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Hiroki Fujino

Publications and source records attributed to Hiroki Fujino.

7 recordsLinked to original sources
arXiv2022

Reflection principles for zero mean curvature surfaces in the simply isotropic 3-space

Zero mean curvature surfaces in the simply isotropic 3-space $\mathbb{I}^3$ naturally appear as intermediate geometry between geometry of minimal surfaces in $\mathbb{E}^3$ and that of maximal surfaces in $\mathbb{L}^3$. In this paper, we investigate reflection principles for zero mean curvature surfaces in $\mathbb{I}^3$ as with the above surfaces in $\mathbb{E}^3$ and $\mathbb{L}^3$. In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $\mathbb{I}^3$, along which the induced metrics become singular.

math.DG↗
arXiv2022

Extension of Krust theorem and deformations of minimal surfaces

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space $\mathbb{E}^3$ is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space $\mathbb{L}^3$. In this article, we introduce a new deformation family that continuously connects minimal surfaces in $\mathbb{E}^3$ and maximal surfaces in $\mathbb{L}^3$, and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space $\mathbb{I}^3$ appear in the middle of the above deformation family. We also prove another type of Krust's theorem for this family, which implies that the graphness of such minimal surfaces in $\mathbb{I}^3$ strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.

math.DG↗
arXiv2020

Reflection principle for lightlike line segments on maximal surfaces

As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment $L$, the surface is invariant under the $180^\circ$-rotation with respect to $L$. However, such a reflection property does not hold for lightlike line segments on the boundaries of maximal surfaces in general. In this paper, we show some kind of reflection principle for lightlike line segments on the boundaries of maximal surfaces when lightlike line segments are connecting shrinking singularities. As an application, we construct various examples of periodic maximal surfaces with lightlike lines from tessellations of $\mathbb{R}^2$.

math.DG↗
arXiv2019

Duality of boundary value problems for minimal and maximal surfaces

In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between solutions of infinite boundary value problems for minimal surfaces and those of lightlike line boundary problems for maximal surfaces in the Lorentz-Minkowski spacetime. We also investigate some symmetry relations associated with the above correspondence together with their conjugations, and observe function theoretical aspects of the geometry of these surfaces. Finally, a reflection property along lightlike line segments on boundaries of maximal surfaces is discussed.

math.DG↗
arXiv2016

Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence

In this paper, we consider the $q \rightarrow 0$ limit of the deformed Virasoro algebra and that of the level 1, 2 representation of Ding-Iohara-Miki algebra. Moreover, 5D AGT correspondence at this limit is discussed. This specialization corresponds to the limit from Macdonalds functions to Hall-Littlewood functions. Using the theory of Hall-Littlewood functions, some problems are solved. For example, the simplest case of 5D AGT conjectures is proven at this limit, and we obtain a formula for the 4-point correlation function of a certain operator.

math-ph↗
arXiv2014

The existence of quasiconformal homeomorphism between planes with countable marked points

We consider quasiconformal deformations of $\mathbb{C}\setminus\mathbb{Z}$. We give some criteria for infinitely often punctured planes to be quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$. In particular, we characterize the closed subsets of $\mathbb{R}$ whose compliments are quasiconformally equivalent to $\mathbb{C}\setminus\mathbb{Z}$.

math.DG↗