New metric for probability measures connects physics and geometry.
problem Developing a new metric for probability measures.
method Transport Hessian metric, formulated dynamical systems.
result Connections to physics equations and mathematical models.
The paper describes flat Hessian metrics on surfaces and their potentials.
problem Understanding Hessian metrics on surfaces.
method Theoretical description and explicit construction using integrable systems.
result Explicit construction of potentials for flat Hessian metrics on surfaces.
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 anal…
Curved Frobenius manifolds link to Hessian metrics in geometry.
problem Understanding curved Frobenius manifolds and their relation to Hessian metrics.
method Analyzing the relationship between curved Frobenius structures and Hessian metrics on spaces with non-vanishing curvature.
result Consistent curved Frobenius structures on constant curvature spaces are linked to Hessian metrics.
Kunneth formula derived for flat affine manifolds and applied to Hessian metrics.
problem Cohomology of flat affine manifolds and Hessian metrics.
method Proved Künneth formula for Dolbeault--Koszul cohomology of flat affine manifolds and applied to Hessian metrics.
result Kunneth formula for Hessian metrics on products and manifolds with hyperbolic manifolds.
The paper introduces a new system of equations for Hessian-cscK metrics.
problem Finding constant scalar curvature Kähler metrics.
method Proposes a coupled system of complex Hessian equations and shows it can be variational.
result Proves a C0-estimate for the system that depends on entropy. New formulas for Riemannian gradient and Hessian on manifold metrics.
problem Evaluate Riemannian gradient and Hessian for various metrics on manifolds.
method Explicit formulas derived from Levi-Civita connection and projection.
result Derives new metrics and optimization frameworks on manifolds.
New geometric structures derived from Hessian metrics and tensors.
problem Constructing geometric structures from Hessian metrics and tensors.
method Introduced a family of Hessian metrics and constructed a (1,1)-tensor field. result Derived golden and metallic structures from the tensor field.
We study the Hessian geometry of toric Gibbons-Hawking metrics and their phase change phenomena via the images of their moment maps.
Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.
problem Rigidity of Ricci curvature on Hessian manifold leaves.
method Analysis of Ricci curvature properties of Hessian metrics on foliation leaves.
result Non-negative Ricci curvature on a single leaf forces the Hessian metric to be flat and yields bounds on the first Betti number.
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.
We extend the correspondence between Hessian and Kähler metrics and curvatures to Lagrange spaces.
Study of pseudo-Riemannian metrics related to Monge-Ampère structures.
problem Properties of pseudo-Riemannian metrics and Hessian structures.
method Analysis of Monge-Ampère structures and Ricci flat solutions.
result Description of a family of Ricci flat solutions parametrized by six coefficients.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
The paper studies hybrid connections on Hessian manifolds and their properties.
problem Investigating hybrid connections on Hessian manifolds.
method Defining and analyzing hybrid connections as incompressible affine connections projective to a flat connection D. result The difference abla−D is determined by the logarithmic differential of a Hessian potential function. Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.
problem Understanding the relationship between parameter magnitudes and Hessian curvature in deep learning models.
method Developed a matrix-free algorithm based on sketched SVDs to measure similarity between parameter masks and Hessian eigenspaces.
result Top Hessian eigenvectors tend to be concentrated around larger parameters, indicating a connection between parameter magnitudes and loss curvature.
Study classifies special metrics on specific surfaces.
problem Classifying extremal Kähler metrics with singularities.
method Analyzes Hessian of the Curvature of the Metric on K-surfaces.
result Identifies non-CSC HCMU metrics on S{α}2 and S{α,β}2. Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.
problem Establishing mixed Hessian inequalities on Hermitian manifolds.
method Weak convergence theorem of complex Hessian operators and general mixed Hessian inequality.
result Existence of bounded solutions of complex Hessian equations.
Locally conformally Hessian manifolds are dense in radiant ones of rank 1.
problem Characterizing locally conformally Hessian manifolds and their properties.
method Analyzing quotient spaces of Hessian manifolds and using statistical manifold theory.
result The set of radiant l.c.H. metrics of rank 1 is dense in all radiant l.c.H. metrics.
Study of manifolds with flat connections and diagonal metrics leading to vanishing Euler characteristic.
problem Understanding the geometry and topology of affine-orthogonal manifolds.
method Deformation of flat connections into Levi-Civita connections and analysis of Euler characteristic.
result Deformations force the Euler characteristic to vanish, supporting Chern's conjecture.
