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Suppose we have a compact Riemannian manifold $(M,g)$, and a Morse function $f : M \rightarrow \mathbb{R}$. Suppose we consider the gradient flow generated by this function, i.e. $$\dot{x_t} = - \nabla f(x_t).$$

By Morse theory, this flow will converge to a critical point of $f$, and only local minima of $f$ will have basins of attraction with positive volume. How does one go about estimating the actual volume of these basins in terms of $f$ or its derivatives?

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    $\begingroup$ Just a remark: any closed manifold will admit a morse function with one minimum. The volume of the basin is the total volume of the manifold. The function can be chosen arbitrarily small in the $C^k$ sense. $\endgroup$
    – Thomas Rot
    Commented Feb 17, 2017 at 8:39
  • $\begingroup$ Though of course one can normalize by introducing things like $(\max f - \min f)^{-1}$ in the expressions. $\endgroup$
    – Willie Wong
    Commented Feb 17, 2017 at 16:13

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