Superfícies de Weingarten algébricas regulares especiais no espaço euclidiano

Abstract

In this dissertation we will be interested on the classification of special algebraic Weingarten surfaces in $\mathbb{R}^3$. The classification of Weingarten surfaces in the general setting remains mostly open to this day. After initial work in the 1950s, led by Chern, Hopf, Voss, Hartman, Winter, among others, there has been recent progress in this theory, especially when the Weingarten relation takes the form $H = f(H^2 - K)$ , where $f$ is a function $C^1$ defined on $[0, +\infty)$. Attention is directed to functions that satisfy the condition $4t(f'(t))^2 < 1$ on the interval $t \in [0, +\infty)$ called the elliptic condition.

Manfredo do Carmo and João Lucas Barbosa, showed in “On regular algebraic surfaces of R3 with constant mean curvature” that the only regular algebraic surfaces with constant mean curvature are:

- Spheres, $\left(x-x_0\right)^2+\left(y-y_0\right)^2+\left(z-z_0\right)^2=r^2$;

- Cylinders, $\left(x-x_0\right)^2+\left(y-y_0\right)^2=r^2$;

- Plans, $ax+by+cz+d=0$.

The first two cases being when the mean curvature $H$ differs from zero and the last when it is identically zero.

The discovery that there are only three types of regular algebraic surfaces of constant mean curvature—planes, spheres, and right cylinders—is a surprising and significant result. As the family of Weingarten surfaces encompasses the family of surfaces with constant mean curvature, it is natural to ask whether when analyzing the Weingarten there are other regular algebraic besides these three. In this context, we present the following original results.

Theorem: Let $\Sigma$ be a regular algebraic surface in $\mathbb{R}^3$. Suppose that $\Sigma$ is a special Weingarten surface satisfying the relation $aH + bK = 1$, with $a > 0$ and $b \geq 0$. Then $\Sigma$ is either a sphere or a cylinder.

Theorem: The only regular algebraic special Weingarten surface of minimal type is the plane.

These results are fundamental for understanding the geometry and classification of Weingarten surfaces in $\mathbb{R}^3$, significantly contributing to the advancement of knowledge regarding the geometric properties of these special surfaces.