Geometry of branched minimal surfaces of finite index

Abstract

Given I , B ∈ N ∪ { 0 } $I,B\in \mathbb{N}\cup \left\{0\right\}$ , we investigate the existence and geometry of complete finitely branched minimal surfaces M in R 3 ${\mathbb{R}}^{3}$ 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 R 3 ${\mathbb{R}}^{3}$ 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 R 3 ${\mathbb{R}}^{3}$ that are stable and non-orientable; these examples generalize the classical Henneberg minimal surface.