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Erhan Güler

Publications and source records attributed to Erhan Güler.

10 recordsLinked to original sources
arXiv2026

Schwarz-Type Null Curves in $\mathbb{C}^4$: Symmetry and Period Reduction

We study a genus-three hyperelliptic holomorphic null-curve family in $\mathbb{C}^4$, modelled on the algebraic data of the classical Schwarz P/D family, whose real parts define minimal immersions into $\mathbb{R}^4$. For the order-four automorphism $ω\mapsto iω$ of the underlying Schwarz curve, we compute explicitly its action on the four holomorphic Weierstrass $1$-forms and derive the resulting identities for all real period vectors. Consequently, for every lattice invariant under the induced target rotation, torus-period closure can be checked on one representative from each orbit of a symmetry-stable homology generating set. We also identify precisely when the additional parameter produces a nondegenerate codimension-two deformation. The paper does not claim the construction of a new embedded periodic minimal surface in $\mathbb{R}^4$; rather, it provides an explicit symmetry reduction of the period problem associated with this family.

math.GM↗
arXiv2026

Gauss Maps in Hyperbolic Surface Theory:A Unified Perspective

Immersed surfaces in hyperbolic three-space carry several natural Gauss-type maps with distinct geometric roles. The hyperbolic Gauss maps record the ideal endpoints of oriented normal geodesics; the Legendre Gauss lift retains the position-normal data and its contact structure; adjusted Gauss maps arise from gauge normalization and Iwasawa splitting in Weierstrass--Kenmotsu representations; and the conformal Gauss map encodes the mean-curvature sphere congruence in Möbius geometry. We present these constructions in a common framework, emphasizing their target spaces, analytic properties, and mutual relations. Particular attention is given to the generalized DPW method, the necessity of flatness in adjusted rank-one data, and the harmonic-map characterization of Willmore surfaces. The resulting viewpoint distinguishes the asymptotic, contact, integrable, and conformal information carried by an immersed surface in \(\mathbb{H}^{3}(-1)\).

math.DG↗
arXiv2026

A Weierstrass-Kenmotsu Type Representation for CMC $0\le H<1$ in \$\mathbb{H}^3(-1)$

We develop a Weierstrass-Kenmotsu type representation for conformal immersions of constant mean curvature $0\le H<1$ in hyperbolic $3$-space $\HH$. The construction is based on the Hermitian model of $\HH$, a balanced spectral deformation, and Iwasawa splitting of $\SL$. We show that such immersions arise locally from a rank-one $(1,0)$-form $η$ and a constant complex parameter $λ\in\C^*$ through a flat $\SL$-connection of the form \[ S^{-1}dS=η-λ\,η^*, \] with mean curvature \[ H=\frac{1-|λ|^2}{1+|λ|^2}. \] Conversely, every conformal CMC immersion with $0\le H<1$ is locally obtained from such flat rank-one data. We establish an explicit correspondence with the representation of Aiyama and Akutagawa via a gauge transformation, and interpret the construction in terms of Kokubu's adjusted normal Gauss map. We further discuss the role of the flatness condition, present simple local and cylindrical model examples, and outline aspects of monodromy and numerical implementation within this framework.

math.DG↗
arXiv2026

Explicit Minimal Surface Models in $\mathbb{R}^5$ via Holomorphic Null Curves

We study explicit conformal minimal immersions into $\mathbb{R}^5$ obtained from holomorphic null curves in $\mathbb{C}^5$. Although the general correspondence between conformal minimal immersions in $\mathbb{R}^n$ and holomorphic null data in $\mathbb{C}^n$ is classical, our aim here is different. We isolate the five-dimensional case and develop a concrete, self-contained account that emphasizes explicit formulas, integral-free constructions, and coordinate expressions suitable for computation and visualization. Starting from a Weierstrass-type representation in $\mathbb{R}^5$, we derive a family of conformal minimal immersions depending on a single holomorphic seed function and two real parameters. The resulting formulas allow the immersion and the induced metric to be written in closed form. We then examine polynomial seeds in detail, derive their polar and Cartesian expansions, and discuss the geometric information carried by natural coordinate projections. We reinterpret the construction in the language of moving frames, the generalized Gauss map, and a local DPW-type scheme. This provides a conceptual bridge between explicit holomorphic formulas and the Cartan-integrable-systems viewpoint. The discussion is local and formula-driven; global questions such as periods, completeness, and embeddedness lie beyond the present scope. We also briefly clarify why the complex-analytic structure underlying the representation is essential, and why it cannot be replaced by a naive quaternionic formalism, due to the loss of commutativity, holomorphic structure, and compatibility with the null-curve framework.

math.DG↗
arXiv2025

The higher-order Henneberg-type minimal surfaces family in $\mathbb{R}^4$

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its differential geometric characteristics, including the normal vector fields $\mathbf{n}_1$ and $\mathbf{n}_2$, as well as the Gauss curvature. Furthermore, by projecting these parametric forms from four to three dimensions, we generate visualizations that reveal the geometric structure of the Henneberg-type minimal surface. In addition, we examine the integral-free form and derive the corresponding algebraic function for this family of surfaces.

math.DG↗
arXiv2024

Rotational hypersurfaces family satisfying $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$ in ${\mathbb{E}}^{n}$

In this paper, we investigate rotational hypersurfaces family in $n$ -dimensional Euclidean space ${\mathbb{E}}^{n}$. Our focus is on studying the Gauss map $\mathcal{G}$ of this family with respect to the operator $\mathbb{L}_{k}$, which acts on functions defined on the hypersurfaces. The operator $\mathbb{L}_{k}$ can be viewed as a modified Laplacian and is known by various names, including the Cheng--Yau operator in certain cases. Specifically, we focus on the scenario where $k=n-3$ and $n\geq 3$. By applying the operator $\mathbb{L}_{n-3}$ to the Gauss map $\mathcal{G}$, we establish a classification theorem. This theorem establishes a connection between the $n\times n$ matrix $\mathcal{A}$, and the Gauss map $\mathcal{G}$ through the equation $\mathbb{L}_{n-3}\mathcal{G}=\mathcal{A}\mathcal{G}$.

math.DG↗
arXiv2020

Fourth Fundamental Form and $i$-th Curvature Formulas in ${\mathbb{E}}^{4}$

We introduce fourth fundamental form $IV,$ and $i$-th curvature formulas of hypersurfaces in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}$. Defining fourth fundamental form and $i$-th curvatures for hypersurfaces, we calculate them on rotational hypersurface. In addition we study rotational hypersurface satisfying $Δ^{IV}\mathbf{x=Ax}$ for some $4\times 4$ matrix $\mathbf{A.}$

math.DG↗
arXiv2016

Isometric Surfaces and the Third Laplace-Beltrami

In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are obtained.

math.DG↗
arXiv2016

Generalized Bour's theorem in Minkowski 3-space

We obtain isometric minimal helicoidal and rotational surfaces using generalized Bour's theorem in three dimensional Minkowski space. In addition, we show that the surfaces preserve minimality when their Gauss maps identically equal, choosing any diffentiable functions on the profile curve.

math.DG↗