Hessian Manifolds Research Articles

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Convergence of the Hesse–Koszul flow on compact Hessian manifolds

We study the long time behavior of the Hesse–Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse–Einstein metric. We also derive a convergence result for a twisted Hesse–Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse–Einstein metric by Cheng–Yau and Caffarelli–Viaclovsky as well as the real Calabi theorem by Cheng–Yau, Delanoë and Caffarelli–Viaclovsky.

General Chen Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ(2,2)-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end.

Geodesic Walks in Polytopes

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induced by the Hessian of a convex function; each step is the solution of an ordinary differential equation (ODE). The resulting sampling algorithm for polytopes mixes in O*(mn^{3/4}) steps. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of O*(mn) by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes. Our analysis of the geodesic walk for general Hessian manifolds does not assume positive curvature and might be of independent interest.

Selfsimilar Hessian manifolds

A selfsimilar manifold is a Riemannian manifold (M,g) endowed with a homothetic vector field ξ. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian manifold or a Euclidean space. A radiant Hessian manifold is a selfsimilar Hessian manifold (M,∇,g,ξ) such that ∇ξ=λId, where λ is a constant. We prove that any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds and describe the local structure of radiant selfsimilar Hessian manifolds.

Self-similar solutions to the Hesse flow

We define a Hesse soliton, that is, a self-similar solution to the Hesse flow on Hessian manifolds. On information geometry, the e-connection is important, which does not coincide with the Levi–Civita one. Therefore, it is interesting to consider a Hessian manifold with a flat connection which does not coincide with the Levi–Civita one. We call it a proper Hessian manifold. In this paper, we show that any compact proper Hesse soliton is expanding and any non-trivial compact gradient Hesse soliton is proper. Furthermore, we show that the dual space of a Hesse–Einstein manifold can be understood as a Hesse soliton.

Wintgen inequality for surfaces in Hessian manifolds

Wintgen inequality for surfaces in Hessian manifolds

The δ(2,2)-Invariant on Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature.

We establish Chen inequality for the invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Recently, in co-operation with Chen, we proved a Chen first inequality for such submanifolds. The present authors previously initiated the investigation of statistical submanifolds in Hessian manifolds of constant Hessian curvature; this paper represents a development in this topic.

A Chen First Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

The study of statistical submanifolds in Hessian manifolds of constant Hessian curvature was started by two of the present authors. We continue this work and establish a Chen first inequality for such submanifolds.

From Hessian to Weitzenböck: manifolds with torsion-carrying connections

We investigate affine connections that have zero curvature but not necessarily zero torsion. Slightly generalizing from what is known as Weitzenbock connections, such non-flat connections (we call “pseudo-Weitzenbock connections”) can be constructed from any frame (a set of linearly independent vector fields), not just the orthonormal frames, with their torsions vanishing if and only if the frame is a coordinate frame. In such situations, the notion of biorthogonal frames generalizes the notion of biorthogonal coordinates of a Hessian manifold, with respect to any given Riemannian metric g (not necessarily Hessian). Our main theorem shows that the pair of pseudo-Weitzenbock connections, each adapted to one of the pair of g-biorthogonal frames, are g-conjugate to each other. As a result, the pseudo-Weitzenbock connection pair generalize dually flat connections characteristic of Hessian manifolds, by being both curvature-free yet admitting (generally unequal) torsions. These results allow us to construct a pseudo-Weitzenbock connection for the manifold of parametric statistical models and treat it as a “statistical manifold admitting torsion”.

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

We show vanishing theorems of $L^2$-cohomology groups of Kodaira–Nakano type on complete Hessian manifolds by introducing a new operator $\partial'_F$. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}_{\bar{\partial}}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng–Yau metric for $p>q$.

An Optimal Transport Approach to Monge\u2013Amp\xe8re Equations on Compact Hessian Manifolds

In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.

Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold.

Curvature Inequalities between a Hessian Manifold with Constant Curvature and its Submanifolds

In this paper after a short description of Hessian manifolds, we establish new curvature inequalities between a Hessian manifold and its submanifolds.

Long-time existence of a geometric flow on closed Hessian manifolds

In this paper we study Hessian manifolds. We define a flow, which we call the Hessian flow, to study the existence of Einstein–Hessian metrics on Hessian manifolds. The flow will considered as a real Monge–Ampére equation and we prove the short-time existence, the global existence and the uniqueness of it.

Some Curvature Characterizations for Hessian Manifolds with Constant Hessian Sectional Curvature

N , Hessian manifoldunun kompakt bir hiperyuzeyi olsun. Once Hessian manifold konsepti hakkinda bilgi verecegiz sonra ikinci temel tensoru kullanarak sabit Hessian kesit egrilikli Hessian manifoldlari icin bazi egrilik karakterizasyonalari elde edecegiz

Curvature of Hessian manifolds

We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all analytic Riemannian 2-metrics are Hessian metrics.

Hessian manifolds of nonpositive constant Hessian sectional curvature

We classify the maximal Hessian manifolds of constant Hessaian sectional curvature nonpositive.

ON DUAL HOLOMORPHIC B-TYPE HESSIAN METRICS

The main purpose of the present paper is to construct some B-type Hessian manifolds.