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Shimpei Kobayashi

Publications and source records attributed to Shimpei Kobayashi.

At least 19 recordsLinked to original sources
arXiv2026

Minimal Lagrangian surfaces in the two-dimensional complex hyperbolic quadric via the loop group method

We study minimal Lagrangian surfaces in the complex hyperbolic quadric. We show that minimality of a Lagrangian surface is characterized by a loop of flat connections, which yields an associated $\mathbb S^1$-family of isometric deformations. We also establish a correspondence with spacelike maximal surfaces in anti-de Sitter $3$-space via the Gauss map. Using the resulting harmonic map into the hyperbolic two-space, we develop a DPW-type representation and construct explicit examples, including $\mathbb{R}$-equivariant and radially symmetric surfaces. In particular, under suitable conditions, the $\mathbb{R}$-equivariant family contains catenoid-type examples.

math.DG↗
arXiv2026

A duality for minimal surfaces in the Heisenberg group

We introduce and study the notion of a transformation surface associated with a nowhere-vertical minimal surface in the three-dimensional Heisenberg group, and prove its minimality and duality. Furthermore, by using the logarithmic derivative of the moving frame with respect to the spectral parameter, we derive the Sym formula for the dual minimal surface.

math.DG↗
arXiv2026

Two classes of Willmore Surfaces in $\mathbb{S}^2\times \mathbb{S}^2$

We establish two classification theorems for Willmore surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$. Firstly, we prove that a Willmore surface which is also minimal must be either a special complex curve given by a slice or a diagonal; or, a minimal surface in a totally geodesic submanifold $\mathbb{S}^2 \times \mathbb{S}^1$ described by a solution of the sinh-Gordon equation in one variable. Secondly, we demonstrate that a Willmore surface is of product type if and only if it is the product of an elastic curve in $\mathbb{S}^2$ and a great circle.

math.DG↗
arXiv2026

Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method

We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric $Q_{2}\cong \mathbb S^{2}\times \mathbb S^{2}$, formulated via a flat family of connections $\{\nabla^λ\}_{λ\in \mathbb S^{1}}$ on a trivial bundle. We prove that minimality is equivalent to the flatness of $\nabla^λ$ for all $λ$, describe the associated isometric $\mathbb S^{1}$-family, and establish a precise correspondence with minimal surfaces in $\mathbb S^{3}$ through their Gauss maps. Our framework unifies and streamlines earlier constructions (e.g., Castro--Urbano) and yields explicit families including $\mathbb R$-equivariant, radially symmetric, and trinoid-type examples.

math.DG↗
arXiv2025

A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space

We classify weakly complete constant Gaussian curvature $-1 -1$ and $K \neq 0$ via the harmonicity of the Lagrangian and Legendrian Gauss maps. We then show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into the hyperbolic two-space for $-1< K<0$ or the two-sphere for $K>0$, respectively. Consequently, weakly complete constant Gaussian curvature surfaces with $-1 < K <0$ are in one-to-one correspondence with holomorphic quadratic differentials on the unit disk or the complex plane.

math.DG↗
arXiv2025

The evolution of a curve induced by the Pohlmeyer-Lund-Regge equation

This paper investigates the evolution of space curves governed by the Pohlmeyer-Lund-Regge (PLR) equation, an integrable extension of the sine-Gordon equation. We examine a specific type of curve evolution, known as the Lund-Regge evolution, and derive its representation in the Frenet frame. We show the Frenet frame evolution aligns with the Lax system of the PLR equation and develop a construction method for curve families via the Sym formula. In conclusion, we describe the Lund-Regge evolution corresponding the Date multi-soliton solutions to the PLR equation, with illustrations of curves and surfaces.

math.DG↗
arXiv2025

The complex landslide flow and the method of integrable systems

We investigate a connection between the complex landslide flow, defined on a pair of Teichmüller spaces, and the integrable system approach to harmonic maps into a symmetric space. We will prove that the holonomy of the complex landslide flow can be derived from the holonomy of the family of flat connections determined by a harmonic map into the hyperbolic two-space.

math.DG↗
arXiv2024

Gauss maps of Möbius surfaces in the $n$-dimensional sphere

In this note we discuss Gauss maps for Möbius surfaces in the $n$-sphere, and their applications in the study of Willmore surfaces. One such ``Gauss map'', naturally associated to a Willmore surface that has a dual Willmore surface, is the Lorentzian $2$-plane bundle given by a lift of the suface and its dual. More generally, we define the concept of a Lorentzian $2$-plane lift for an arbitrary Möbius surface, and show that the conformal harmonicity of this lift is equivalent to the Willmore condition for the surface. This clarifies some previous work of F. Hélein, Q. Xia-Y Shen, X. Ma and others, and, for instance, allows for the treatment of the Björling problem for Willmore surfaces in the presence of umbilics.