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 g…
Adapts PDE method to prove L∞ estimates for complex Hessian equations.
problem Proving L∞ estimates for complex Hessian equations on transverse Kähler manifolds. method Adapts PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18].
result Obtains L∞ estimate for transverse complex Monge-Ampère equations. The paper explores the geometric structure of cost functions in multiple dimensions.
problem Understanding the geometric properties of cost functions in multidimensional settings.
method Analyzes the Hessian metric and geodesics in logarithmic and original coordinates.
result The geometry is one-dimensional in logarithmic coordinates but effectively (n−1)-dimensional in original coordinates. We characterize those complete K{ä}hler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.
New proof for stability estimates in complex equations without pluripotential theory.
problem Stability estimates for complex Monge-Ampère and Hessian equations.
method New proof using general degenerations of background metrics.
result Uniform stability estimates for both equations under various degenerations.
Proves triviality and nonexistence of gradient Ricci solitons as warped metrics.
problem Proving triviality and nonexistence of gradient Ricci solitons as warped metrics.
method Proved through the construction of gradient Ricci solitons as warped products and studying Ricci-Hessian type manifolds.
result Gradient Ricci solitons are trivial and non-existent as warped metrics.
New Einstein metrics identified on a specific full flag manifold.
problem Investigating Einstein metrics on a full flag manifold.
method Analyzing G-stability of Einstein metrics on M=G/K. result Identified four new Einstein metrics on F(5), confirming pairwise non-homotheticity. In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold (M,g) corresponding to e…
Hessian metrics on a sphere with specific properties.
problem Existence and properties of Hessian metrics on a sphere with singularities.
method Construction of pseudo-metric tensor with distribution coefficients.
result Existence of Hessian metrics on a sphere with specific properties.
The paper proves conjectures about Minkowski norms with specific symmetry groups.
problem Proving conjectures about Minkowski norms with certain symmetries.
method Analyzing isometries of the Hessian metric for Minkowski norms invariant under SO(k)imesSO(n−k). result Proves Laugwitz and Landsberg Unicorn conjectures for Minkowski norms with the specified symmetry.
Researchers solve metric curvature equations on manifolds with boundary.
problem Finding complete conformal metrics with specific curvature functions.
method Revealed algebraic structure of fully nonlinear equations; used topological obstructions.
result Solved a class of fully nonlinear equations for conformal metrics.
The Ricci Calabi functional is a functional on the space of Kähler metrics of Fano manifolds. Its critical points are called generalized Kähler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized Kähler Einstein metrics. As its application, we give a…
New approach connects Finsler geometry's metric and connections.
problem Deriving Finsler geometry's metric and connections from compatibility axioms.
method Compatibility axioms between metric and Finsler connection.
result Metrical formulation of Finsler geometry for field theory.
We establish an unexpected relation among the Weil-Petersson metric, the generalized Hodge metrics and the BCOV torsion. Using this relation, we prove that certain kind of moduli spaces of polarized Calabi-Yau manifolds do not admit complete subvarieties. That is, there is no complete family for certain class of polari…
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global lower bound.
We show vanishing theorems of L2-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of L2-cohomology groups L2Hp,q(Ω) on a regular convex cone Ω with the Cheng-Yau metric for p>q.
This paper proves a Nakai-Moishezon criterion for complex Hessian equations.
problem The solvability of complex Hessian equations on Kähler manifolds.
method Establishing a Nakai-Moishezon criterion for Kähler classes on analytic Kähler varieties.
result Proves Lejmi-Szekelyhidi's conjecture for the J-equation. We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this …
Study equivalence between Hessian and Born structures on tangent bundles.
problem Equivalence between Hessian and Born structures on tangent bundles.
method Analyzing conditions for Hessian structures and integrability of induced almost Born structures.
result Conditions for equivalence between Hessian and Born structures are established.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a f…
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
Study proves structure results for homogeneous spaces supporting specific equations.
problem Proving structure results for homogeneous spaces supporting specific equations.
method Analyzing homogeneous spaces with non-constant solutions to two general classes of equations involving the Hessian and an invariant 2-tensor.
result Generalizes rigidity results for gradient Ricci solitons and warped product Einstein metrics.
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
Theory developed for complex Hessian measures on Hermitian manifolds.
problem Defining and analyzing complex Hessian measures on Hermitian manifolds.
method Potential theory for m-subharmonic functions with respect to a Hermitian metric.
result Equivalence between polar sets and negligible sets for m-subharmonic functions.
Paper finds local normal forms for wavefronts in flat coordinates.
problem Understanding local diffeomorphic types of wavefronts.
method Using connections and the metric, criteria for wavefront types are derived in affine flat coordinates.
result Local normal forms of e/m-wavefronts in affine flat coordinates are derived.