Renzo Ilario Caddeo

Biharmonic submanifolds in spheres

We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereSn. In the case of curves inSn we solve explicitly the biharmonic equation.

BIHARMONIC SUBMANIFOLDS OF S 3

We explicitly classify the nonharmonic biharmonic submanifolds of the unit three-dimensional sphere

SO(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces

{\mathbb S}^3

The möbius strip and Viviani’s windows

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Biharmonic maps between Riemannian Manifolds

We study SO(2)-invariant minimal and constant mean curvature surfaces in R3 endowed with a homogenous Riemannian metric whose group of isometries has dimension greater or equal to 4.

On biharmonic maps

The space curves called “Viviani’s windows” are curves that solved a celebrated geometric puzzle: “Aenigma Geometricum de miro opificio Testudinis Quadrabilis Hemisphaericae” (Geometric enigma on the remarkable realization of a squarable hemispherical vault). This is a (pseudo)-architectural problem proposed by Vincenzo Viviani, a disciple of Galileo, in 1692 (see [L], page 201, and [R1, R2] for a complete and detailed treatment) formulated as follows: build on a hemispherical cupola four equal windows of such a size that the remaining surface can be exactly squared. Among several known solutions, the following was found by Viviani and by other eminent mathematicians of that time: the four windows are the intersections of a hemisphere of radius α with two cylinders of radius α 2 that have in common only a ruling containing a diameter of the hemisphere.