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In this note, we prove a family of sharp weighed inequalities which involve weighted k-th mean curvature integral and two distinct quermassintegrals for closed hypersurfaces in \mathbb {R}^n. This inequality generalizes the corresponding result of Wei and Zhou [Bull. Lond. Math. Soc. 55 (2023), pp. 263–281] where their proof is based on earlier results of Kwong-Miao [Pacific J. Math. 267 (2014), pp. 417–422; Commun. Contemp. Math. 17 (2015), p. 1550014]. Here we present a proof which does not rely on Kwong-Miao's results. [ABSTRACT FROM AUTHOR]

In this short note we show that Doyle-Grigor'yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor'yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range. [ABSTRACT FROM AUTHOR]

The set of degrees of maps D(M,N), where M,N are closed oriented n-manifolds, always contains 0 and the set of degrees of self-maps D(M) always contains 0 and 1. Also, if a,b\in D(M), then ab\in D(M); a set A\subseteq \mathbb {Z} so that ab\in A for each a,b\in A is called multiplicative. On the one hand, not every infinite set of integers (containing 0) is a mapping degree set (Neofytidis, Wang, and Wang [Bull. Lond. Math. Soc. 55 (2023), pp. 1700–1717]) and, on the other hand, every finite set of integers (containing 0) is the mapping degree set of some 3-manifolds (Costoya, Muñoz and Viruel [ Finite sets containing zero are mapping degree sets , arXiv: 2301.13719 ]). We show the following: Not every multiplicative set A containing 0,1 is a self-mapping degree set. For each n\in \mathbb {N} and k\geq 3, every D(M,N) for n-manifolds M and N is D(P,Q) for some (n+k)-manifolds P and Q. As a consequence of (ii) and Costoya, Muñoz and Viruel, every finite set of integers (containing 0) is the mapping degree set of some n-manifolds for all n\neq 1,2,4,5. [ABSTRACT FROM AUTHOR]