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Results tagged with harmonic-functions
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user 85934
For questions regarding harmonic functions.
2
votes
1
answer
220
views
When is this differential form harmonic?
Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions …
- 2,095
6
votes
1
answer
178
views
Tangential harmonic $1$-forms are pullbacks of harmonic functions
This question has also been posted on MSE, but maybe here is the right place to obtain an answer.
Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The Hodg …
- 2,095
6
votes
1
answer
416
views
Every homotopy class contains at least a harmonic representative
Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is harmoni …
- 2,095