Right Conoids Demonstrating a Time-like Axis within Minkowski Four-Dimensional Space

Abstract

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored.

1. Introduction

The parametric equation

$$\begin{array}{ccc}\hfill \mathfrak{r}(u,v)& =& \alpha \left(v\right)+u\beta \left(v\right)\hfill \\ & =& \left(0,0,g\left(v\right)\right)+u\left(cos\left(f\left(v\right)\right),sin\left(f\left(v\right)\right),0\right)\hfill \end{array}$$

determines a ruled surface in three-dimensional Minkowski space ${\mathbb{L}}^3$. This specific surface, distinguished by its unique geometric properties, is accurately identified as a right conoid. It features a time-like axis defined by the vector $(0,0,1)$ and adheres to the metric signatures $(+,+,-)$ inherent to Minkowski space.

To unravel the intricacies inherent in this parametric representation, let us break down its components. The curve $\alpha \left(v\right)$ acts as a foundational axis along the z-direction. Concurrently, the vector function $\beta \left(v\right)$ defines the generating vector shaping the ruled surface. The parameter u moves dynamically along this generating vector, while v parametrizes the curve $\alpha \left(v\right)$, influencing the overall structure of the ruled surface.

The essential geometric features of the right conoid are encapsulated by this parametric formulation, which also serves as a mathematical framework for understanding its behavior within ${\mathbb{L}}^3$. A foundation is laid by the intricate relationship among the reference curve, generating vector, and parametric variables, allowing for the exploration of the geometric intricacies and properties of this ruled surface within the context of Minkowski space.

This classification gains significance as the surface is generated by moving a straight line intersecting a predetermined linear reference, ensuring that perpendicular alignment between these lines is maintained throughout the generation process. By defining the $xy$-plane as the orthogonal plane and the z-axis as the baseline, we derive the parametric equation that defines the right conoid.

Moreover, identifying this ruled surface as a right conoid with a time-like axis in ${\mathbb{L}}^3$ emphasizes its geometric finesse and structural integrity. This classification not only emphasizes its importance within the context of Minkowski space but also enables a deeper understanding of its curvature characteristics and spatial configuration.

This study aims to thoroughly investigate the core attributes of hypersurfaces categorized as right conoids with a time-like axis (RCH-T) within the expansive realm of four-dimensional Minkowski space ${\mathbb{L}}^4$. Our primary objective is to compute differential geometric objects crucial to these hypersurfaces. Leveraging the robust framework provided by the Cayley–Hamilton theorem, our aim is to scrutinize and quantify the intrinsic curvatures inherent in these particular hypersurfaces.

In addition to exploring curvature properties, a core objective of our research is to establish criteria that govern minimization within this particular geometric framework. This entails a thorough examination of factors influencing the reduction in specific geometric attributes, thereby enriching our understanding of the dynamics of RCH-T in four-dimensional Minkowski space.

An essential focus of our investigation is the intricate relationship between RCH-T and the Laplace–Beltrami ($\mathcal{L}$–$\mathcal{B}$) operator in the expansive domain of ${\mathbb{L}}^4$. By exploring this connection, our aim is to gain insights into the intrinsic geometric properties of these hypersurfaces, thereby deepening our understanding of their behavior within the Minkowski space framework.

Furthermore, our analysis extends to include the elucidation of the umbilical condition, which reveals specific geometric characteristics that RCH-T may manifest. To provide a thorough overview of these geometric properties, we introduce the Willmore functional for RCH-T, enabling a quantitative assessment of their shape attributes.

This multifaceted approach not only advances our theoretical understanding but also lays the foundation for practical applications and further advancements in the study of hypersurfaces within Minkowski space.

In Section 2, we delve into the foundational principles and concepts of four-dimensional Minkowski geometry. Section 3 focuses on detailing curvature formulas applicable to hypersurfaces in ${\mathbb{L}}^4$. Moving forward to Section 4, we provide a comprehensive definition of right conoid hypersurfaces with a time-like axis, highlighting their distinct properties and characteristics. Section 5 shifts the discussion to the $\mathcal{L}$–$\mathcal{B}$ operator for a smooth function in ${\mathbb{L}}^4$, utilizing the hypersurfaces analyzed previously. Section 6 addresses umbilical right conoid hypersurfaces with a time-like axis in ${\mathbb{L}}^4$. The Willmore functional of these hypersurfaces is introduced in Section 7. Finally, the present study concludes in the last section.

2. Preliminaries

In Minkowski $(n+1)$-space ${\mathbb{L}}^{n+1}$, the symbol $s_j$ is expressed as ${\sigma }_j$, which denotes the j-th elementary symmetric function defined by

$${\sigma }_j(a_1,a_2,\cdots ,a_n)=\sum _{1\le i_1<i_2<\cdots <i_j\le n}{a}_{i_1}{a}_{i_2}\cdots a_{i_j},$$

and we use the notation

$$r_i^j={\sigma }_j(k_1,k_2,\cdots ,k_{i-1},k_{i+1},k_{i+2},\cdots ,k_n).$$

An oriented hypersurface M is considered within the intricate domain of Minkowski four-dimensional space ${\mathbb{L}}^4$. The shape operator of this hypersurface, denoted by $\mathcal{S}={\left({\mathfrak{s}}_{ij}\right)}_{3\times 3}$, is employed in the exploration of various geometric measures.

Let $r_i^j={\sigma }_j(k_1,k_2,k_3)$ with the defined condition, leading to $r_i^0=1$. The function $s_k$ is recognized as the k-th average curvature of the oriented surface M. Specifically, the average curvature H is given by $H={\displaystyle \frac{1}{3}}{s}_1$. Moreover, the Gauss–Kronecker ($\mathcal{G}$–$\mathcal{K}$) curvature of M is denoted as $K=s_3$.

These curvature terms play crucial roles in differential geometry, particularly in understanding the geometric properties of hypersurfaces. The mean curvature H represents the average of the principal curvatures and is fundamental in various geometric and physical contexts. For M, H is directly related to the first mean curvature $s_1$.

