Mathematics – Differential Geometry
Scientific paper
2011-01-31
Mathematics
Differential Geometry
42 pages, 1 figure
Scientific paper
Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip $U_L$ with the standard Gaussian probability measure induced by the $L^2$-metric on $U_L$, and we denote by $N_L$ the expected number of critical points of a random function in $U_L$. We prove that $N_L\sim C_m\dim U_L$ as $L\rightarrow \infty$, where $C_m$ is an explicit positive constant that depends only on the dimension $m$ and satisfying the asymptotic estimate $\log C_m\sim\frac{m}{2}\log m$ as $m\ra \infty$.
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