Let $M = \Gamma \backslash H$ be a geometrically finite hyperbolic surface, realized as the quotient of the hyperbolic upper half plane $H$ by a geometrically finite discrete group of isometries acting on $H$. To a parabolic element of the uniformizing group $\Gamma$, there is an associated 1-form parabolic Eisenstein series. To a primitive hyperbolic element, then, following ideas due to Kudla-Millson, there is a corresponding 1-form hyperbolic Eisenstein series. In this article, we study the limiting behavior of these hyperbolic Eisenstein series on a degenerating family of hyperbolic Riemann surfaces of finite type, using basically the limiting behavior of counting functions associated to degenerating hyperbolic Riemann surfaces. In this sense, we generalize the results obtained in Garbin, Jorgenson and Munn (Comment Math Helv 83:701-721, 2008) to the case of geometrically finite hyperbolic surfaces of infinite volume and form-valued parabolic and hyperbolic Eisenstein series.

Let $M \subset \mathbf{R}^{n+2}$ be a two-dimensional complete intersection. We show how to check whether a mapping $f : M \rightarrow \mathbf{R}^2$ is 1-generic with only folds and cusps as singularities. In this case we give an effective method to count the number of positive and negative cusps of a polynomial $f$, using the signatures of some quadratic forms.

In this paper, we deal with complete linear Weingarten submanifolds $M^n$ immersed with parallel normalized mean curvature vector field in a Riemannian space form $\mathbf{Q}_c^{n+p}$ of constant sectional curvature $c$. Under an appropriated restriction on the norm of the traceless part of the second fundamental form, we show that such a submanifold $M^n$ must be either totally umbilical or isometric to a Clifford torus, if $c = 1$, a circular cylinder, if $c = 0$, or a hyperbolic cylinder, if $c = −1$. We point out that our results are natural generalizations of those ones obtained in [2] and [6].

In this paper, we investigate the boundary value problem of the following operator $$\left\{\begin{array}{l}\Delta^{2}u-a\rm div\it A\nabla{u}+Vu=\rho\lambda{u}\ \ \hbox{in} \ \rm\Omega, \\ u|_{\partial \Omega}=\frac{\partial u}{\partial v}|_{\partial \Omega}=0,\\ \end{array}\right.$$ where $\Omega$ is a bounded domain in an $n$-dimensional complete Riemannian manifold $M^n$, $A$ is a positive semidefinite symmetric (1,1)-tensor on $M^n$, $V$ is a non-negative continuous function on $\Omega$, $v$ denotes the outwards unit normal vector field of $\partial \Omega$ and $\rho$ is a weight function which is positive and continuous on $\Omega$. By the Rayleigh-Ritz inequality, we obtain universal inequalities for the eigenvalues of these operators on bounded domain of complete manifolds isometrically immersed in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.

We prove some weighted inequalities for delta derivatives acting on products and compositions of functions on time scales and apply them to obtain generalized dynamic Opial-type inequalities. We also employ these inequalities to establish some new dynamic Lyapunov-type inequalities, which are essential in studying disfocality, disconjugacy, lower bounds of eigenvalues, and distance between generalized zeros for half-linear dynamic equations. In particular, we solve an open problem posed by Saker in [Math. Comput. Modelling 58 (2013), 1777-1790]. Moreover, the results presented in this paper generalize, improve, extend, and unify most of known results not only in the discrete and continuous analysis but also on time scales.

The analytic part of a planar harmonic mapping plays a vital role in shaping its geometric properties. For a normalized analytic function $f$ defined in the unit disk, define an operator $\Phi[f](z) = f(z) + \overline{f(z)-z}$. In this paper, necessary and sufficient conditions on $f$ are determined for the harmonic function $\Phi[f]$ to be univalent and convex in one direction. Similar results are obtained for $\Phi[f]$ to be starlike and convex in the unit disk. This results in the coefficient estimates, growth results and convolution properties of $\Phi[f]$. In addition, various radii constants associated with $\Phi[f]$ have been computed.

We consider laws of the iterated logarithm and the rate function for sample paths of random walks on random conductance models under the assumption that the random walks enjoy long time sub-Gaussian heat kernel estimates.

The asymptotic behavior of conformal metrics with negative curvatures near an isolated singularity was described up to the second order derivatives by Kraus and Roth, 2008. We refine Kraus and Roth's result for the second order mixed derivatives and give estimates for higher order derivatives near an isolated singularity. We also compute the Minda-type limits for SK-metrics near the singularity. Combining these limits with Ahlfors' lemma, we provide two observations for SK-metrics.

In this paper, we consider $p$-harmonic functions on complete Riemannian manifolds and give different proofs of some main theorems by Chang-Chen-Wei, in [3]. Moreover, we are able to refine their results in case of weakly $p$-harmonic functions. Some applications to study the connectedness at infinity of stable minimal hypersurfaces are also given.

Let $M$ and $N$ be connected and orientable, closed surfaces. For a stable map $\varphi : M \rightarrow N$, denote by $c(\varphi)$ and $n(\varphi)$ the numbers of cusps and nodes of $\varphi$ respectively. In this paper, we determine the minimal number $c(\varphi) + n(\varphi)$ among the apparent contours of degree $d$ stable maps $M \rightarrow N$ whose singular points set consists of one component.

The paper addresses two long-standing problems: of extending the second main theorem of Nevanlinna to the case of small functions, and of finding an upper limit for the number of asymptotic functions of a function of finite lower order. Upper estimates of the sum of deviations and the numbers of strong asymptotic functions and strong functional asymptotic spots of meromorphic functions of finite lower order are presented. The structure of the set of Petrenko's deviations from small functions for meromorphic functions of finite lower order is examined. An analogue of Denjoy's question for strong asymptotic small functions of meromorphic functions of finite lower order is also considered.