In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to asmooth variation through harmonic maps). In this paper, in contrast, we show that there are (non-full) harmonic maps from the 2-sphere to the 3-sphere and 4-sphere which have non-integrable Jacobi fields. This is particularly surprising in the case of the 3-sphere where the space of harmonic maps of any degree is a smooth manifold, each map having image in a totally geodesic 2-sphere.

We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.

In this paper, we shall study large deviation principle for random upper semicontinuous functions, and obtain Cramér type theorems for those whose underlying space is a separable Banach space of type $p$.

In this paper we are concerned with Sobolev type inequalities for Riesz potentials of functions in Orlicz classes. As an application, we study continuity properties of Riesz potentials.

For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kähler classes containing Kähler metrics of constant scalar curvature but does not admit any Kähler-Einstein metric.

In this paper we continue to explore the index of elliptic units. In a previous article we determined the asymptotic behavior in $\boldsymbol{Z}_p$-extensions of the $p$-part of this index divided by the $p$-part of the ideal class number. We proved the existence of an invariant $\mu_\infty$ which governs this behavior, and gave sufficient conditions for the vanishing of $\mu_\infty$. Here we give examples with nonzero $\mu_\infty$, especially in the case of anticyclotomic $\boldsymbol{Z}_p$-extensions.

Let $B$ be a normal affine $\boldsymbol{C}$-domain and let $A$ be a $\boldsymbol{C}$-subalgebra of $B$ such that $B$ is a finite $A$-module. Let $\delta$ be a locally nilpotent derivation on $A$. Then $\delta$ lifts uniquely to the quotient field $L$ of $B$, which we denote by $\Delta$. We consider when $\Delta$ is a locally nilpotent derivation of $B$. This is a classical subject treated in [17, 19, 16]. We are interested in the case where $A$ is the $G$-invariant subring of $B$ when a finite group $G$ acts on $B$. As a related topic, we treat in the last section the finite coverings of log affine pseudo-planes in terms of the liftings of the $\boldsymbol{A}^1$-fibrations associated with locally nilpotent derivations.

In this paper, we consider a nonlinear elliptic equation on the plane away from the origin, which arises from the spherical Onsager vortex theory in physics or the problem of prescribing Gaussian curvature in geometry. Depending on various situations for the prescribed function in the nonlinear term, the complete structure of radial solutions in terms of initial data will be offered.