The Gauss–Kronecker curvature K, also known as the Gaussian curvature in dimension three, characterizes the intrinsic curvature of M independent of its embedding in higher-dimensional space. It is given by the third mean curvature $s_3$.

Together, these curvatures provide comprehensive insights into the geometry of M, facilitating analyses ranging from surface classification to applications in fields such as physics and materials science.

The characteristic polynomial, denoted as ${\displaystyle \sum _{k=0}^3}{\left(-1\right)}^k{s}_k{\lambda }^{3-k}=0$, of $\mathcal{S}$ is indicated by the equation

$$det(\mathcal{S}-\lambda {\mathcal{I}}_3)=0.$$

(1)

Here, ${\mathcal{I}}_3$ denotes the identity matrix. The curvature relations are revealed as $\left({\displaystyle \genfrac{}{}{0pt}{}{3}{i}}\right){\mathbb{K}}_i=s_i$, highlighting the intrinsic geometric properties of the hypersurface.

In this investigation, a vector is deemed identical to its transpose. We analyze the embedding of $\mathfrak{x}=\mathfrak{x}(u,v,w)$ from $M^3\subset {\mathbb{E}}^3$ into ${\mathbb{L}}^4$.

Definition 1.

An inner product for two vectors ${\mathbf{t}}^1=(t_1^1,t_2^1,t_3^1,t_4^1),$ ${\mathbf{t}}^2=(t_1^2,t_2^2,t_3^2,t_4^2)$ of ${\mathbb{L}}^4$ is achieved through the expression

$$〈{\mathbf{t}}^1,{\mathbf{t}}^2〉=t_1^1{t}_1^2+t_2^1{t}_2^2+t_3^1{t}_3^2-t_4^1{t}_4^2.$$

Definition 2.

A triple vector product for three vectors ${\mathbf{t}}^1=(t_1^1,t_2^1,t_3^1,t_4^1),$ ${\mathbf{t}}^2=(t_1^2,t_2^2,t_3^2,t_4^2),$ ${\mathbf{t}}^3=(t_1^3,t_2^3,t_3^3,t_4^3)$ in ${\mathbb{L}}^4$ is given by the determinant

$${\mathbf{t}}^1\times {\mathbf{t}}^2\times {\mathbf{t}}^3=\left|\begin{array}{cccc}{\mathfrak{e}}_1& {\mathfrak{e}}_2& {\mathfrak{e}}_3& -{\mathfrak{e}}_4\\ t_1^1& t_2^1& t_3^1& t_4^1\\ t_1^2& t_2^2& t_3^2& t_4^2\\ t_1^3& t_2^3& t_3^3& t_4^3\end{array}\right|.$$

Here, the base components in ${\mathbb{L}}^4$ are determined by ${\mathfrak{e}}_k$.

Definition 3.

The shape operator matrix $\mathcal{S}$ of the hypersurface $\mathfrak{x}$ in Minkowski four-dimensional space ${\mathbb{L}}^4$ is indicated by ${\left({\mathfrak{g}}_{ij}\right)}^{-1}$·$\left({\mathfrak{h}}_{ij}\right)$, where ${\left({\mathfrak{g}}_{ij}\right)}_{3\times 3}$ and ${\left({\mathfrak{h}}_{ij}\right)}_{3\times 3}$ represent the first and the second fundamental form matrices, respectively. The components of the matrices are defined as ${\mathfrak{g}}_{ij}=〈{\mathfrak{x}}_i,{\mathfrak{x}}_j〉,$ ${\mathfrak{h}}_{ij}=〈{\mathfrak{x}}_{ij},\mathbb{G}〉,$ for $i,j=1,2,3$, and the Gauss map of $\mathfrak{x}$ is obtained through the expression

$$\mathbb{G}={\displaystyle \frac{{\mathfrak{x}}_u\times {\mathfrak{x}}_v\times {\mathfrak{x}}_w}{∥{\mathfrak{x}}_u\times {\mathfrak{x}}_v\times {\mathfrak{x}}_w∥}}.$$

(2)

Refer to Chen et al. [1], Li, Güler, and Toda [2], and O’Neill [3] for detailed explanations.

3. Curvatures in ${\mathbb{L}}^{\mathbf{4}}$

In this section, we explore the curvature expressions associated with a hypersurface characterized by $\varphi =\varphi (u,v,w)$ within the context of ${\mathbb{L}}^4$. We delve into the geometric properties of the hypersurface, considering its embedding into the space ${\mathbb{L}}^4$.

Proposition 1.

The following curvature formulas are associated with a hypersurface ϕ in ${\mathbb{L}}^4$: ${\mathbb{K}}_0=1$ by definition,

$$3{\mathbb{K}}_1=-{\displaystyle \frac{{\mathfrak{d}}_2}{{\mathfrak{d}}_3}},3{\mathbb{K}}_2={\displaystyle \frac{{\mathfrak{d}}_1}{{\mathfrak{d}}_3}},{\mathbb{K}}_3=-{\displaystyle \frac{{\mathfrak{d}}_0}{{\mathfrak{d}}_3}},$$

(3)

where ${\mathfrak{d}}_3{\lambda }^3+{\mathfrak{d}}_2{\lambda }^2+{\mathfrak{d}}_1\lambda +{\mathfrak{d}}_0=0$ describes the characteristic polynomial Eq. $P_{\mathcal{S}}\left(\lambda \right)=0$ of the shape operator matrix $\mathcal{S}$, ${\mathfrak{d}}_3=det\left({\mathfrak{g}}_{ij}\right)$, ${\mathfrak{d}}_0=det\left({\mathfrak{h}}_{ij}\right)$, and $\left({\mathfrak{g}}_{ij}\right)$,$\left({\mathfrak{h}}_{ij}\right)$ denote the first and the second fundamental form matrices, respectively.

Proof. 

The curvatures are indicated by the characteristic polynomial equation of $\mathcal{S}$ in ${\mathbb{L}}^4$:

$$\begin{array}{ccc}\hfill {\mathbb{K}}_0& =& 1,\hfill \\ \hfill 3{\mathbb{K}}_1& =& k_1+k_2+k_3=-{\displaystyle \frac{{\mathfrak{d}}_2}{{\mathfrak{d}}_3}},\hfill \\ \hfill 3{\mathbb{K}}_2& =& k_1{k}_2+k_1{k}_3+k_2{k}_3={\displaystyle \frac{{\mathfrak{d}}_1}{{\mathfrak{d}}_3}},\hfill \\ \hfill {\mathbb{K}}_3& =& k_1{k}_2{k}_3=-{\displaystyle \frac{{\mathfrak{d}}_0}{{\mathfrak{d}}_3}}.\hfill \end{array}$$

□

Definition 4.

A hypersurface ϕ in ${\mathbb{L}}^4$ is termed j-minimal when ${\mathbb{K}}_{j=1,2,3}=0$.

Proposition 2.

The following relation between the curvatures and the fundamental forms holds for a hypersurface $\varphi =\varphi (u,v,w)$ in ${\mathbb{L}}^4$:

$${\mathbb{K}}_0\left({\mathfrak{f}}_{ij}\right)-3{\mathbb{K}}_1\left({\mathfrak{t}}_{ij}\right)+3{\mathbb{K}}_2\left({\mathfrak{h}}_{ij}\right)-{\mathbb{K}}_3\left({\mathfrak{g}}_{ij}\right)={\mathcal{O}}_3,$$

where $\left({\mathfrak{g}}_{ij}\right),\left({\mathfrak{h}}_{ij}\right),\left({\mathfrak{t}}_{ij}\right),\left({\mathfrak{f}}_{ij}\right)$ give the first, second, third, and fourth fundamental form matrices, and ${\mathcal{O}}_3$ determines the zero matrix.

4. RCH-T in ${\mathbb{L}}^{\mathbf{4}}$

In this section, the right conoid hypersurface with a time-like axis is described, and its geometric qualities are examined in Minkowski four-dimensional space ${\mathbb{L}}^4$.

In ${\mathbb{L}}^4$, attention is given to a ruled hypersurface determined by

$$\begin{array}{ccc}\hfill \varphi (u,v,w)& =& \alpha \left(v,w\right)+u\beta \left(v,w\right)\hfill \\ & =& \left(0,0,0,\Psi \right)+u\left(cos\wp cos\rho ,sin\wp cos\rho ,sin\rho ,0\right).\hfill \end{array}$$

Here, $\alpha ,\beta$ denote the surfaces, $u\in \mathbb{R}-\left\{0\right\}$, $〈\beta ,\beta 〉=1$, $\wp =\wp \left(v\right),$ $\rho =\rho \left(w\right),$ $\Psi =\Psi \left(v,w\right)$ describe differentiable functions, and $0\le \wp ,\rho <2\pi .$

The following definition supplies the hypersurface $\varphi (u,v,w)=\mathbf{R}·{\eta }^T$, where the profile hypersurface $\eta =\left(u,0,0,\Psi \right)$ rotates around the time-like axis $\ell =\left(0,0,0,1\right)$ through

$$\mathbf{R}(v,w)=\left(\begin{array}{cccc}cos\wp cos\rho & -sin\wp & -cos\wp sin\rho & 0\\ sin\wp cos\rho & cos\wp & -sin\wp sin\rho & 0\\ sin\rho & 0& cos\rho & 0\\ 0& 0& 0& 1\end{array}\right),$$

with $\mathbf{R}\in SO\left(4\right),$ $\mathbf{R}·{\ell }^T={\ell }^T,$ ${\mathbf{R}}^T·\epsilon ·\mathbf{R}=\mathbf{R}$·$\epsilon$·${\mathbf{R}}^T=\epsilon ,$ $\epsilon =$ diag $(1,1,1,-1).$

Definition 5.

A right conoid hypersurface with a time-like axis is immersed by ϕ: $M^3$ $\subset {\mathbb{E}}^3⟶$ ${\mathbb{L}}^4$, parametrized by

$$\varphi (u,v,w)=\left(\begin{array}{c}{\varphi }_1(u,v,w)\\ {\varphi }_2(u,v,w)\\ {\varphi }_3(u,v,w)\\ {\varphi }_4(u,v,w)\end{array}\right)=\left(\begin{array}{c}ucos\wp cos\rho \\ usin\wp cos\rho \\ usin\rho \\ \Psi \end{array}\right).$$

(4)

Here, $u\in \mathbb{R}-\left\{0\right\}$, $\wp =\wp \left(v\right)$, $\rho =\rho \left(w\right)$, $0\le \wp ,\rho <2\pi ,$ and $\Psi =\Psi \left(v,w\right)$.

For brevity, the following notations are chosen: $c=cos\wp ,$ $s=sin\wp ,$ $C=cos\rho ,$ and $S=sin\rho .$

The computation of $\left({\mathfrak{g}}_{ij}\right)$ is determined by the first derivatives of the RCH-T, as given in Equation (4) with respect to $u,v,$ and $w,$ resulting in the following matrix:

$$\left({\mathfrak{g}}_{ij}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& u^2{C}^2{\wp }_v^2-{\Psi }_v^2& -{\Psi }_v{\Psi }_w\\ 0& -{\Psi }_v{\Psi }_w& u^2{\rho }_w^2-{\Psi }_w^2\end{array}\right),$$

(5)

and ${\wp }_v={\displaystyle \frac{\partial \wp }{\partial v}},$ ${\wp }_v^2={\displaystyle \frac{{\partial }^2\wp }{\partial v^2}},$ etc. Therefore, $det\left({\mathfrak{g}}_{ij}\right)=u^2\mathbb{W},$ where $\mathbb{W}={\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)C^2-{\rho }_w^2{\Psi }_v^2.$ The characterization of the RCH-T described by Equation (4) as a space-like (resp., time-like, light-like) hypersurface is contingent upon whether $\mathbb{W}>0$ (resp., $\mathbb{W}<0,$ $\mathbb{W}=0$).

Applying the Gauss map formula (Formula ), the Gauss map of the RCH-T, as determined by Equation (4), is derived as follows:

$$\mathbb{G}={\displaystyle \frac{1}{{\left|\mathbb{W}\right|}^{1/2}}}\left(\begin{array}{c}-s{\rho }_w{\Psi }_v-cSC{\wp }_v{\Psi }_w\\ c{\rho }_w{\Psi }_v-sSC{\wp }_v{\Psi }_w\\ C^2{\wp }_v{\Psi }_w\\ Cu{\wp }_v{\rho }_w\end{array}\right).$$

(6)

By taking the second derivatives with respect to $u,v,$ and w of the RCH-T defined by Equation (4) and employing the Gauss map as given by Equation (6), the subsequent matrix is acquired:

$$\left({\mathfrak{h}}_{ij}\right)=\left(\begin{array}{ccc}0& {\displaystyle \frac{C{\wp }_v{\rho }_w{\Psi }_v}{{\left|\mathbb{W}\right|}^{1/2}}}& {\displaystyle \frac{C{\wp }_v{\rho }_w{\Psi }_w}{{\left|\mathbb{W}\right|}^{1/2}}}\\ {\displaystyle \frac{C{\wp }_v{\rho }_w{\Psi }_v}{{\left|\mathbb{W}\right|}^{1/2}}}& {\displaystyle \frac{Cu\left(CS{\wp }_v^3{\Psi }_w+{\rho }_w\left({\wp }_{vv}{\Psi }_v-{\wp }_v{\Psi }_{vv}\right)\right)}{{\left|\mathbb{W}\right|}^{1/2}}}& -{\displaystyle \frac{u{\wp }_v{\rho }_w\left(C{\Psi }_{vw}+{\rho }_{w}S{\Psi }_v\right)}{{\left|\mathbb{W}\right|}^{1/2}}}\\ {\displaystyle \frac{C{\wp }_v{\rho }_w{\Psi }_w}{{\left|\mathbb{W}\right|}^{1/2}}}& -{\displaystyle \frac{u{\wp }_v{\rho }_w\left({\rho }_{w}S{\Psi }_v+C{\Psi }_{vw}\right)}{{\left|\mathbb{W}\right|}^{1/2}}}& {\displaystyle \frac{Cu{\wp }_v\left({\rho }_{ww}{\Psi }_w-{\rho }_w{\Psi }_{ww}\right)}{{\left|\mathbb{W}\right|}^{1/2}}}\end{array}\right),$$

(7)

where ${\wp }_{uu}={\displaystyle \frac{{\partial }^2\wp }{\partial u^2}},$ ${\wp }_{uv}={\displaystyle \frac{{\partial }^2\wp }{\partial u\partial v}},$ etc. By employing (5) and (7), the computation of the shape operator matrix $\mathcal{S}={\left({\mathfrak{s}}_{ij}\right)}_{3\times 3}$ for (4) is carried out. Therefore, utilizing (3) in conjunction with (5) and (7), the curvatures of the RCH-T are determined.

Theorem 1.

Let ϕ be an RCH-T defined by the equation in (4) within ${\mathbb{L}}^4$. The curvatures associated with ϕ are outlined, with ${\mathbb{K}}_0=1$ by default, as follows:

$$\begin{array}{ccc}\hfill {\mathbb{K}}_1& =& {\displaystyle \frac{1}{3u{\left(C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2\right)}^{3/2}}}\left[C{\wp }_v{\rho }_w\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}+2C{\wp }_v{\rho }_w{\Psi }_v{\Psi }_w{\Psi }_{vw}\right.\hfill \\ & & +C{\wp }_v{\rho }_w\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+{\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2{\Psi }_w+C\left({\wp }_{vv}{\rho }_w{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w^2\hfill \\ & & \left.-C{u}^2{\wp }_{vv}{\rho }_w^3{\Psi }_v-C^2{u}^2{\wp }_v^3\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w\right],\hfill \\ & & \\ \hfill {\mathbb{K}}_2& =& {\displaystyle \frac{{\wp }_v}{3{\left(C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2\right)}^2}}\left[-C^2{\rho }_w\left({\rho }_w{\wp }_{vv}{\Psi }_v-{\rho }_w{\wp }_v{\Psi }_{vv}+CS{\wp }_v^3{\Psi }_w\right){\Psi }_{ww}\right.\hfill \\ & & -C{\wp }_v{\rho }_w^2\left(2S{\rho }_w{\Psi }_v+C{\Psi }_{vw}\right){\Psi }_{vw}-C^2{\wp }_v{\rho }_w{\rho }_{ww}{\Psi }_w{\Psi }_{vv}-{\wp }_v{\rho }_w^4{\Psi }_v^2\hfill \\ & & \left.+C^3{\wp }_v^3\left(S{\rho }_{ww}-C{\rho }_w^2\right){\Psi }_w^2+C^2{\wp }_{vv}{\rho }_w{\rho }_{ww}{\Psi }_v{\Psi }_w\right],\hfill \\ & & \\ \hfill {\mathbb{K}}_3& =& {\displaystyle \frac{C^2{\wp }_v^2{\rho }_w^2}{u{\left(C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2\right)}^{5/2}}}\left[-C{\rho }_w{\wp }_v\left({\Psi }_w^2{\Psi }_{vv}-2{\Psi }_v{\Psi }_w{\Psi }_{vw}+{\Psi }_v^2{\Psi }_{ww}\right)\right.\hfill \\ & & \left.+\left({\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2+C\left({\rho }_w{\wp }_{vv}{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w\right){\Psi }_w\right],\hfill \end{array}$$

Proof. 

By applying the theorem of Cayley–Hamilton, the curvatures ${\mathbb{K}}_i$ associated with $\varphi$ are determined through the characteristic polynomial, mentioned by Equation (1), of the RCH-T defined by Equation (4):

$${\mathbb{K}}_0{\lambda }^3-3{\mathbb{K}}_1{\lambda }^2+3{\mathbb{K}}_2\lambda -{\mathbb{K}}_3=0.$$

□

Corollary 1.

Let ϕ be an RCH-T mentioned by Equation (4) in ${\mathbb{L}}^4$. ϕ is deemed 1-minimal if the ensuing partial differential equation emerges:

$$\begin{array}{c}C{\wp }_v{\rho }_w\left[\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}+2{\Psi }_v{\Psi }_w{\Psi }_{vw}+\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}\right]\hfill \\ +{\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2{\Psi }_w+C\left({\wp }_{vv}{\rho }_w{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w^2\hfill \\ -C{u}^2{\wp }_{vv}{\rho }_w^3{\Psi }_v-C^2{u}^2{\wp }_v^3\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w=0,\hfill \end{array}$$

where $u\ne 0,$ $C{\wp }_v{\left(u^2{\rho }_w^2-{\Psi }_w^2\right)}^{1/2}\ne \pm {\rho }_w{\Psi }_v.$

Corollary 2.

Consider ϕ as an RCH-T indicated by Equation (4) within ${\mathbb{L}}^4$. ϕ is regarded as 2-minimal if the subsequent partial differential equation is satisfied:

$$\begin{array}{c}-C^2{\rho }_w\left({\wp }_{vv}{\rho }_w{\Psi }_v-{\wp }_v{\rho }_w{\Psi }_{vv}+CS{\wp }_v^3{\Psi }_w\right){\Psi }_{ww}\hfill \\ -C{\wp }_v{\rho }_w^2\left(2S{\rho }_w{\Psi }_v+C{\Psi }_{vw}\right){\Psi }_{vw}-C^2{\wp }_v{\rho }_w{\rho }_{ww}{\Psi }_w{\Psi }_{vv}\hfill \\ -{\wp }_v{\rho }_w^4{\Psi }_v^2+C^3{\wp }_v^3\left(S{\rho }_{ww}-C{\rho }_w^2\right){\Psi }_w^2+C^2{\wp }_{vv}{\rho }_w{\rho }_{ww}{\Psi }_v{\Psi }_w=0,\hfill \end{array}$$

where $C{\wp }_v{\left(u^2{\rho }_w^2-{\Psi }_w^2\right)}^{1/2}\ne \pm {\rho }_w{\Psi }_v.$

Corollary 3.

Let ϕ be an RCH-T given by Equation (4) in ${\mathbb{L}}^4$. ϕ is deemed 3-minimal if the following partial differential equation is provided:

$$\begin{array}{c}-C{\rho }_w{\wp }_v\left({\Psi }_v^2{\Psi }_{ww}-2{\Psi }_v{\Psi }_w{\Psi }_{vw}+{\Psi }_w^2{\Psi }_{vv}\right)\hfill \\ +\left({\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2+C\left({\rho }_w{\wp }_{vv}{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w\right){\Psi }_w=0,\hfill \end{array}$$

where $u\ne 0,$ $C{\wp }_v{\left(u^2{\rho }_w^2-{\Psi }_w^2\right)}^{1/2}\ne \pm {\rho }_w{\Psi }_v.$

It should be noted that the resolutions for $\Psi$ in the corollaries are challenges yet to be addressed.

5. The Laplace–Beltrami Operator of the RCH-T in ${\mathbb{L}}^{\mathbf{4}}$

In this section, the focus shifts to the application of the $\mathcal{L}$–$\mathcal{B}$ operator on a smooth function in ${\mathbb{L}}^4$. The subsequent steps involve calculating this operator using the RCH-T defined by the equation in (4).

Definition 6.

The Laplace–Beltrami operator is defined for a smooth function $\varsigma =\varsigma \left(x^1,x^2,x^3\right){\mid }_{\mathcal{D}}\left(\mathcal{D}\subset {\mathbb{R}}^3\right)$ of class $C^3$ and depends on the first fundamental form $\left({\mathfrak{g}}_{ij}\right)$:

$$\Delta \varsigma ={\displaystyle \frac{1}{{\mathbf{g}}^{1/2}}}\sum _{i,j=1}^3{\displaystyle \frac{\partial }{\partial x^i}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{ij}{\displaystyle \frac{\partial \varsigma }{\partial x^j}}\right),$$

(8)

where $\left({\mathfrak{g}}^{ij}\right)={\left({\mathfrak{g}}_{kl}\right)}^{-1}$ and $\mathbf{g}=det\left({\mathfrak{g}}_{ij}\right).$

For comprehensive insights into the Laplace–Beltrami operator, refer to the works of Chen et al. [1] and Lawson [4]. Hence, the following is provided.

Theorem 2.

The Laplace–Beltrami operator of the RCH-T ϕ expressed by Equation (4) is formulated as $\Delta \varphi =3{\mathbb{K}}_1\mathbb{G}$, where ${\mathbb{K}}_1$ represents the mean curvature and $\mathbb{G}$ depicts the Gauss map of ϕ.

Proof. 

The $\mathcal{L}$–$\mathcal{B}$ operator applied to the RCH-T as described by Equation (4) is represented as follows:

$$\begin{array}{ccc}\hfill \Delta \varphi & =& {\displaystyle \frac{1}{{\mathbf{g}}^{1/2}}}\left[{\displaystyle \frac{\partial }{\partial u}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{11}{\displaystyle \frac{\partial \varphi }{\partial u}}\right)+{\displaystyle \frac{\partial }{\partial v}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{22}{\displaystyle \frac{\partial \varphi }{\partial v}}\right)+{\displaystyle \frac{\partial }{\partial v}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{23}{\displaystyle \frac{\partial \varphi }{\partial w}}\right)\right.\hfill \\ & & \left.+{\displaystyle \frac{\partial }{\partial w}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{32}{\displaystyle \frac{\partial \varphi }{\partial v}}\right)+{\displaystyle \frac{\partial }{\partial w}}\left({\mathbf{g}}^{1/2}{\mathfrak{g}}^{33}{\displaystyle \frac{\partial \varphi }{\partial w}}\right)\right],\hfill \end{array}$$

(9)

where

$$\left({\mathfrak{g}}^{ij}\right)={\displaystyle \frac{1}{u^2\mathbb{W}}}\left(\begin{array}{ccc}1& 0& 0\\ 0& u^2{\rho }_w^2-{\Psi }_w^2& {\Psi }_v{\Psi }_w\\ 0& {\Psi }_v{\Psi }_w& C^2{u}^2{\wp }_v^2-{\Psi }_v^2\end{array}\right),$$

(10)

and $\mathbb{W}=C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2$. Through the substitution of the derivatives of the components, as determined by (10), into the formula presented by (9), the formation of $\Delta \varphi =\left(\Delta {\varphi }_1,\Delta {\varphi }_2,\Delta {\varphi }_3,\Delta {\varphi }_4\right)$ along with its individual components is accomplished.

$$\begin{array}{c}\Delta {\varphi }_1={\displaystyle \frac{1}{u{\mathbb{W}}^2}}(-C{\rho }_w{\wp }_v\left(s{\Psi }_v{\rho }_w+CSc{\Psi }_w{\wp }_v\right)\left(\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+2{\Psi }_v{\Psi }_w{\Psi }_{vw}+\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}\right)\hfill \\ -c{C}^3{S}^2{\wp }_v^4{\Psi }_w^4-C\left(s{\wp }_{vv}{\rho }_w^2+2c{S}^2{\wp }_v^2{\rho }_w^2+cCS{\wp }_v^2{\rho }_{ww}\right){\Psi }_v^2{\Psi }_w^2+c{C}^{3}S{u}^2{\wp }_v^4\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w^2\hfill \\ +C^2{u}^2{\wp }_v{\rho }_w\left(sS{\wp }_v^2{\rho }_w^2+cS{\wp }_{vv}{\rho }_w^2+Cs{\wp }_v^2{\rho }_{ww}\right){\Psi }_v{\Psi }_w-s{\wp }_v{\rho }_w\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^3{\Psi }_w\hfill \\ -C^{2}S{\wp }_v{\rho }_w\left(c{\wp }_{vv}+s{\wp }_v^2\right){\Psi }_v{\Psi }_w^3+sC{u}^2{\wp }_{vv}{\rho }_w^4{\Psi }_v^2),\hfill \\ \\ \Delta {\varphi }_2={\displaystyle \frac{1}{u{\mathbb{W}}^2}}(C{\rho }_w{\wp }_v\left(c{\Psi }_v{\rho }_w-CSs{\Psi }_w{\wp }_v\right)\left(\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+2{\Psi }_v{\Psi }_w{\Psi }_{vw}+\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}\right)\hfill \\ -s{C}^3{S}^2{\wp }_v^4{\Psi }_w^4-C\left(-c{\wp }_{vv}{\rho }_w^2+2s{S}^2{\wp }_v^2{\rho }_w^2+sCS{\rho }_{ww}{\wp }_v^2\right){\Psi }_v^2{\Psi }_w^2+s{C}^{3}S{u}^2{\wp }_v^4\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w^2\hfill \\ -C^2{u}^2{\wp }_v{\rho }_w\left(cS{\wp }_v^2{\rho }_w^2+cC{\wp }_v^2{\rho }_{ww}-sS{\wp }_{vv}{\rho }_w^2\right){\Psi }_v{\Psi }_w+c{\wp }_v{\rho }_w\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^3{\Psi }_w\hfill \\ -C^{2}S{\wp }_v{\rho }_w\left(s{\wp }_{vv}-c{\wp }_v^2\right){\Psi }_v{\Psi }_w^3-cC{u}^2{\wp }_{vv}{\rho }_w^4{\Psi }_v^2),\hfill \\ \\ \Delta {\varphi }_3={\displaystyle \frac{1}{u{\mathbb{W}}^2}}(C^3{\wp }_v^2{\rho }_w\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_w{\Psi }_{vv}+2C^3{\wp }_v^2{\rho }_w{\Psi }_v{\Psi }_w^2{\Psi }_{vw}\hfill \\ +C^3{\wp }_v^2{\rho }_w\left(-{\Psi }_v^2+sCS{u}^2{\wp }_v^2\right){\Psi }_w{\Psi }_{ww}-s{C}^{4}S{u}^2{\wp }_v^4{\rho }_{ww}{\Psi }_w^2+C^{4}S{\wp }_v^4\left({\Psi }_w^2-u^2{\rho }_w^2\right){\Psi }_w^2\hfill \\ -C^2{\wp }_v\left(-C{\wp }_v{\rho }_{ww}{\Psi }_v{\Psi }_w-2S{\wp }_v{\rho }_w^2{\Psi }_v{\Psi }_w-C{\wp }_{vv}{\rho }_w{\Psi }_w^2+C{u}^2{\rho }_w^3{\wp }_{vv}\right){\Psi }_v{\Psi }_w),\hfill \\ \\ \Delta {\varphi }_4={\displaystyle \frac{1}{u^2{\mathbb{W}}^2}}(C^2{\wp }_v^2{\rho }_w^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+2C^2{\wp }_v^2{\rho }_w^2{\Psi }_v{\Psi }_w{\Psi }_{vw}+C^2{\wp }_v^2{\rho }_w^2\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}\hfill \\ +C{\rho }_w{\wp }_v\left(2S{\wp }_v{\rho }_w^2{\Psi }_v^2+C{\wp }_v{\rho }_{ww}{\Psi }_v^2+C{\rho }_w{\wp }_{vv}{\Psi }_v{\Psi }_w+C^{2}S{\wp }_v^3{\Psi }_w^2-C^2{u}^2{\wp }_v^3\left(C{\rho }_{ww}+S{\rho }_w^2\right)\right){\Psi }_w\hfill \\ -C^2{u}^2{\wp }_v{\wp }_{vv}{\rho }_w^4{\Psi }_v).\hfill \end{array}$$

□

Definition 7.

The hypersurface ϕ is termed harmonic if each component of $\Delta \varphi$ is null.

Example 1.

Upon substituting $\wp \left(v\right)=v,$ $\rho \left(w\right)=w,$ and $\Psi \left(v,w\right)=w$ into an RCH-T defined by Equation (4), the expression is simplified to

$$\varphi (u,v,w)=\left(ucosvcosw,usinvcosw,usinw,w\right).$$

Its Gauss map and the shape operator matrix are characterized by

$$\mathbb{G}={\displaystyle \frac{1}{{\left(1-u^2\right)}^{1/2}}}\left(-cosvsinw,-sinvsinw,cosw,u\right),$$

$$\mathcal{S}=\left(\begin{array}{ccc}0& 0& {\displaystyle \frac{1}{{\left(1-u^2\right)}^{1/2}}}\\ 0& {\displaystyle \frac{tanw}{u{\left(1-u^2\right)}^{1/2}}}& 0\\ -{\displaystyle \frac{1}{{\left(1-u^2\right)}^{3/2}}}& 0& 0\end{array}\right).$$

Subsequently, the principal curvatures are $k_1={\displaystyle \frac{i}{u^2-1}}=-k_2$ and $k_3={\displaystyle \frac{tanw}{u{\left(1-u^2\right)}^{1/2}}},$ and the curvatures are determined through

$$\begin{array}{ccc}\hfill {\mathbb{K}}_1& =& {\displaystyle \frac{tanw}{3u{\left(1-u^2\right)}^{1/2}}},\hfill \\ \hfill {\mathbb{K}}_2& =& {\displaystyle \frac{1}{3{\left(1-u^2\right)}^2}},\hfill \\ \hfill {\mathbb{K}}_3& =& {\displaystyle \frac{tanw}{u{\left(1-u^2\right)}^{5/2}}}.\hfill \end{array}$$

Hence,

$$\Delta \varphi ={\displaystyle \frac{tanw}{u\left(1-u^2\right)}}\left(-cosvsinw,-sinvsinw,cosw,u\right).$$

In conclusion, the hypersurface is neither minimal nor harmonic.

Example 2.

Upon choosing $\wp \left(v\right)=v,$ $\rho \left(w\right)=w,$ and $\Psi \left(v,w\right)=v$ for an RCH-T defined by Equation (4), the RCH-T is parametrized by

$$\varphi (u,v,w)=\left(ucosvcosw,usinvcosw,usinw,v\right).$$

ϕ possesses the Gauss map

$$\mathbb{G}={\displaystyle \frac{1}{{\left(1-u^2{cos}^{2}w\right)}^{1/2}}}\left(-sinv,cosv,0,ucosw\right).$$

Therefore, the shape operator matrix is determined by

$$\mathcal{S}=\left(\begin{array}{ccc}0& {\displaystyle \frac{cosw}{{\left(1-u^2{cos}^{2}w\right)}^{1/2}}}& 0\\ -{\displaystyle \frac{cosw}{{\left(1-u^2{cos}^{2}w\right)}^{3/2}}}& 0& {\displaystyle \frac{usinw}{{\left(1-u^2{cos}^{2}w\right)}^{3/2}}}\\ 0& -{\displaystyle \frac{sinw}{u{\left(1-u^2{cos}^{2}w\right)}^{1/2}}}& 0\end{array}\right).$$

The curvatures are given by

$$\begin{array}{ccc}\hfill {\mathbb{K}}_1& =& 0,\hfill \\ \hfill {\mathbb{K}}_2& =& {\displaystyle \frac{1}{3{\left(1-u^2{cos}^{2}w\right)}^2}},\hfill \\ \hfill {\mathbb{K}}_3& =& 0.\hfill \end{array}$$

Finally, $\Delta \varphi =\left(0,0,0,0\right).$ This implies that the hypersurface is 1-minimal, 3-minimal, and harmonic.

A comprehensive understanding of the diverse expressions of the geometric structures of the RCH-T indicated by Equation (4) is provided by exploring the hypersurfaces classified as minimal, non-minimal, harmonic, and non-harmonic.

6. The Umbilical Condition of the RCH-T in ${\mathbb{L}}^{\mathbf{4}}$

In this section, the umbilical condition for the RCH-T defined by Equation (4) is disclosed.

Definition 8.

A point $\mathbf{p}$ on a hypersurface is termed umbilical if and only if $k_1=k_2=k_3.$ This implies ${\mathbb{K}}_1{ }^3={\mathbb{K}}_3$.

As a result, we offer the following, laying down an essential foundation for deciphering the fundamental principles behind our findings.

Theorem 3.

A point $\mathbf{p}$ on the hypersurface of the right conoid, $\varphi :M^3\subset {\mathbb{E}}^3⟶{\mathbb{L}}^4$, that features a time-like axis, is considered umbilic if and only if it satisfies the partial differential equation ${\mathbb{K}}_1{ }^3-{\mathbb{K}}_3=0$. Here, ${\mathbb{K}}_1$ and ${\mathbb{K}}_3$ indicate the mean and Gauss–Kronecker curvatures, respectively.

Proof. 

By using the curvatures ${\mathbb{K}}_1$ and ${\mathbb{K}}_3$ of the right conoid hypersurface with the time-like axis $\varphi$, the following partial differential equation arises:

$$\begin{array}{ccc}\hfill {\mathsf{\Phi }}^3-27C^2{u}^2{\mathbb{W}}^2{\wp }_v^2{\rho }_w^2\mathsf{\Theta }& =& 0,\hfill \end{array}$$

(11)

where

$$\begin{array}{ccc}\hfill \mathsf{\Phi }& =& C{\wp }_v{\rho }_w\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}+2C{\wp }_v{\rho }_w{\Psi }_v{\Psi }_w{\Psi }_{vw}\hfill \\ & & +C{\wp }_v{\rho }_w\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+{\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2{\Psi }_w\hfill \\ & & +C\left({\wp }_{vv}{\rho }_w{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w^2-C{u}^2{\wp }_{vv}{\rho }_w^3{\Psi }_v\hfill \\ & & -C^2{u}^2{\wp }_v^3\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w,\hfill \\ \hfill \mathsf{\Theta }& =& -C{\rho }_w{\wp }_v\left({\Psi }_w^2{\Psi }_{vv}-2{\Psi }_v{\Psi }_w{\Psi }_{vw}+{\Psi }_v^2{\Psi }_{ww}\right)\hfill \\ & & +\left({\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2+C\left({\rho }_w{\wp }_{vv}{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w\right){\Psi }_w,\hfill \\ \hfill \mathbb{W}& =& C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2.\hfill \end{array}$$

□

This theorem highlights a significant geometric property of the points on the hypersurface. It states that a point on the right conoid hypersurface with a time-like axis is umbilic exactly when it satisfies the given partial differential equation. The primary challenge is to find solutions $\Psi$ that fulfil the partial differential equations described in Equation (11).

7. The Willmore Functional of the RCH-T in ${\mathbb{L}}^{\mathbf{4}}$

In this section, we introduce the Willmore property of the RCH-T.

Definition 9.

Let s: M $\subset {\mathbb{E}}^2⟶$ ${\mathbb{R}}^n$ be a smooth immersion such that $W\left(s\right)<\infty$. The function s is deemed a critical point for W if

$$\forall n\in C^{\infty }(M,{\mathbb{R}}^n),{\displaystyle \frac{d}{dq}}W(s+qn)\left|{ }_{q=0}\right.=0.$$

An immersion satisfying this condition is referred to as Willmore.

Refer to the studies conducted by Li and Yau [5], Toda [6], and the research carried out by Willmore [7,8] for a thorough exploration of Euclidean properties. Hence, the Willmore functional is employed for the RCH-T given by Equation $\left(\right)$ in Minkowski four-dimensional space.

Theorem 4.

An immersion $\varphi :M^3\subset {\mathbb{E}}^3⟶{\mathbb{L}}^4$ is considered Willmore if and only if it satisfies the partial differential equation

$$\Delta {\mathbb{K}}_1+3{\mathbb{K}}_1({\mathbb{K}}_1{ }^3-{\mathbb{K}}_3)=0,$$

where Δ denotes the Laplace–Beltrami operator, and ${\mathbb{K}}_1$ and ${\mathbb{K}}_3$ represent the mean and Gauss–Kronecker curvatures, respectively.

Proof. 

By using the curvatures determined by Theorem 1, and by utilizing the expression $\Delta {\mathbb{K}}_1={\left({\mathbb{K}}_1\right)}_{uu}+{\left({\mathbb{K}}_1\right)}_{vv}+{\left({\mathbb{K}}_1\right)}_{ww}$ and incorporating the umbilical condition ${\mathbb{K}}_1{ }^3-{\mathbb{K}}_3=0,$ the following partial differential equation is unveiled:

$${\left({\mathbb{K}}_1\right)}_{uu}+{\left({\mathbb{K}}_1\right)}_{vv}+{\left({\mathbb{K}}_1\right)}_{ww}+\mathsf{\Phi }\left({\mathsf{\Phi }}^3-27C^2{u}^2{\mathbb{W}}^2{\wp }_v^2{\rho }_w^2\mathsf{\Theta }\right)=0,$$

(12)

where

$$\begin{array}{ccc}\hfill \mathsf{\Phi }& =& C{\wp }_v{\rho }_w\left(C^2{u}^2{\wp }_v^2-{\Psi }_v^2\right){\Psi }_{ww}+2C{\wp }_v{\rho }_w{\Psi }_v{\Psi }_w{\Psi }_{vw}\hfill \\ & & +C{\wp }_v{\rho }_w\left(u^2{\rho }_w^2-{\Psi }_w^2\right){\Psi }_{vv}+{\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2{\Psi }_w\hfill \\ & & +C\left({\wp }_{vv}{\rho }_w{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w^2-C{u}^2{\wp }_{vv}{\rho }_w^3{\Psi }_v\hfill \\ & & -C^2{u}^2{\wp }_v^3\left(C{\rho }_{ww}+S{\rho }_w^2\right){\Psi }_w,\hfill \\ \hfill \mathsf{\Theta }& =& -C{\rho }_w{\wp }_v\left({\Psi }_w^2{\Psi }_{vv}-2{\Psi }_v{\Psi }_w{\Psi }_{vw}+{\Psi }_v^2{\Psi }_{ww}\right)\hfill \\ & & +\left({\wp }_v\left(C{\rho }_{ww}+2S{\rho }_w^2\right){\Psi }_v^2+C\left({\rho }_w{\wp }_{vv}{\Psi }_v+CS{\wp }_v^3{\Psi }_w\right){\Psi }_w\right){\Psi }_w,\hfill \\ \hfill \mathbb{W}& =& C^2{\wp }_v^2\left(u^2{\rho }_w^2-{\Psi }_w^2\right)-{\rho }_w^2{\Psi }_v^2.\hfill \end{array}$$

□

A profound connection between the immersion of the hypersurface and its Willmore property is established by this theorem. It is asserted that the immersion $\varphi$ from ${\mathbb{E}}^3$ into ${\mathbb{L}}^4$ achieves the status of being Willmore if and only if it aligns with the specified equation. The identification of solutions $\Psi$ for the partial differential equation referred to as (12) remains an unresolved matter.

8. Conclusions

In conclusion, this study has offered a detailed exploration of hypersurfaces identified as right conoids with a time-like axis (RCH-Ts) in four-dimensional Minkowski space ${\mathbb{L}}^4$. Through meticulous analysis and computation of essential matrices—specifically, the fundamental form, Gauss map, and shape operator—specific curvatures intrinsic to these geometrical entities have been uncovered. The application of the Cayley–Hamilton theorem has been instrumental in shedding light on the complex geometric properties of RCH-T hypersurfaces.

The foundational investigation into four-dimensional Minkowski geometry laid a solid groundwork, elucidating crucial curvature formulas applicable to hypersurfaces within this space. The comprehensive definition of RCH-Ts has highlighted their unique characteristics and has distinguished them within the realm of hypersurfaces.

Moreover, practical implications of the Laplace–-Beltrami operator on these hypersurfaces have been explored, demonstrating its computational utility and furthering our understanding of their geometric behavior. The introduction of umbilical RCH-T hypersurfaces has expanded our exploration, revealing additional geometric insights.

Lastly, the presentation and analysis of the Willmore functional has provided a quantitative measure to evaluate the shape characteristics of RCH-T hypersurfaces, contributing to both theoretical advancements and potential practical applications.

In summary, this comprehensive investigation not only deepens our theoretical understanding of hypersurfaces in Minkowski space but also sets the stage for future research and applications in geometry and mathematical physics.