Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\ldots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito [dB90] to higher dimensions.

As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Chern's conjecture.

Motivated by [5], we obtain new equivalent conditions for two-sided heat kernel estimates of symmetric non-local Dirichlet forms in terms of resistance forms. Characterizations for upper bounds of heat kernel estimates as well as near diagonal lower bounds of Dirichlet heat kernel estimates are also established. These results can be seen as a complement of the recent studies on heat kernel estimates and parabolic Harnack inequalities for symmetric non-local Dirichlet forms in [10, 11].

Let $D$ be an open subset of an $n$-dimensional Stein manifold, where $n\geq 2$. Assume that the canonical map $H^{n-1}(D,\mathcal{O})\to H^{n-1}(D,\mathcal{M})$ is injective. Then, we prove that $D$ is pseudoconvex of order 1, which generalizes the well-known theorem of Cartan-Behnke-Stein.

Schmidt and Stix proved that every smooth variety over a field finitely generated over the field of rational numbers has an open basis for the Zariski topology consisting of “anabelian” varieties. This was predicted by Grothendieck in his letter to Faltings. In the present paper, we generalize this result to smooth varieties over generalized sub-$p$-adic fields. Moreover, we also discuss an absolute version of this result.

In this article, we are concerned with radial solutions for the best constant of the Cafarelli-Kohn-Nirenberg inequality. Firstly, we classify the radial solutions according to its asymptotic behavior as $r \to 0$ and $r \to \infty$. Secondly, we investigate the structure of radial singular solutions. Lastly, we briefly discuss the Neumann problem related to the Cafarelli-Kohn-Nirenberg inequality.

In this paper we generalize the characterization for Sobolev spaces with constant exponent via averages on balls to the variable exponent setting. This method can be used to define Sobolev spaces with variable exponent on metric spaces.

We show that a closed almost Kähler 4-manifold of pointwise constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is $J$-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.

In this paper we extend Gazaki's results on the Chow groups of abelian varieties to the higher Chow groups. We introduce a Gazaki type filtration on the higher Chow group of zero-cycles on an abelian variety, whose graded quotients are connected to the Somekawa type $K$-group. Via the étale cycle map, we will compare this filtration with a filtration on the étale cohomology induced by the Hochschild-Serre spectral sequence. As an application over local fields, we obtain an estimate of the kernel of the reciprocity map.

We establish pointwise growth estimates for the spherical derivative of solutions of the first order algebraic differential equations. A generalization of this result to higher order equations is also given. We discuss the related question of when for a given class $X$ of meromorphic functions in the unit disc, defined by means of the spherical derivative, and $m\in \mathbb{N}\setminus\{1\}$, $f^m\in X$ implies $f\in X$. An affirmative answer to this is given for example in the case of $\mathord{\rm UBC}$, the $\alpha$-normal functions with $\alpha\ge 1$ and certain (sufficiently large) Dirichlet type classes.

In this paper, we introduce a modified Picard-Mann hybrid iterative process for a finite family of mappings in the framework of hyperbolic spaces. Furthermore, we establish $\Delta$-convergence and strong convergence results for a sequence generated by a modified Picard-Mann hybrid iterative process involving mappings satisfying the condition $(E)$ in the setting of hyperbolic spaces which more general than one mapping in the setting of CAT(0) spaces in Ritika and Khan [19]. Our results are the extension and improvement of the results in Ritika and Khan [19]. Moreover, in the numerical example we also illustrate an example for supporting our main result.