Let f : X .仺 Y be a holomorphic map of complex manifolds, which is proper, K丯ahler, and surjective with connected fibers, and which is smooth over Y \ Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X. In our previous paper, we discussed the Nakano semi-positivity of Rqf(KX/Y . E) for q . 0 with respect to the so-called Hodge metric, when the map f is smooth. Here we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(Rqf(KX/Y . E)) 乬over Y \ Z乭 as a singular Hermitian metric with semi-positive curvature 乬over Y 乭. As a particular consequence, if Y is projective, Rqf(KX/Y . E) is weakly positive over Y \ Z in the sense of Viehweg.

An n dimensional minimal submanifold $\sum$  of $R^{n+m}$ is called nonparametric if $\sum$  can be represented as the graph of a vector-valued function $f : D \subset \mathbb{ R}^n \to \mathbb{R}^n$. This note provides a sufficient condition for the stability of such $\sum$ in terms of the norm of the differential $df$.

We derive a gradient estimate for positive functions, in particular for positive solutions to the heat equation, on finite or locally finite graphs. Unlike the well known Li-Yau estimate, which is based on the maximum principle, our estimate follows from the graph structure of the gradient form and the Laplacian operator. Though our assumption on graphs is slightly stronger than that of Bauer et al. (J Differ Geom 99:359–405, 2015), our estimate can be easily applied to nonlinear differential equations, as well as differential inequalities. As applications, we estimate the greatest lower bound of Cheng’s eigenvalue and an upper bound of the minimal heat kernel, which is recently studied by Bauer et al. (Preprint, 2015) by the Li-Yau estimate. Moreover, generalizing an earlier result of Lin and Yau (Math Res Lett 17:343–356, 2010), we derive a lower bound of nonzero eigenvalues by our gradient estimate.

The universal Teichmu¨ller space is an infinitely dimensional generalization of the classical Teichmu¨ller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator ˜ Q of the universal Teichmu¨ller space with the Hilbert structure, and prove the following: (i) ˜ Q is non-positive definite. (ii) ˜ Q is a bounded operator. (iii) ˜ Q is not compact; the set of the spectra of ˜ Q is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmu¨ller space endowed with the Weil-Petersson metric.

In this paper we study constant mean curvature surfaces  in a product space, M2 × R, where M2 is a complete Riemannian manifold. We assume the angle function ν = hN, @ @t i does not change sign on . We classify these surfaces according to the infimum c() of the Gaussian curvature of the projection of . When H 6= 0 and c() ≥ 0, then  is a cylinder over a complete curve with curvature 2H. If H = 0 and c() ≥ 0, then  must be a vertical plane or  is a slice M2 ×{t}, or M2 ≡ R2 with the flat metric and  is a tilted plane (after possibly passing to a covering space). When c() < 0 and H > p −c()/2, then  is a vertical cylinder over a complete curve of M2 of constant geodesic curvature 2H. This result is optimal. We also prove a non-existence result concerning complete multigraphs in M2 × R, when c(M2) < 0.