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Natural algebraic representation formulas for curves in ℂ³

Hubert Gollek (2002)

Banach Center Publications

We consider several explicit examples of solutions of the differential equation Φ₁’²(z) + Φ₂’²(z) + Φ₃’²(z) = d²(z) of meromorphic curves in ℂ³ with preset infinitesimal arclength function d(z) by nonlinear differential operators of the form (f,h,d) → V(f,h,d), V = (Φ₁,Φ₂,Φ₃), whose arguments are triples consisting of a meromorphic function f, a meromorphic vector field h, and a meromorphic differential 1-form d on an open set U ⊂ ℂ or, more general, on a Riemann surface Σ. Most of them are natural...

New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space ℍ n + 1 , that is, complete hypersurfaces of ℍ n + 1 whose mean curvature H and normalized scalar curvature R satisfy R = a H + b for some a , b ∈ ℝ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of ℍ n + 1 . Furthermore, a rigidity result...

New stability results for spheres and Wulff shapes

Julien Roth (2018)

Communications in Mathematics

We prove that a closed convex hypersurface of the Euclidean space with almost constant anisotropic first and second mean curvatures in the L p -sense is W 2 , p -close (up to rescaling and translations) to the Wulff shape. We also obtain characterizations of geodesic hyperspheres of space forms improving those of [Ro1] and [Ro].

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