Geometry of branched minimal surfaces of finite index

Abstract

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.