International Journal of Mathematics

We prove some lacunary convergent series representations for $\zeta (n)$. For example, we prove that

This paper deals with the isoperimetric deficit upper bound for the convex domain in a surface ${𝕏}_{𝜖}^2$ of constant curvature $𝜖$ by the containment measure of a convex domain to contain another convex domain in integral geometry. Some reverse Bonnesen-style inequalities are obtained. In particular, two of them strengthen Zhou’s result in ${𝕏}_{𝜖}^2$ and Bottema’s result in the Euclidean plane ${𝔼}^2$.

Kishimoto raised the problem to classify all compactifications of contractible affine $3$-folds into smooth Fano $3$-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this paper, we show that there are $14$ deformation equivalence classes of smooth Fano $3$-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.

Let $\varphi$ be a globally defined real Kähler potential on a domain $\mathrm{\Omega }\subset {ℂ}^d$, and $g_F$ be a Kähler metric on the Hartogs domain $M=\{(z,w)\in \mathrm{\Omega }\times {ℂ}^{d_0}:\parallel w{\parallel }^2<e^{-\varphi (z)}\}$ associated with the Kähler potential ${\mathrm{\Phi }}_F(z,w)=\varphi (z)+F(\varphi (z)+ln\parallel w{\parallel }^2)$. First, we obtain explicit formulas of the coefficients $a_j(j=1,2)$ of the Bergman function expansion for the Hartogs domain $(M,g_F)$ in a momentum profile $\phi$. Second, using explicit expressions of $a_j(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $a_j(j=1,2)$ to be constants. Finally, we obtain all the invariant complete Kähler metrics on Cartan–Hartogs domains such that their the coefficients $a_j(j=1,2)$ of the Bergman function expansions are constants.

We obtain Cheng type inequality, Cheeger type inequality, Faber–Krahn type inequality and McKean type inequality of $p$-Laplacian on a Finsler manifold. These generalize the corresponding theorems in Riemannian geometry and sharpen some results in recent literatures. Moreover, for a complete noncompact Finsler manifold with negative constant flag curvature and vanishing $S$ curvature, the first eigenvalue is calculated.