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Distributed branch points and the shape of elastic surfaces with constant negative curvature. (English) Zbl 1479.53071

The paper studies surfaces of constant negative curvature with branch points. Such surfaces are not of class \(C^2\) at the branch points, but have a well defined tangent plane, and are therefore of class \(C^{1,1}\). The authors define a topological index for the branch points and prove its robustness. There is also a discrete version of such surfaces based on discrete differential geometry.
The authors argue that these surfaces are good candidates for minimizing bending energy. By using the discrete model, they made numerical experiments that corroborate this expectation. They also show numerically that fractal-like patterns appear in the distribution of branch points, and so this model can partially explain the appearance of buckling patterns in hyperbolic surfaces, which is the motivating question of the paper.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A70 Discrete differential geometry
52B70 Polyhedral manifolds
53Z50 Applications of differential geometry to data and computer science
35Q74 PDEs in connection with mechanics of deformable solids
74K99 Thin bodies, structures
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References:

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