Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a non-flat two-torus?
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$\begingroup$ For torus or revolution it should be easy. Is that what you want? $\endgroup$– ε-δCommented Apr 18, 2012 at 2:42
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$\begingroup$ It's done here: proquest.umi.com/… $\endgroup$– Ian AgolCommented Apr 18, 2012 at 3:34
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3$\begingroup$ It's not clear to me what your starting data are. For example, every smooth Riemannian metric on a torus can be described as being of the form $F(z)\ dz\circ d\bar z$ where $F:\mathbb{C}\to\mathbb{R}^+$ is $L$-periodic for some lattice $L\subset\mathbb{C}$. Given this description, the harmonic $1$-forms are the constant linear combinations of the real and imaginary parts of $dz$. However, if you just have a surface that is known to be a $2$-torus and are given some random metric on it with no symmetries, explicitly finding the harmonic $1$-forms is going to be a matter of luck. $\endgroup$– Robert BryantCommented Apr 18, 2012 at 10:44
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$\begingroup$ Ooops! Sorry, Robert et al! Yes, the question I wrote is trivial. I meant to say the eigenforms of the Hodge Laplacian! Stupid me. Will update. $\endgroup$– Eric ZaslowCommented Apr 18, 2012 at 11:43
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1 Answer
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This is not so much an answer as a few remarks and a caution. If I understand your request correctly, I think that it is unlikely that you are going to find a truly explicit example.
First, let me remark that, since we are considering a compact, oriented surface $T^2$, it would be enough to know all of the eigenfunctions of the Laplacian on $0$-forms, since, by the Hodge decomposition theorem, for positive eigenvalues $\lambda$, any eigenfunction of the Hodge Laplacian on $2$-forms with eigenvalue $\lambda$ would be of the form $\ast f$ where $f$ is a $\lambda$-eigenfunction, and any eigenform of degree $1$ with eigenvalue $\lambda$ would be of the form $df_0 + \ast d f_1$ for some $f_0$ and $f_1$ that are $\lambda$-eigenfunctions.
Second, if we assume that the metric is given in the form $g = F(z) dz\circ d\bar z$ where $F$ is a positive, $L$-periodic function on $\mathbb{C}$, then this metric will have nonconstant Gaussian curvature if and only if $F$ is not constant. In any case, a $\lambda$-eigenfunction will be an $L$-periodic function $f$ on $\mathbb{C}$ that satisfies $$ 4f_{z\bar z} + \lambda F f = 0. $$ So you are asking for a method of explicitly describing all of the $L$-periodic solutions of this equation. I am not aware of any positive, nonconstant $F$ for which this is known.
If you go down a dimension and ask for the list of positive, $\pi$-periodic functions $F$ on the real line for which all of the $\pi$-periodic solutions of $$ f''(x) + \lambda F(x) f(x) = 0 $$ are explicitly known for each $\lambda>0$, I think you will find that this list is very short. It makes me suspect that, in the $2$-dimensional case, the list is not longer.
answered Apr 20, 2012 at 11:40
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$\begingroup$ Thanks, Robert! That's exactly where I had arrived, having considered a torus with $S^1$ isometry. For particular $F$ you can make it look a lot like, but not precisely equal to, the "quantum pendulum." That system is described by the well-studied Matthieu equation, but again it's not precisely the correct equation (or at least I can't make it be). $\endgroup$ Commented Apr 20, 2012 at 15:44