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First variation of the general curvature-dependent surface energy

Günay Doğan, Ricardo H. Nochetto (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation...

Flat 3-webs of degree one on the projective plane

A. Beltrán, M. Falla Luza, D. Marín (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this work is to study global 3 -webs with vanishing curvature. We wish to investigate degree 3 foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree 3 foliations whose Legendre transform are webs with zero curvature.

∇-flat functions on manifolds

Wojciech Kozłowski (2004)

Annales Polonici Mathematici

We investigate ∇-flat and pointwise-∇-flat functions on affine and Riemannian manifolds. We show that the set of all ∇-flat functions on (M,∇) is a ring which has interesting properties similar to the ring of polynomial functions.

Flats in 3-manifolds

Michael Kapovich (2005)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Flats in Spaces with Convex Geodesic Bicombings

Dominic Descombes, Urs Lang (2016)

Analysis and Geometry in Metric Spaces

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes...

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