## 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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