A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.
Minimal area surfaces and fibered hyperbolic 3-manifolds
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic 3 3 -manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic 3 3 -manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
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- NSF-PAR ID:
- 10466057
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 150
- Issue:
- 761
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 4931 to 4946
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/15998
