arXiv subjects

Hanzhang Yin

Publications and source records attributed to Hanzhang Yin.

8 recordsLinked to original sources

arXiv2026

On solitons of a geometric flow on Hessian manifolds

In this work, we introduce the Koszul solitons of the Hesse-Koszul flow, i.e., self-similar solutions for a geometric flow on Hessian manifolds. We obtain some partial differential equations and basic properties of Koszul solitons, and provide some examples. The key idea is to investigate the relationship between affine manifolds and complex manifolds, this method can also be used to study some other geometric problems on Hessian manifolds. It is worth noting that Hessian manifolds, as a special case of affine manifolds, have wide applications in statistics and information geometry.

math.DG↗

arXiv2026

Kleber's conjecture and complementary products of symmetric functions

We prove Kleber's rectangular-complement conjecture for Schur functions over an arbitrary commutative ring $R$, showing that, for a fixed rectangle, the products $s_\lambda s_{\lambda^\vee}$, indexed by unordered complementary pairs, are linearly independent in $\Lambda_R$. The proof rests on a general independence theorem for componentwise splittings, which asserts that for every partition $\theta$, the products $s_\alpha s_\beta$ are linearly independent as $\{\alpha,\beta\}$ ranges over unordered pairs of partitions satisfying $\alpha+\beta=\theta$. The independence of the products $s_\lambda s_{\lambda^\vee}$ also yields linear independence of the Koike--Terada universal-character products over any field, answering a question of Gao--Orelowitz--Yong. We also prove the analogous result for monomial symmetric functions over fields of characteristic zero, as well as integral linear independence over $\mathbb{Z}$.

math.CO↗

arXiv2026

On Ricci forms of canonical metrics over noncompact complex manifolds

In this paper, we study several types of geometric problems related to the Ricci curvature on noncompact complex manifolds, such as the existence of K\"{a}hler-Einstein metrics on complete K\"{a}hler manifolds with negative Ricci curvature, which can be seen as an improvement of the main theorem in Cheng-Yau [4]; the existence of canonical Hermitian metrics with prescribed Ricci curvature on complete Hermitian manifolds, which can be regarded as noncompact versions of the Gauduchon conjecture on certain complete complex surfaces. Our method can also be used to construct Hesse-Einstein metrics in affine differential geometry.

math.DG↗

arXiv2026

The record statistic and forward stability of Schubert products

We initiate a probabilistic study of forward stability for products of Schubert polynomials through the record statistic (left-to-right maxima) of permutations. Building on the explicit record formula for forward stability obtained by Hardt and Wallach, we study random pairs of permutations drawn from three natural families: uniform permutations, Grassmannian permutations, and Boolean permutations. For each family, we determine record probabilities and use them to analyze the asymptotic behavior of forward stability. For uniform and Grassmannian permutations, we obtain asymptotics for the mean together with limiting distribution results. For Boolean permutations, we prove linear-order growth of the mean, and our analysis also produces an explicit time-inhomogeneous Markov chain that yields an exact linear-time uniform sampler. Beyond these cases, we prove that the record-set statistic is equidistributed on the avoidance classes of $132$ and $231$, and consequently the corresponding forward stability distributions coincide. We conclude with conjectures for numerous further permutation classes and a conjectural recursive criterion for when two avoidance classes have the same record-set distribution.

math.CO↗

arXiv2025

Fully non-linear elliptic equations on noncompact complex manifolds

In this paper, we establish a priori estimates and existence results for solutions of a general class of fully non-linear equations on noncompact K\"{a}hler and Hermitian manifolds. As geometric applications, we construct complete K\"{a}hler metrics with prescribed volume forms on strictly pseudoconvex domains, as well as find Einstein metrics on complete noncompact K\"{a}hler manifolds and Hessian manifolds with negative first Chern class.

math.DG↗

arXiv2025

A non-iterative straightening algorithm and orthogonality for skew Schur modules

We generalize Fulton's determinantal construction of Schur modules to the skew setting, providing an explicit and functorial presentation using only elementary linear algebra and determinantal identities, in parallel with the partition case. Building on the non-iterative straightening formula of the first author for partition shapes, we develop a non-iterative straightening algorithm for skew Schur modules that expresses arbitrary elements in a new D-basis with an explicit closed coefficient formula. We then show that this D-basis is the result of applying Gram-Schmidt orthogonalization to the semistandard tableau basis, which identifies a natural inner product on the skew Schur module and recasts straightening as an orthogonal projection.

math.CO↗

arXiv2025

A topological rigidity theorem on noncompact Hessian manifolds

In this work, we obtain a short time solution for a geometric flow on noncompact affine Riemannian manifolds. Using this result, we can construct a Hessian metric with nonnegative bounded Hessian sectional curvature on some Hessian manifolds with nonnegative Hessian sectional curvature. Our results can be regarded as a real version of Lee-Tam \cite{LT20}. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature is diffeomorphic to $\mathbb{R}^n$ if its tangent bundle has maximal volume growth. This is an improvement of Theorem 1.3 in Jiao-Yin \cite{JY25}.

math.DG↗

arXiv2024

A geometric flow on noncompact affine Riemannian manifolds

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature and bounded geometry is diffeomorphic to $\mathbb{R}^n$ if its tangent bundle has maximal volume growth.

math.DG↗