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arXiv2024

Half-dimensional immersions into the para-complex projective space and Ruh-Vilms type theorems

In this paper we study isometric immersions $f:M^n \to {\mathbb {C}^{\prime}}\!P^n$ of an $n$-dimensional pseudo-Riemannian manifold $M^n$ into the $n$-dimensional para-complex projective space ${\mathbb {C}^{\prime}}\!P^n$. We study the immersion $f$ by means of a lift $\mathfrak f$ of $f$ into a quadric hypersurface in ${S^{2n+1}_{n+1}}$. We find the frame equations and compatibility conditions. We specialize these results to dimension $n = 2$ and a definite metric on $M^2$ in isothermal coordinates and consider the special cases of Lagrangian surface immersions and minimal surface immersions. We characterize surface immersions with special properties in terms of primitive harmonicity of the Gauss maps.

math.DG↗
arXiv2024

A characterization of the alpha-connections on the statistical manifold of multivariate normal distributions

We study a statistical manifold $(\mathcal{N}, g^F, \nabla^{A}, \nabla^{A*})$ of multivariate normal distributions, where $g^F$ is the Fisher metric and $\nabla^{A}$ is the Amari-Chentsov connection and $\nabla^{A*}$ is its conjugate connection. We will show that it admits a solvable Lie group structure and moreover the Amari-Chentsov connection $\nabla^{A}$ on $(\mathcal{N}, g^F)$ will be characterized by the conjugate symmetry, i.e., a curvatures identity $R=R^*$ of a connection $\nabla$ and its conjugate connection $\nabla^*$.

math.DG↗
arXiv2023

Geodesics of multivariate normal distributions and a Toda lattice type Lax pair

We study geodesics of multivariate normal distributions with respect to the Fisher metric. First it will be shown that a computational formula for geodesics can be understood using the block Cholesky decomposition and a natural Riemannian submersion. Next a mid point algorithm for geodesics will be obtained. And finally a new Toda lattice type Lax pair will be derived from the geodesic and the block Cholesky decomposition.

math.DG↗
arXiv2023

Maximal surfaces in the Lorentzian Heisenberg group

The 3-dimensional Heisenberg group can be equipped with three different types of left-invariant Lorentzian metric, according to whether the center of the Lie algebra is spacelike, timelike or null. Using the second of these types, we study spacelike surfaces of mean curvature zero. These surfaces with singularities are associated with harmonic maps into the 2-sphere. We show that the generic singularities are cuspidal edge, swallowtail and cuspidal cross-cap. We also give the loop group construction for these surfaces, and the criteria on the loop group potentials for the different generic singularities. Lastly, we solve the Cauchy problem for harmonic maps into the 2-sphere using loop groups, and use this to give a geometric characterization of the singularities. We use these results to prove that a regular spacelike maximal disc with null oundary must have at least two cuspidal cross-cap singularities on the boundary.

math.DG↗
arXiv2022

Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group

We study symmetric minimal surfaces in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal surfaces in $\mathrm{Nil}_3$ with non-trivial topology. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group $\mathrm{Iso}_{\circ}(\mathrm{Nil}_3)$ of $\mathrm{Nil}_3$.

math.DG↗
arXiv2022

Minimal cylinders in the three-dimensional Heisenberg group

We study minimal cylinders in the three-dimensional Heisenberg group ${\rm Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders can also be obtained.

math.DG↗
arXiv2022

Timelike minimal surfaces in the three-dimensional Heisenberg group

Timelike surfaces in the three-dimensional Heisenberg group with left invariant semi-Riemannian metric are studied. In particular, non-vertical timelike minimal surfaces are characterized by the non-conformal Lorentz harmonic maps into the de Sitter two phere. On the basis of the characterization, the generalized Weierstrass type representation will be established through the loop group decompositions.

math.DG↗
arXiv2022

On a constant curvature statistical manifold

We will show that a statistical manifold $(M, g, \nabla)$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $\nabla$ is projectively flat and the curvatures satisfies $R=R^*$, where $R^*$ is the curvature of the dual connection $\nabla^*$. Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $-1$ statistical structures and vice versa.

math.DG↗
arXiv2020

The Gauss maps of Demoulin surfaces with conformal coordinates

Demoulin surfaces in real projective $3$-space are investigated. Our result enable us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian $6$-symmetric space.

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