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On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

Let ( M , g ) be a compact Riemannian manifold and P g an elliptic, formally self-adjoint, conformally covariant operator of order m acting on smooth sections of a bundle over M . We prove that if P g has no rigid eigenspaces (see Definition 2.2), the set of functions f ∈ C ∞ ( M , ℝ ) for which P e f g has only simple non-zero eigenvalues is a residual set in C ∞ ( M , ℝ ) . As a consequence we prove that if P g has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the C ∞ -topology....

On the nonexistence of stable minimal submanifolds and the Lawson-Simons conjecture

Ze-Jun Hu, Guo-Xin Wei (2003)

Colloquium Mathematicae

Let M̅ be a compact Riemannian manifold with sectional curvature K M ̅ satisfying 1 / 5 < K M ̅ ≤ 1 (resp. 2 ≤ K M ̅ < 10 ), which can be isometrically immersed as a hypersurface in the Euclidean space (resp. the unit Euclidean sphere). Then there exist no stable compact minimal submanifolds in M̅. This extends Shen and Xu’s result for 1/4-pinched Riemannian manifolds and also suggests a modified version of the well-known Lawson-Simons conjecture.

On the normality of an almost contact 3 -structure on Q R -submanifolds

Shoichi Funabashi, Jin Suk Pak, Yang Jae Shin (2003)

Czechoslovak Mathematical Journal

We study n -dimensional Q R -submanifolds of Q R -dimension ( p - 1 ) immersed in a quaternionic space form Q P ( n + p ) / 4 ( c ) , c ≧ 0 , and, in particular, determine such submanifolds with the induced normal almost contact 3 -structure.

On the notion of Jacobi fields in constrained calculus of variations

Enrico Massa, Enrico Pagani (2016)

Communications in Mathematics

In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called second variation. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as extremaloids. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of...

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S 3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

On the Paneitz energy on standard three sphere

Paul Yang, Meijun Zhu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We prove that the Paneitz energy on the standard three-sphere S3 is bounded from below and extremal metrics must be conformally equivalent to the standard metric.

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