International Journal of Mathematics

By using some line integrals in terms of the radial $m$-Bakry–Émery and radial $m$-modified Bakry–Émery Ricci curvatures, we give various compactness criteria for complete Riemannian manifolds when $m$ is a positive constant, a negative constant, or infinity. Our results not only guarantee the compactness of complete Riemannian manifolds allowing the presence of negative amounts of the radial $m$-Bakry–Émery and radial $m$-modified Bakry–Émery Ricci curvatures, but also give new Myers-type theorems via $m$-Bakry–Émery and $m$-modified Bakry–Émery Ricci curvatures even the line integrals are reduced to pointwise positive lower bounds on the $m$-Bakry–Émery and $m$-modified Bakry–Émery Ricci curvatures. The key ingredients in proving our results are Riccati inequalities obtained from Bochner–Weitzenböck formulas via $m$-Bakry–Émery and $m$-modified Bakry–Émery Ricci curvatures.

Given an extension $0\to V\to G\to Q\to 1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta :Q\to \widehat{V}$, we define a dual unitary $2$-cocycle on $G$ and show that the associated deformation of $Ĝ$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q⋉\widehat{V}$. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^2(Q)$ and a unitary quantization map $L^2(G)\to HS(L^2(Q))$ of Kohn–Nirenberg type.

We prove that the automorphism group of a general complete intersection $X$ in ${ℂ\mathbb{P}}^n$ is trivial with a few well-understood exceptions. We also prove that the automorphism group of a complete intersection $X$ acts on the cohomology of $X$ faithfully with a few well-understood exceptions.

In this paper, we study the stability of the volume preserving mean curvature flow of closed hypersurfaces in the hyperbolic space. We prove that an $L^2$-almost umbilical hypersurface will be deformed to a totally umbilical hypersurface along the flow. Our result removes the assumption on the mean curvature in the theorems of Huang-Lin-Zhang [Peking J. Math. (2023)] and Leng-Xu-Zhao [Int. J. Math. (2014)].

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the $1$-dimensional heat equation $u_t=u_{xx}$ on ${ℝ}^2$ with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the $n$-dimensional heat equation $u_t=△u$ on ${ℝ}^{n+1}$ with zero initial data (i.e. vanishing on the hyperplane $P:\{t=0\}$) can be easily obtained as a consequence of the examples on ${ℝ}^2.$ Finally, given an arbitrary hyperplane $P\subset {ℝ}^{n+1}$, we can construct a family of nonzero solutions which vanish on $P\subset {ℝ}^{n+1}.$

In this paper, we prove two separate lower bounds — one for nondegenerate convex domains and the other for nondegenerate $ℂ$-convex (but not necessarily convex) domains — for the squeezing function that hold true for all domains in ${ℂ}^n$, for a fixed $n\ge 2$, of the stated class. We provide explicit expressions in terms of $n$ for these estimates.