The Second Variational Formula for Statistical Biharmonic Maps

2025/09/05 by Ueno, Ryu
#53A15 #53B12 #53C43 #58E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.04807

Abstract

Recently, the statistical bi-energy functional and its first variational formula were introduced by the author and H. Furuhata. The Maps satisfying the corresponding Euler-Lagrange equation are called statistical biharmonic maps. We present the second variational formula for the statistical bi-energy functional and introduce the notion of stability. If the target statistical manifold is a Hessian manifold, it turns out that the second variational formula can be represented using the Hessian curvature. We provide examples of statistical biharmonic maps into Hessian manifolds that are significant in Hessian and information geometry. We also determine the stability of these statistical biharmonic maps, including the improper affine sphere.

Citations

• Stable proper biharmonic maps in Euclidean spheres
• Morse Index Stability of Biharmonic Maps in Critical Dimension
• A Variation Problem for Mappings between Statistical Manifolds
• On a constant curvature statistical manifold
• A characterization of the alpha-connections on the statistical manifold of normal distributions
• A note on equivariant biharmonic maps and stable biharmonic maps
• Biharmonic maps into a Riemannian manifold of non-positive curvature
• Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature
• On the generalized Chen's conjecture on biharmonic submanifolds

Related

• A Variation Problem for Mappings between Statistical Manifolds
• The second variational formula of the variational problems for mappings between statistical manifolds
• Statistical Biharmonicity of Identity Maps
• Energy identity of approximate biharmonic maps to Riemannian manifolds and its application
• The biharmonic stress-energy tensor and the Gauss map