Timeline for High multiplicity eigenvalue implies symmetry?
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| Mar 18, 2012 at 0:32 | comment | added | Liviu Nicolaescu | The Poincare sphere is also a Seifert manifold. I have to admit that I am a bit confused about how the Seifert structure fits in the above picture. | |
| Mar 17, 2012 at 22:50 | comment | added | Noam D. Elkies | The Poincaré homology sphere $P$ is a counterexample, right? It's the quotient of $S^3$ by a finite group $G=2.A_5$ acting freely by isometries. Any automorphism of $P$ lifts to its universal cover $S^3$,and is thus a coset of $2.A_5$ in the normalizer of $2.A_5$ in ${\rm Aut}(S^3) = O_4({\bf R})$; but this normalizer is $2.A_5$ itself, so ${\rm Aut(P)}$ is trivial. But for even $d$ the space $H_d$ of harmonic polynomials of degree $d$ has a $2.A_5$ invariant subspace that is an eigenspace for the Laplacian on $P$ of dimension asymptotic to $\dim(H_d) / 60 \rightarrow \infty$ with $d$. | |
| Mar 16, 2012 at 22:49 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |