Some classification of affine homothetical surfaces of finite type in 𝕀

A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian Δ. In this paper, we classify two types of affine homothetical surfaces of finite type in isotropic 3-space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕀</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {I³} under the condition <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> {Δ rᵢ=λᵢrᵢ} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>r</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(rᵢ)} is the i -component function of the position vector r ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> </m:mrow> </m:mrow> </m:math> {i=1,2,3} ), <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>λ</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> {λᵢ} and Δ denotes the Laplace operator.