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 onetoone correspondence between homogeneous multiaffine stable polynomials and PSDstable 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 wellknown Fisher–Hadamard and Koteljanskii inequalities from determinants to PSDstable 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 3manifolds
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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 NSFPAR ID:
 10466057
 Date Published:
 Journal Name:
 Proceedings of the American Mathematical Society
 Volume:
 150
 Issue:
 761
 ISSN:
 00029939
 Page Range / eLocation ID:
 4931 to 4946
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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