Geometry of branched minimal surfaces of finite index
Metadatos
Afficher la notice complèteEditorial
De Gruyter
Materia
Constant mean curvature Finite index H-surfaces Area estimates for constant mean curvature surfaces
Date
2024-03-12Referencia bibliográfica
Meeks, W. & Pérez, J. (2024). Geometry of branched minimal surfaces of finite index. Advanced Nonlinear Studies, 24(1), 206-221. https://doi.org/10.1515/ans-2023-0118
Patrocinador
CNPq - Brazil, grant no. 400966/2014-0.; MINECO/MICINN/FEDER grant no. PID2020-117868GB-I00; Junta de Andalucía grant no. P18-FR-4049Résumé
Given I, B ∈ ℕ ∪ {0}, we investigate the existence and geometry of complete finitely branched minimal
surfaces M in ℝ3 with Morse index at most I and total branching order at most B. Previous works of
Fischer-Colbrie (“On complete minimal surfaces with finite Morse index in 3-manifolds,” Invent. Math., vol. 82,
pp. 121–132, 1985) and Ros (“One-sided complete stable minimal surfaces,” J. Differ. Geom., vol. 74, pp. 69–92,
2006) explain that such surfaces are precisely the complete minimal surfaces in ℝ3 of finite total curvature and
finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such
an M with estimates that are given in terms of I and B. In order to obtain some of our main results for these
special surfaces, we obtain general intrinsic monotonicity of area formulas for m-dimensional submanifolds
Σ of an n-dimensional Riemannian manifold X, where these area estimates depend on the geometry of X and
upper bounds on the lengths of the mean curvature vectors of Σ.We also describe a family of complete, finitely
branched minimal surfaces in ℝ3 that are stable and non-orientable; these examples generalize the classical
Henneberg minimal surface.