arXiv research

Study the Hessian geometry of an ideal gas in a centrifuge.

New metric for probability measures connects physics and geometry.

The target space geometry of abelian vector multiplets in ${\cal N}= 2$ theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…

Study on manifolds with density using modified Hessians for curvature comparison.

Curved Frobenius manifolds link to Hessian metrics in geometry.

We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry is an odd-dimensional counterpart of the Kähler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous to the one between Hessian and Kähler manifolds. In particular, we construct a Sasaki…

We extend the correspondence between Hessian and Kähler metrics and curvatures to Lagrange spaces.

We study the Hessian geometry of toric multi-Taub-NUT metrics and their phase change phenomena via the images of their moment maps. This generalizes an earlier paper on toric Gibbons-Hawking metrics.

Solves critical LYZ equation in Kähler geometry.

A family of probability distributions parametrized by an open domain $Λ$ in $R^n$ defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannia…

The paper quantizes Hessian structures on R^2 using KV-algebras.

Study of pseudo-Riemannian metrics related to Monge-Ampère structures.

The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.

We compute the Hessian of quantized Ding functionals and give an elementary proof for the convexity of quantized Ding functionals along Bergman geodesics from the view point of projective geometry. We study also the asymptotic behavior of the Hessian using the Berezin-Toeplitz quantization.

Optimizes quadratic bandits with tight Hessian-dependent sample complexity bounds.

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

Study of contravariant pseudo-Hessian manifolds and their Poisson structures.

New geometric structures derived from Hessian metrics and tensors.

The paper explores the geometric structure of cost functions in multiple dimensions.

Paper finds local normal forms for wavefronts in flat coordinates.

New Hessian estimators for Riemannian manifolds with reduced bias.

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

New approach connects Finsler geometry's metric and connections.

The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surf…

Introduces Legendre bundle for dually flat manifolds and quantum field theories.

Paper analyzes latent space geometry in generative models using Fisher information.

CWGD measures gradient diversity weighted by curvature, improving SGD convergence.

The paper proves properties of geometric flows on noncompact manifolds.

Author presents the second variational formula for statistical biharmonic maps.

The paper defines Hesse solitons and explores their properties on Hessian manifolds.

We study the Kepler metrics on Kepler manifolds from the point of view of Sasakian geometry and Hessian geometry. This establishes a link between the problem of classical gravity and the modern geometric methods in the study of AdS/CFT correspondence in string theory.

The paper studies the smoothness of critical points of variational integrals on Hessian spaces.

The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.

Derives estimates for geometric elliptic equations on complex manifolds.

Recently, the first named author together with Xinan Ma \cite{ma2015neumann}, have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniforml…

Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.

Study the geometry of a Lie group using Hessian and flat affine structures.

The study explores generalized divergences and exponential families with a focus on sufficient conditions and laws of large numbers.

In this brief survey, we will remark the interaction among the Hessian tensor on a semi-Riemannian manifold and some of the several questions in Lorentzian (and also in semi-Riemannian) geometry where this 2-covariant tensor is involved. In particular, we deal with the characterization of Killing vector fields and the …

The paper finds convex hypersurfaces with specific curvature properties.

New method computes affine normal directions efficiently for sparse polynomials.

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …

Introduces a new geometric structure for statistical manifolds with degenerate metrics.

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is o…

New method proves asymptotic normality for matrix sensing problems.

Noise injection regularizes Hessian, improving neural network training and generalization.

Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.