We construct complete, embedded minimal annuli asymptotic to vertical planes in the Riemannian 3-manifold
PSL
~
2
(
R
,
τ
)
. The boundary of these annuli consists of 4 vertical lines at infinity. ...They are constructed by taking the limit of a sequence of compact minimal annuli. The compactness is obtained from an estimate of curvature which uses foliations by minimal surfaces. This estimate is independent of the index of the surface. We also prove the existence of a one-periodic family of Riemann’s type examples. The difficulty of the construction comes from the lack of symmetry of the ambient space
PSL
~
2
(
R
,
τ
)
.
We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3-manifold \mathcal {N}. We also obtain a least area, incompressible, properly embedded, finite ...topology, 2-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This determines its asymptotic behavior. Some rigidity theorems are obtained.
Funicular structures can resist a given load with pure axial forces and therefore tend to use material very efficiently. One main challenge in their design is form finding, which often requires ...advanced numerical methods. In this article, we show analytically that a very common family of curves, conics, is funicular for a particular load case: a uniform radial load emanating from a focus. The result is a generalization of the well-known funicularity of parabolas and arcs of circles, respectively, under uniform vertical load and constant normal pressure. It can be used to design self-stressed structures by hand without the need for calculations. Portions of conics can be combined to obtain original shapes.
In this paper, we develop the theory of properly immersed minimal surfaces in the quotient space
(
H
2
×
R
)
/
G
,
where
G
is a subgroup of isometries generated by a vertical translation and a ...horizontal isometry (without fixed points) in
H
2
. The horizontal isometry can be either a parabolic translation along horocycles in
H
2
or a hyperbolic translation along a geodesic in
H
2
.
We prove that if a properly immersed minimal surface in
(
H
2
×
R
)
/
G
has finite total curvature, then its total curvature is a multiple of
2
π
and, moreover, we understand the geometry of the ends. The results hold true more generally for properly immersed minimal surfaces in
M
×
S
1
,
where
M
is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces
(
H
2
×
R
)
/
G
.
The design of curved structural building envelopes is challenging as it requires to account for a wide variety of constraints. In particular, the shape must be mechanically efficient, fabricable, and ...fit the site geometry. This article shows how a family of surfaces, called isotropic Linear Weingarten surfaces (nicknamed i-liwien), may fulfil all these constraints together, and be used as an intuitive design tool. We start by showing that these shapes are funicular for a uniform vertical load, and that principal stress lines form a conjugate net. This allows in particular for the design of gridshells with planar faces and low bending moments or for the design of self-stressed cable nets cladded by planar glass panels. We then propose a discrete model based on recent advances in discrete differential geometry. We use this model to propose a generation method from boundary curves with two additional control parameters. We demonstrate the shape potential on several examples. The application to shells, membranes, cable-nets and gridshells are discussed.
We construct a family of compact free boundary minimal annuli immersed in the unit ball
B
3
of
R
3
, the first such examples other than the critical catenoid. This solves a problem formulated by ...Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in
B
3
foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in
B
3
. Their existence solves in the negative a problem proposed by Wente in 1995.
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