Sharp lower bound on the curvatures of ASD connections over the cylinder

We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder.Comment: 5 page

Double variational principle for mean dimension with potential

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory.Comment: 46 pages, 3 figures. arXiv admin note: text overlap with arXiv:1901.0562

Strongly quasi-hereditary algebras and rejective subcategories

Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.Comment: 20 pages, to appear in Nagoya Math.

Magnetic field and early evolution of circumstellar disks

The magnetic field plays a central role in the formation and evolution of circumstellar disks. The magnetic field connects the rapidly rotating central region with the outer envelope and extracts angular momentum from the central region during gravitational collapse of the cloud core. This process is known as magnetic braking. Both analytical and multidimensional simulations have shown that disk formation is strongly suppressed by magnetic braking in moderately magnetized cloud cores in the ideal magnetohydrodynamic limit. On the other hand, recent observations have provided growing evidence of a relatively large disk several tens of astronomical units in size existing in some Class 0 young stellar objects. This introduces a serious discrepancy between the theoretical study and observations. Various physical mechanisms have been proposed to solve the problem of catastrophic magnetic braking, such as misalignment between the magnetic field and the rotation axis, turbulence, and non-ideal effect. In this paper, we review the mechanism of magnetic braking, its effect on disk formation and early evolution, and the mechanisms that resolve the magnetic braking problem. In particular, we emphasize the importance of non-ideal effects. The combination of magnetic diffusion and thermal evolution during gravitational collapse provides a robust formation process for the circumstellar disk at the very early phase of protostar formation. The rotation induced by the Hall effect can supply a sufficient amount of angular momentum for typical circumstellar disks around T Tauri stars. By examining the combination of the suggested mechanisms, we conclude that the circumstellar disks commonly form in the very early phase of protostar formation.Comment: 17 pages, 12 figures. accepted for publication in PASA as part of the special issue on "Disc dynamics and planet formation". Journal reference is adde

Strong deflection limit analysis and gravitational lensing of an Ellis wormhole

Observations of gravitational lenses in strong gravitational fields give us a clue to understanding dark compact objects. In this paper, we extend a method to obtain a deflection angle in a strong deflection limit provided by Bozza [Phys. Rev. D 66, 103001 (2002)] to apply to ultrastatic spacetimes. We also discuss on the order of an error term in the deflection angle. Using the improved method, we consider gravitational lensing by an Ellis wormhole, which is an ultrastatic wormhole of the Morris-Thorne class.Comment: 23 pages, 1 figure, minor correction, title changed, accepted for publication in Physical Review

A packing problem for holomorphic curves

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study the relation between theta functions and our packing problem. Applying the Nevanlinna theory, we investigate the packing densities of entire holomorphic curves in the complement of hyperplanes.Comment: 33 page

Moduli space of Brody curves, energy and mean dimension

We study the mean dimension of the moduli space of Brody curves. We introduce the notion of "mean energy" and show that this can be used to estimate the mean dimension.Comment: 24 page

Gluing an infinite number of instantons

This paper is one step toward infinite energy gauge theory and the geometry of infinite dimensional moduli spaces. We generalize a gluing construction in the usual Yang-Mills gauge theory to an ``infinite energy'' situation. We show that we can glue an infinite number of instantons, and that the resulting instantons have infinite energy in general. Moreover we show that they have an infinite dimensional parameter space. Our construction is a generalization of Donaldson's ``alternating method''.Comment: Some explanations are adde

Deformation of Brody curves and mean dimension

The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.Comment: 18 page

Mean dimension of full shifts

Let $K$ be a finite dimensional compact metric space and $K^\mathbb{Z}$ the full shift on the alphabet $K$. We prove that its mean dimension is given by $\dim K$ or $\dim K-1$ depending on the "type" of $K$. We propose a problem which seems interesting from the view point of infinite dimensional topology.Comment: 9 page