Journal of Differential Geometry

Comparing the Morse index and the first Betti number of minimal hypersurfaces

Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp

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Abstract

By extending and generalizing previous work by Ros and Savo, we describe a method to show that in certain positively curved ambient manifolds the Morse index of every closed minimal hypersurface is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques–Neves for a wide class of ambient spaces.

Article information

Source
J. Differential Geom., Volume 108, Number 3 (2018), 379-410.

Dates
Received: 4 February 2016
First available in Project Euclid: 2 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1519959621

Digital Object Identifier
doi:10.4310/jdg/1519959621

Mathematical Reviews number (MathSciNet)
MR3770846

Zentralblatt MATH identifier
1385.53051

Citation

Ambrozio, Lucas; Carlotto, Alessandro; Sharp, Ben. Comparing the Morse index and the first Betti number of minimal hypersurfaces. J. Differential Geom. 108 (2018), no. 3, 379--410. doi:10.4310/jdg/1519959621. https://projecteuclid.org/euclid.jdg/1519959621


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