Topology of the nodal set of random equivariant spherical harmonics on $S^3$

Junehyuk Jung; Steve Zelditch

doi:10.1093/imrn/rnz348

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arXiv · 1908.00979

Topology of the nodal set of random equivariant spherical harmonics on $S^3$

Junehyuk Jung·Steve Zelditch

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Abstract

We show that real and imaginary parts of equivariant spherical harmonics on $S^3$ have almost surely a single nodal component. Moreover, if the degree of the spherical harmonic is $N$ and the equivariance degree is $m$, then the expected genus is proportional to $m \left(\frac{N^2 - m^2}{2} + N\right) $. Hence if $\frac{m}{N}= c $ for fixed $0 < c < 1$, the genus has order $N^3$.

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Junehyuk Jung, Steve Zelditch. 2019-08-02. Topology of the nodal set of random equivariant spherical harmonics on $S^3$. https://doi.org/10.1093/imrn%2Frnz348

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