• Corpus ID: 247450591

Curvature Varifolds with Orthogonal Boundary

@inproceedings{Kuwert2022CurvatureVW,
  title={Curvature Varifolds with Orthogonal Boundary},
  author={Ernst Kuwert and Marius Muller},
  year={2022}
}

• E. Kuwert, Marius Muller
• Published 15 March 2022
• Mathematics

We consider the class Sm ⊥ (Ω) of m-dimensional surfaces in Ω ⊂ Rn which intersect S = ∂Ω orthogonally along the boundary. A piece of an affine m-plane in Sm ⊥ (Ω) is called an orthogonal slice. We prove estimates for the area by the Lp-integral of the second fundamental form in three cases: first when Ω admits no orthogonal slices, second for m = p = 2 if all orthogonal slices are topological disks, and finally for all Ω if the surfaces are confined to a neighborhood of S. The orthogonality constraint has a weak formulation for curvature varifolds, we classify those varifolds of vanishing curvature. As an application, we prove for any Ω the existence of an orthogonal 2-varifold which minimizes the L2 curvature in the integer rectifiable class. MSC 2020 Subject Classification. Primary 28A75, Secondary 49Q15, 49Q20.