Cambridge Journal of Mathematics

Volume 4 (2016)

Number 4

Morse index and multiplicity of min-max minimal hypersurfaces

Pages: 463 – 511



Fernando C. Marques (Department of Mathematics, Princeton University, Princeton New Jersey, U.S.A.)

André Neves (Imperial College London, United Kingdom)


The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the minmax minimal hypersurface.

We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.

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Published 31 October 2016