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Based on ideas of L. Alías, D. Impera and M. Rigoli developed in [13], we present a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice Theorem of [13, Thm. 16 and 21], see Theorems 5 and 6.

Maximum principle for semi-elliptic trace operators and geometric applications