Abstract The Hessian of the entropy function can be thought of as a metric tensor on state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study these geometries for some families of black holes and find tha...

We present a second order numerical scheme to compute capillary bridges between arbitrary solids by minimizing the total energy of all interfaces. From theoretical point view, this approach can be interpreted as computation generalized minimal surfaces using Newton-scheme utilizing shape Hessian. In particular, we give an explicit representation Hessian for functionals on shells involving norma...

We revisit the problem of prescribing negative Gauss curvature for graphs embedded in $ \mathbb R^{n+1} when n\geq 2 $. The reduces to solving a fully nonlinear Monge–Ampère equation that becomes hyperbolic case curvature. show linearization around graph with Lorentzian Hessian can be written as geometric wave suitable metric dimensions 3 Using energy estimates linearized and version Nash–Moser...

In this article, we obtain bounds for Ricci curvature doubly warped products pointwise bi-slant submanifolds in generalized complex space forms and discuss the equality case of inequality. We also derive non-existence such immersions. Finally, construct some applications result terms Harmonic function, Hessian tensor, Dirchilet energy function.

The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that structure a para-Kähler-like manifold constant curvature in Kurose's sense Hessian structure. also derive properties submersions which are compatible results illustrated by several nontrivial examples.

A Finsler metric is of sectional flag curvature if its flag curvature depends only on the section. In this article, we characterize Randers metrics of sectional flag curvature. It is proved that any non-Riemannian Randers metric of sectional flag curvature must have constant flag curvature if the dimension is greater than two. 0. Introduction Finsler geometry has a long history dated from B. Ri...

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in [Ch1]. This new inequality is shown to be connected with Perelman’s entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate a...

The state space of a finite quantum system is identified with the set of positive semidefinite matrices of trace 1. The set of all strictly positive definite matrices of trace 1 becomes naturally a differentiable manifold and the Bogoliubov-Kubo-Mori scalar product defines a Riemannian structure on it. Reference [4] tells about the relation of this metric to the von Neumann entropy functional. ...