When is a closed differential form harmonic relative to some metric? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:32:48Z http://mathoverflow.net/feeds/question/75038 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric When is a closed differential form harmonic relative to some metric? Slobodan Simi&#263; 2011-09-09T21:34:09Z 2012-05-13T03:40:31Z <p>Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$. </p> <p><strong>Question</strong>: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic, i.e., $\Delta_g \omega = 0$? </p> <p>Here $\Delta_g$ is the Laplace-deRham operator, defined as usual by $\Delta_g = d \delta + \delta d$, where $\delta$ is the $g$-codifferential. Note that non-exactness is important, since if $\omega$ were to be exact and harmonic, then by the Hodge decomposition theorem $\omega = 0$.</p> <p>For instance, if $\omega$ is a 1-form on the unit circle, then it is not hard to see that $\omega$ is harmonic with respect to some metric $g$ if and only if it is a volume form (i.e., it doesn't vanish). This observation generalizes to forms of top degree on any $M$.</p> <p>What can be said in general for forms which are not of top degree?</p> http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric/75043#75043 Answer by aglearner for When is a closed differential form harmonic relative to some metric? aglearner 2011-09-09T22:10:05Z 2011-09-10T01:40:37Z <p>This question is quite subtle, I don't believe the answer is known in the general situation. But if you consider the case of $1$-forms on surfaces, one can completely characterise those that are harmonic. They form a "space" of finite dimension modulo self-diffeos of the surface. They are called <em>minimal</em>, and they can all be represented as real parts of some holomorphic $1$-forms. Minimal 1-forms on a surface $S$ are characterised by the property that for each point $x\in S$, where the one-form is non-vanishing there exists a circle $S^1$ on $S$ such that the one-form restricted to $S^1$ has not zeros while $x\in S^1$.</p> <p>For example, in the case of a $T^2$ a one-form is harmonic for some metric iff it has no zeros at all.</p> http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric/75136#75136 Answer by Dan Fox for When is a closed differential form harmonic relative to some metric? Dan Fox 2011-09-11T10:41:44Z 2011-09-11T10:41:44Z <p>A closed $k$-form is called <em>instrinsically harmonic</em> if there is some Riemannian metric with respect to which it is harmonic. E. Calabi ("An instrinsic characterization of harmonic 1-forms", 1969) showed that a one-form having non-degenerate zeros on a compact manifold without boundary is intrinsically harmonic if and only if it satisfies a property called <em>transitivity</em>. The precise statement and proof can be found in chapter 9 of M. Farber's book "Topology of closed one-forms". In what comes I am following Farber. That a closed one-form $\omega$ have non-degenerate zeros means that near each zero it can be written in the form $\omega = df$ with $f$ a Morse function. For such a one-form, the additional assumption of harmonicity means that the Morse index of a zero cannot be $0$ or $n$ (write $\omega = df$ near the zero; because $\omega$ is co-closed, $f$ is harmonic, so by the maximum principle cannot have a max or min at the zero). That $\omega$ be <em>transitive</em> means that for any point $p$ of $M$ which is not a zero of $\omega$ there is a smooth $\omega$-positive loop $\gamma: [0, 1] \to M$; that is, $\gamma(0) = p = \gamma(1)$, and $\omega(\dot{\gamma}(t)) > 0$ for $t \in [0, 1]$. Then Calabi's theorem states that a closed one-form with non-degenerate zeros is instrinsically harmonic if and only if it is transitive. Near a non-degenerate index $0$ zero of a closed one-form the one-form can be written in the form $\delta_{ij}x^{i}dx^{j}$, for which it can be checked there are no positive loops beginning sufficiently near the origin.</p> <p>(If one can handle $k$-forms then by Hodge duality one expects to be able to get somewhere with $(n-k)$-forms. The intrinsic harmonicity of $(n-1)$-forms was characterized in terms of transitivity in the thesis of Ko Honda, available on his web page).</p> <p>E. Volkov's <em>Characterization of intrinsically harmonic forms. J. Topol. 1 (2008), no. 3, 643–650,</em> weakens the non-degeneracy condition, replacing it with the condition that the closed one-form be <em>locally instrinsically harmonic</em> - that is, the restriction of the form to a suitable open neighborhood of its zero set is instrinsically harmonic.</p> <p>As far as I know, for higher degree forms nothing much is known at all, though for some special cases, like $2$-forms on $4$-manifolds, something more has been said. One imagines that with further assumptions on the form, perhaps more can be said - for example a symplectic form is always intrinsically harmonic (use the metric determined by a compatible almost complex structure). On the other hand, Volkov's paper exhibits a closed $2$-form of rank $2$ on a $4$-manifold which is transitive but not intrinsically harmonic.</p> http://mathoverflow.net/questions/75038/when-is-a-closed-differential-form-harmonic-relative-to-some-metric/96761#96761 Answer by Stephen for When is a closed differential form harmonic relative to some metric? Stephen 2012-05-12T10:30:35Z 2012-05-13T03:40:31Z <p>(Removed by author: off-point. Sorry).</p>