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The number of cusps of complete Riemannian manifolds with finite volume. (English) Zbl 1419.53050

Let \(E\) be an end of a Riemannian manifold \(M_n\). Then the number of ends (cusps) can be counted via the bottom of Neumann spectrum \(\mu_1(M)\) as the following theorem shows: Theorem 1.2. Let \((M,g,e_f,dv)\) be a complete smooth metric measure space. Assume for some nonnegative constants \(\alpha\) and \(\beta\), \[ |f|(x)\leq \alpha r(x)+\beta \] for \(x\in M\) with Ricci curvature bounded from below by \(\mathrm{Ric}_f\geq-(n-1)\). Assume that \(M\) has finite volume given by \(V_f\), and \[ \mu_1(M)\geq \dfrac{(n-1 +\alpha)^2}{4}. \] Let us denote \(N(M)\) to be the number of ends (cusps) of \(M\). Then there exists a constant \(C(n)>0\) depending only on \(n\), such that, \[ N(M)\leq C(n) (\dfrac{V_f}{V_{o,f}(1)})^2 \ln (\dfrac{V_f}{V_{o,f}(1)}), \] where \(V_{o,f}(1)\) denotes the \(f\)-volume of the unit ball centered at any fixed point \(o\in M\).

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C24 Rigidity results
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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