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''Definition:'' Let ''M'' be a Riemannian manifold, and <math>f:\; M \to \R</math> an [[upper semicontinuous]] function. Assume that for any open subset <math>U\subset M</math>, and any [[harmonic function]] ''f''<sub>1</sub> on ''U'', such that <math>f_1 \geq f</math> on the boundary of ''U'', the inequality <math>f_1 \geq f</math> holds on all ''U''. Then ''f'' is called ''subharmonic''.
This definition is equivalent to one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality <math>\Delta f \geq 0</math>, where <math>\Delta</math> is the usual [[Laplace–Beltrami_operator|Laplacian]].<ref>{{Cite journal | author = Greene, R. E. | year = 1974 | title = Integrals of subharmonic functions on manifolds of nonnegative curvature | journal = Inventiones Mathematicae | volume = 27 | pages = 265–298 | doi = 10.1007/BF01425500 | last2 = Wu | first2 = H. |
==See also==
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