c\| f(x) \|^2 \Big) d\mu(x).$$
When $c=0$, ...
- 169
8
votes
1
answer
375
views
Harmonic functions on complete Riemannian manifolds
I have started reading a paper of Colding and Minicozzi, where they prove that on a complete Riemannian manifold $M$ of non-negative Ricci curvature, the space of harmonic functions of growth order at ...
- 81
5
votes
0
answers
160
views
Potential theory as a tool in extrinsic flows
Let $M \subseteq \mathbb{R}^n$ be a submanifold. For a point $x$ disjoint from $M$, we can define the electric potential $\Phi(x) = \int_M \frac{dM}{|x-m|^{n-2}}$, which is smooth and harmonic where ...
- 217
5
votes
1
answer
580
views
A question on the monotonicity formula for minimal submanifolds
I'm reading the proof of monotonicity formula from A Course in Minimal Surfaces by Colding-Minicozzi. The theorem says
Suppose $\Sigma^k \subset \mathbb{R}^n$ is a minimal submanifold and $x_0\in\...
- 53
7
votes
0
answers
139
views
implicit function theorem and harmonic mapping
We are given two Riemannian manifolds $M,N$ of dimension $m$ and $n$ and a function $G \colon M \times N \to \mathbb{R}^n$ which satisfies the assumptions of the implicit function theorem, meaning ...
1
vote
0
answers
84
views
Existence of nonparabolic ends
Let $M$ a nonparabolic Riemannian manifold. If exists only one nonparabolic end $E$. We would like to know why the subspace of space of bounded harmonic functions with finite Dirichlet integral is the ...
- 381
3
votes
1
answer
874
views
Motivation and examples of parabolic manifolds
Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...
- 381
5
votes
1
answer
305
views
harmonic coordinates on non-compact manifolds
Is it possible to show the existence of harmonic coordinates (e.g., on uniform-sized balls) on certain classes of non-compact Riemannian manifolds? For example, one may expect that such harmonic ...
- 103
2
votes
0
answers
62
views
Does a map which preserve harmonic forms preserve co-closed forms (locally)?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be $d$-dimensional oriented Riemannian manifolds ($d \ge 2$). Let $f:\M \to \N$ be smooth.
Let $1 \le k \le d-1$ be fixed....
- 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
5
votes
1
answer
227
views
Are all the mappings which satisfy this equation scaled isometries?
Let $M,N$ be smooth oriented $d$-dimensional Riemannian manifolds, $\, f:M \to N$ a smooth map. Let $\Omega^1(M,f^*TN)=\Gamma(T^*M \otimes f^*TN)$ be the space of $f^*TN$-valued one-forms.
Let $d$ ...
- 6,741
12
votes
1
answer
644
views
Harmonic functions (eigenfunctions of the Laplace-Beltrami operator) of SO(2n)/U(n)
Have the eigenfunctions of the Laplace-Beltrami operator on $SO(2n)/U(n)$ been worked out explicitly? If not, how does one approach finding them?
(I'm thinking of this as in analogy with the ...
- 343
3
votes
0
answers
241
views
Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity
What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...
- 272
7
votes
1
answer
259
views
Harmonic function with injective boundary conditions is an immersion?
Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are given an immersion $f:M \to \mathbb{R}^n$. (i.e $df$ is invertible at every point $p \in M$, note ...
- 294
-1
votes
1
answer
259
views
An example for affine function [closed]
I'm looking for an example of a non-Euclidean non-compact Riemannian manifold $(M,g)$ such that we could define a non-constant affine function $f:M\rightarrow \mathbb{R}$, namely its gradient vector ...
5
votes
1
answer
342
views
harmonic extension of a curve by different parametrization
Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
- 914
7
votes
0
answers
290
views
Harmonic map heat flow in positive curvature
Suppose I wish to relax/smooth a map $\phi:M\rightarrow N$ between two surfaces $M,N$ embedded in $\mathbb{R}^3$. I could try flowing the map using harmonic heat flow, which (as I understand it) is ...
- 705
6
votes
1
answer
503
views
Recovering Spherical Harmonics from Discrete Samples
Consider a collection of $N$ points on the 2-sphere chosen uniformly at random. Let's say that there's an edge between two such vertices if their geodesic distance is less than $r_N$. The resulting ...
- 7,237