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Consider the following example: On $\mathbb{R}^4$ with coordinates $u^1,u^2,u^3, z^1$, define $z^2 = u^2 - z^1 u^1$ and $z^3 = u^3 - z^1u^2$. We have that $U\cap Z$ is spanned by $\mathrm{d}u^2-z^1\,\... • 97.5k 5 votes Accepted ### Why are we interested in spectral gaps for Laplacian operators A spectral gap gives information on geometry of the manifold via Cheeger's inequality, https://en.wikipedia.org/wiki/Cheeger_constant See also Buser's inequality discussed there. More directly, a ... • 11.3k 2 votes Accepted ### Harmonic polynomials on the sphere I view this as a concatenation of two facts: Fact 1: Let$k$be a field, let$I$be an ideal of$k[x_1, \ldots, x_n]$and let$J$be associated graded ideal of$I$, meaning that a degree$d$... • 138k 5 votes ### Harmonic polynomials on the sphere There is a similar separation of the variables for ellipsoids$x^2/a^2+y^2/b^2+z^2/c^2=1\$ related to ellipsoidal harmonic functions, see Whittaker Watson, Chap. 23. There is a generalization to ...
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