On the topology and index of minimal surfaces

Abstract

We show that for an immersed two-sided minimal surface in $R^3$, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in $R^3$ of index $2$, as conjectured by Choe. Moreover, we show that the index of a immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.

References in corpus (2)

• On the topology and index of minimal surfaces
• Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area

Cited by in corpus (11)

• Minimal hypersurfaces with bounded index
• On the topology and index of minimal surfaces
• Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
• Geometric convergence results for closed minimal surfaces via bubbling analysis
• Index and topology of minimal hypersurfaces in R^n
• Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities
• Compact minimal hypersurfaces of index one and the width of real projective spaces
• Bubbling analysis and geometric convergence results for free boundary minimal surfaces
• On the index of minimal surfaces with free boundary in a half-space
• Equivariant min-max hypersurface in $G$-manifolds with positive Ricci curvature
• Index Estimates for Surfaces with Constant Mean Curvature in $3$-dimensional Manifolds