An index characterization of the catenoid and index bounds for minimal surfaces in R⁴

The index of a minimal surface is defined to be the number of negative eigenvalues of the operator corresponding to second variation of area. In the present paper, we characterize the catenoid as the only complete oriented minimal surface in R³ of index one with embedded ends. We also obtain upper bounds for the index of minimal surfaces in R⁴, in terms of the total curvature of the surface.