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Surfaces of prescribed linear Weingarten curvature in $\mathbb {R}^{3}$
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 4 / August 2023
- Published online by Cambridge University Press:
- 22 July 2022, pp. 1347-1370
- Print publication:
- August 2023
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- Article
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Given $a,\,b\in \mathbb {R}$
and $\Phi \in C^{1}(\mathbb {S}^{2})$
, we study immersed oriented surfaces $\Sigma$
in the Euclidean 3-space $\mathbb {R}^{3}$
whose mean curvature $H$
and Gauss curvature $K$
satisfy $2aH+bK=\Phi (N)$
, where $N:\Sigma \rightarrow \mathbb {S}^{2}$
is the Gauss map. This theory widely generalizes some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$
, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.
