2022/11/07 by William H. Meeks, Meeks, William H., Joaquín Pérez +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2211.03529
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 and Ros 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.