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Small eigenvalues of surfaces of finite type
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 8 / August 2017
- Published online by Cambridge University Press:
- 05 June 2017, pp. 1747-1768
- Print publication:
- August 2017
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- Article
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Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface
$S$ of finite type and Euler characteristic 
$\unicode[STIX]{x1D712}(S)<0$ has at most 
$-\unicode[STIX]{x1D712}(S)$ small eigenvalues.
Counterexamples to Smoothing Convex Functions
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- Journal:
- Canadian Mathematical Bulletin / Volume 29 / Issue 3 / 01 September 1986
- Published online by Cambridge University Press:
- 20 November 2018, pp. 308-313
- Print publication:
- 01 September 1986
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Greene and Wu have shown that any continuous strongly convex function on a Riemannian manifold can be uniformly approximated by infinitely differentiable strongly convex functions. This result is not true if the word “strongly” is omitted; in this paper, we give examples of manifolds on which
convex functions cannot be approximated by 
convex functions (k = 0, 1,2,...).
convex functions cannot be approximated by 
convex functions (