In this paper, we study the irregular multifractal spectrum of the Gauss map, by investigating the upper and lower Birkhoff spectra. For potentials satisfying a weakly tempered distortion property, we establish that the Hausdorff dimensions of the classical, upper, and lower Birkhoff level sets are identical. Furthermore, we derive an exact formula for the dimension of the intersection of the upper and lower Birkhoff level sets, proving that this dimension equals the minimum of the two individual dimensions. These findings broaden the multifractal theory previously developed for conformal repellers and finite branch expanding systems to the infinite branch context of the Gauss map. As applications, we furnish complete multifractal descriptions of the upper and lower Lyapunov spectra of the Gauss map, as well as the upper and lower arithmetic-mean spectra of continued fractions.

We present a representation formula for translating soliton surfaces to the mean curvature flow in Euclidean space ${\mathbb {R}}^{4}$ and give examples of conformal parameterisations for translating soliton surfaces.

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

The classical result of Nevanlinna states that two nonconstantmeromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Here we study the dimension δ(m, X) of the general fibers of the m-Gaussian map of a singular n-dimensional variety X ⊂ Pn. We show that for all integers a, b, c, d with n ≦ a < b ≦ c < d ≦ N − 1 and a + d = b + c we have δ (a, X) + δ(d, X) > δ(b, X) + δ(c, X). If δ(X, N − 1) is very large we give some classification results which extend to the singular case some results of Ein.

We study the product of a Fermat hypersurface $X_0^{p+1}+\dots+X_n^{p+1}=0 \subset \mathbf{P}^n$ with $n \ge 3$ and $\mathbf{P}^1$, embedded in $\mathbf{P}^{2n+1}$ by Segre embedding where $p>0$ is the characteristic of the base field. This smooth variety is nonreflexive and has Gauss map which is an embedding. This gives a negative answer to the following Kleiman–Piene question in any positive characteristic: does the separability of the Gauss map imply reflexivity? The only known smooth examples, which give a negative answer, are given by Kaji in characteristic 2.

For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.