A harmonic map from a Riemannian manifold into a Grassmann manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem (a generalization of theorem of Takahashi) to generalize the theory of do Carmo and Wallach and to describe the moduli space of harmonic maps satisfying the gauge and the Einstein–Hermitian conditions from a compact Riemannian manifold into a Grassmannian. The geometric meaning of the compactification of the moduli space is interpreted and it is shown that the compactified moduli space is connected and convex. As applications, several rigidity results are exhibited and we also construct moduli spaces of holomorphic isometric embeddings of the complex projective line into complex quadrics of low degree. The compactification of the moduli space leads to classification theorems for equivariant harmonic maps.

In this paper we study a specific class of actions of a 2-torus ℤ₂^𝑘 on manifolds, namely, the actions of complexity one in general position. We describe the orbit space of equivariantly formal 2-torus actions of complexity one in general position and restricted complexity one actions in the case of small covers. It is observed that the orbit spaces of such actions are topological manifolds. If the action is equivariantly formal, we prove that the orbit space is a ℤ₂-homology sphere. We study a particular subclass of these 2-torus actions: restrictions of small covers to a subgroup of index 2 in general position. The subgroup of this form exists if and only if the small cover is orientable, and in this case we prove that the orbit space of a restricted 2-torus action is homeomorphic to a sphere.

It is proved that any (repetitive) Riemannian manifold of bounded geometry can be realized as a leaf of some (minimal) Riemannian matchbox manifold without holonomy. Our methods can be adapted to achieve Cantor transversals or a prescribed holonomy covering, but losing the density of our leaf.

We consider the complete system of equations governing the motion of a general compressible, viscous, electrically and heat conductive fluid driven by non-conservative boundary conditions. We show the existence of a bounded absorbing set in the energy space and asymptotic compactness of trajectories. As a corollary, the set of all entire globally bounded solutions is identified as a natural attractor. Examples of boundary conditions giving rise to unbounded solutions are also discussed.

In this paper, we introduce Wick's quantization on groups and discuss its links with Kohn–Nirenberg's. By quantization, we mean an operation that associates an operator to a symbol. The notion of symbols for both quantizations is based on representation theory via the group Fourier transform and the Plancherel theorem. As an application, we prove Gårding inequalities for three global symbolic pseudodifferential calculi on groups.

A telescopic curve is an algebraic curve defined by 𝑚 − 1 equations in the affine space of dimension 𝑚, which can be a hyperelliptic curve and an (𝑛, 𝑠) curve as a special case. If all the coefficients of the power series expansion of an entire function around the origin in the form of the Hurwitz series are contained in a ring, then the function is said to be Hurwitz integral over the ring. In this paper, we show that the sigma function associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and 1/2 over the ring of integers. Further, we show that the square of the sigma function associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over the ring of integers. Our results are a generalization of the results of Y. Ônishi for (𝑛, 𝑠) curves to telescopic curves.

In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle 𝜋 : 𝑀 → 𝑆¹ with fiber 𝑁 and structure group 𝛤 and 𝑟 ∈ ℤ _{≥ 0} ∪ { ∞ } we distinguish an integer 𝑘 = 𝑘(𝜋, 𝑟) ∈ ℤ _{≥ 0} and construct a function 𝜈^∧ : Diff^𝑟_𝜋(𝑀)₀ → ℝ_𝑘. When 𝑘 ≥ 1, it is shown that the bundle diffeomorphism group Diff^𝑟_𝜋(𝑀)₀ is bounded and 𝑐𝑙𝑏_𝜋𝑑 Diff^𝑟_𝜋(𝑀)₀ ≤ 𝑘 + 3, if Diff^𝑟_𝜚,𝑐(𝐸)₀ is perfect for the trivial fiber bundle 𝜚 : 𝐸 → ℝ with fiber 𝑁 and structure group 𝛤. On the other hand, when 𝑘 = 0, it is shown that 𝜈^∧ is a unbounded quasimorphism, so that Diff^𝑟_𝜋(𝑀)₀ is unbounded and not uniformly perfect. We also describe the integer 𝑘 in term of the attaching map 𝜑 for a mapping torus 𝜋 : 𝑀_𝜑 → 𝑆¹ and give some explicit examples of (un)bounded groups.

We determine the Hausdorff dimension of sets of irrationals in (0,1) whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second author [Proc. Amer. Math. Soc., 151 (2023), 3645–3653] and the solution of Hirst's conjecture [B.-W. Wang and J. Wu, Bull. London Math. Soc., 40 (2008), 18–22], both previously obtained for the regular continued fraction. To prove the result, we construct non-autonomous iterated function systems well-adapted to the given restrictions and growth conditions on partial quotients, estimate the associated pressure functions, and then apply Bowen's formula.

The conformal Gauss maps of time-like minimal surfaces in the Lorentz–Minkowski 3-space 𝐸³₁ give sections of the time-like twistor spaces associated with the pull-back bundles such that the covariant derivatives are fully light-like, that is, these are either light-like or zero, and do not vanish at any point. For an oriented neutral 4𝑛-manifold (\tilde{𝑀}, ℎ), if 𝐽 is an ℎ-reversing almost paracomplex structure of \tilde{𝑀} such that ∇𝐽 is locally given by the tensor product of a nowhere zero 1-form and an almost nilpotent structure related to 𝐽, then we will see that ∇𝐽 is valued in a light-like 2𝑛-dimensional distribution 𝒟 such that (\tilde{𝑀}, ℎ, 𝒟) is a Walker manifold and that the square norm ‖ ∇𝐽 ‖² of ∇𝐽 vanishes. We will obtain examples of ℎ-reversing almost paracomplex structures of 𝐸^4𝑛_2𝑛 as above. In addition, we will obtain all the pairs of ℎ-reversing almost paracomplex structures of 𝐸⁴₂ such that each pair gives sections of the two time-like twistor spaces with fully light-like covariant derivatives.