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The Nash-Kuiper process for curves

Vincent Borrelli, Saïd Jabrane, Francis Lazarus, Boris Thibert (2011/2012)

Séminaire de théorie spectrale et géométrie

A strictly short embedding is an embedding of a Riemannian manifold into an Euclidean space that strictly shortens distances. From such an embedding, the Nash-Kuiper process builds a sequence of maps converging toward an isometric embedding. In that paper, we describe this Nash-Kuiper process in the case of curves. We state an explicit formula for the limit normal map and perform its Fourier series expansion. We then adress the question of Holder regularity of the limit map.

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

For two-dimensional, immersed closed surfaces f : Σ → ℝ n , we study the curvature functionals ℰ p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of ℰ p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional condition....

Currently displaying 301 – 320 of 436