arXiv research

Study connects Chern class components to Kato surface germs.

We show that every Kato surface (or surface with a global spherical shell) admits a locally conformally Kaehler metric.

New examples show strong Kato limits can be branching and not satisfy known conditions.

We investigate bi-Hermitian metrics on compact complex surfaces with odd first Betti number producing new examples with connected anti-canonical divisor using the general construction of \cite{abd15}. The result is a complete classification for all \it unbranched \rm Kato surfaces and a classification up to logarithmic…

Study of special Kato manifolds derived from toric geometry.

New inequalities for spectral zeta kernels on spheres and manifolds.

Study of Kato manifolds and their locally conformally Kähler properties.

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb Q)$. We present a way to compute the associated rational coefficients of the first Chern class $c_1(S)$. We get in particular a simple geometric obstruction for $c_1(S)$ to be an integral class, or equivalently index$(S)=1$. We also f…

Proves Kato inequalities for various conformal operators.

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.

Study shows existence and uniqueness of periodic pseudospherical surfaces from Cauchy problems.

Proves Kato manifolds satisfy Hodge decomposition.

Limits of manifolds with Kato bound on Ricci curvature are rectifiable.

Generalizes Bonnet-Myers theorem with small Kato constant.

New criterion for wave operators on Kato-Ricci manifolds.

The study connects Kato bounds to finite-dimensional RCD spaces.

In this paper we study the Kato' inequality on locally finite graph. We also study the application of Kato inequality to Ginzburg-Landau equations on such graphs. Interesting properties of Schrodinger equation and a Liouville type theorem are also derived.

Study on Lee classes of complex surfaces, proving connectedness and bounds.

Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.

Study proves Kato inequality leads to definite 4-manifolds with positive curvature.

We prove a refined Kato inequality for closed and coclosed differential $(p,q)$ forms on a Kahler manifold.

Locally convex classes on manifolds linked to Ricci curvature bounds.

This article shows that if the negative part of Ricci curvature lies in the Kato class, the heat kernel satisfies a Li-Yau type estimate. Additionally, using the resulting heat kernel bound, we show that the obtained heat kernel estimate leads to bounds on the first Betti number only depending on the Kato constant.

We review recent results about heat kernel estimates based on Kato conditions on the negative part of the Ricci curvature.

Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.

We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theo…

The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.

By introducing the concept of \emph{Kato control pairs} for a given Riemannian minimal heat kernel, we prove that on every Riemannian manifold $(M,g)$ the Kato class $\mathcal{K}(M,g)$ has a subspace of the form $\mathsf{L}^q(M,d\varrho)$, where $\varrho$ has a continuous density with respect to the volume measure $μ_g…

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with $…

Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-Δ$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\nabla^*\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bou…

We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti number estimates for closed manifolds whose Ricci curvature is controlled in th…

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.

Study on metric spaces with properties (ETR), (LBD) and their convergence.

Sharp inequality for $p$-harmonic maps with new optimal constant.

Researchers extend regularity of $p$-harmonic maps into spheres for a new range of $p$.

New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which gives the value of the constants appearing in the refined inequalities. These consta…

Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.

Paper estimates curvature of minimal surfaces in a specific geometric space.

We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $ρ$ is such that the Schrödinger operator $(n-2)Δ+ ρ$ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequa…

Proves ε-regularity for capillary surfaces in Riemannian manifolds.

Study of foliations' geometric and topological structures.

Generalizes complex manifolds to manifolds with corners and generalized corners.

New method proves regularity for small energy solutions of Yang-Mills-Higgs equations.

Witten deformation connects manifold spectra to Morse functions.

New framework for higher-order singular-value derivatives of rectangular matrices.

This short note shows how the Novikov conjecture for mapping class groups follows from a theorem of Kato and a result theorem of Hamenstadt.