4
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Consider the Laplace operator $\Delta = d^{*} d + d d^{*}$ on $\Omega^2(S^3)$. What is the minimal eigenvalue of $\Delta$?

(My computations showed that the answer is 4; the eigenforms correspond to forms with constant coefficients in $R^4$. Later I found papers that claim that answer is 3, but I don’t understand what the eigenforms would be in that case.)

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  • $\begingroup$ Since $S^3$ is orientable, its $2$-spectrum coincides with the $1$-spectrum. The smallest eigenvalue in the $1$-spectrum is $3$, which coincides with the smallest positive eigenvalue on $0$-forms (i.e. functions). $\endgroup$
    – emiliocba
    Commented Aug 5, 2023 at 9:39

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