Study of special Kato manifolds derived from toric geometry.
problem Characterize and study properties of Kato manifolds.
method Construction from toric geometry, topological and analytical properties, combinatorial data, flat degenerations, Hermitian geometry.
result No Kato manifold supports balanced or pluriclosed metrics.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
Study connects Chern class components to Kato surface germs.
problem Understanding intermediate Kato surfaces through Chern class components.
method Relates coefficients of Chern classes to birational germs.
result Established relations between Chern class components and Kato surface germs.
Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.
The study examines heat kernel bounds for manifolds with Ricci curvature in the Kato class.
problem Heat kernel estimates for manifolds with Ricci curvature in the Kato class.
method Kato conditions on the negative part of the Ricci curvature.
result Recent results on heat kernel estimates.
Proves Kato manifolds satisfy Hodge decomposition.
problem Proving Hodge decomposition for Kato manifolds.
method Relating cohomology to modification data and studying Bott-Chern and Aeppli cohomology.
result Kato manifolds satisfy Hodge decomposition.
Limits of manifolds with Kato bound on Ricci curvature are rectifiable.
problem Understanding limits of manifolds with specific curvature bounds.
method Proving rectifiability of limits of manifolds with Kato bound on Ricci curvature.
result Rectifiability of limits of manifolds with Kato bound on Ricci curvature.
Generalizes Bonnet-Myers theorem with small Kato constant.
problem Bounding volume of manifolds with small Kato constant.
method Analyzes Ricci curvature in Kato sense and applies to volume bounds.
result Generalizes Bonnet-Myers theorem with new conditions.
The article proves a gradient estimate for manifolds with negative Ricci curvature in the Kato class.
problem Estimating the first Betti number of manifolds with negative Ricci curvature.
method Using the heat kernel and Li-Yau type estimates, the first Betti number is bounded by the Kato constant.
result The first Betti number of manifolds with negative Ricci curvature in the Kato class is bounded by the Kato constant.
New criterion for wave operators on Kato-Ricci manifolds.
problem Existence and completeness of wave operators for Laplace-Beltrami operators.
method Proves L1 criterion using heat semigroup estimates for Kato-Ricci manifolds. result Establishes new conditions for wave operators on Kato-Ricci manifolds.
New examples show strong Kato limits can be branching and not satisfy known conditions.
problem Exploring the boundaries of strong Kato limits and their properties.
method Constructing specific examples of non-collapsed strong Kato limits.
result Found examples of strong Kato limits that are branching and do not satisfy CD(K,∞) or MCP(K,N) conditions. The study connects Kato bounds to finite-dimensional RCD spaces.
problem Understanding the limits of complete Riemannian manifolds with Kato bounds.
method Using the transformation rule of the Bakry-Émery condition under time change.
result Bi-Lipschitz equivalence 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.
Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.
problem Existence of weak fans for non-classical period maps of weight 3 Calabi-Yau type.
method Modified Kato-Nakayama-Usui construction for period maps of weight 3 Calabi-Yau type.
result Existence of weak fans for a large class of period maps of weight 3 Calabi-Yau type.
We show that every Kato surface (or surface with a global spherical shell) admits a locally conformally Kaehler metric.
Study proves Kato inequality leads to definite 4-manifolds with positive curvature.
problem Positive sectional curvature and Kato type inequalities on 4-manifolds.
method Analyzes Kato type inequalities on compact Riemannian 4-manifolds with positive sectional curvature.
result Proves compact Riemannian 4-manifolds satisfying a Kato type inequality are definite.
Locally convex classes on manifolds linked to Ricci curvature bounds.
problem Characterizing Kato and Dynkin classes on manifolds with Ricci curvature bounds.
method Using recent results on spectral negative parts and Gaussian heat kernel bounds, the study establishes local convexity of these classes.
result Local Kato and Dynkin classes are independent of the metric and smooth compactly supported functions are dense in the local Kato class.
We prove a refined Kato inequality for closed and coclosed differential (p,q) forms on a Kahler manifold.
The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.
problem Proving isoperimetric inequalities in manifolds with small negative Ricci curvature.
method Expanding on the ABP method, the paper uses the elliptic Kato constant to control the non-negativity of the Ricci-tensor and applies techniques from Li-Tam and Kasue.
result Sharp isoperimetric inequalities in the limit are proven in the presence of small negative curvature.
New inequalities for spectral zeta kernels on spheres and manifolds.
problem Establishing new inequalities for spectral zeta functions.
method Applying Kato's inequalities and majorisation techniques.
result Generalized Kato's comparison inequalities to higher dimensions.
Researchers extend regularity of p-harmonic maps into spheres for a new range of p.
problem Establishing regularity of p-harmonic maps for a broader range of p. method Combining Morrey's methods with Hardt and Lin's Extension Theorem, and proving a sharp Kato inequality.
result Regularity for p∈[2.961,3] and p∈[2,p0] with p0≈2.366. Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
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.
problem Eigenvalue estimates for manifolds with Ricci curvature conditions.
method Proves eigenvalue estimates using a Kato condition on the negative part of Ricci curvature.
result Optimal eigenvalue estimates for Zhong-Yang type and Cheng-type bounds.
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 K(M,g) has a subspace of the form Lq(M,dϱ), where ϱ has a continuous density with respect to the volume measure $μ_g…
Study of Kato manifolds and their locally conformally Kähler properties.
problem Characterize Kato manifolds and their locally conformally Kähler metrics.
method Revisit Brunella's proof and construct new examples of Kato manifolds.
result Found a class of Kato manifolds that admit locally conformally Kähler metrics and another class that do not.
New method proves regularity for small energy solutions of Yang-Mills-Higgs equations.
problem Proving regularity for small energy solutions of Yang-Mills-Higgs equations.
method Improved Kato inequality and Weitzenböck formulae.
result Obtains bounded curvature without Coulomb gauges.
The paper derives geometric and spectral inequalities using Ricci curvature assumptions.
problem Estimating geometric and spectral properties of Riemannian manifolds under Ricci curvature conditions.
method Using spectral and Kato conditions on Ricci curvature, the paper derives inequalities.
result The lowest eigenvalue of the Ricci tensor leads to finite fundamental group and isoperimetric inequalities.
Let (M,g) be a Riemannian manifold with Laplace-Beltrami operator −Δ and let E→M be a Hermitian vector bundle with a Hermitian covariant derivative ∇. Furthermore, let H(0) denote the Friedrichs realization of ∇∗∇ and let V be a potential. We prove that V− is H(0)-form bounded with bou…
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…
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.
problem Understanding orientability and convergence of metric measure spaces.
method Analysis of Gromov-Hausdorff and intrinsic flat convergence for spaces satisfying (ETR), (LBD).
result The pointed Gromov-Hausdorff limit coincides with the local flat limit.
Sharp inequality for p-harmonic maps with new optimal constant.
problem Deriving the sharp vectorial Kato inequality for p-harmonic mappings. method Analyzing the inequality for p-harmonic mappings and comparing with scalar valued cases. result Established the optimal constant for p-harmonic maps and enhanced the range of p values for regularity. New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
problem Vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
method Utilized refined Kato type inequalities and Böchner technique to generalize results to Lp-integrable pluriharmonic functions and harmonic 1-forms. result Proved vanishing property of pluriharmonic functions with finite Lp energy on complete 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.
problem Understanding structure of limits of manifolds with Ricci curvature bounds.
method Mosco convergence of Dirichlet energies to Cheeger energy, introduction of monotone quantities, volume convergence.
result Volume convergence to Hausdorff n-measure in limits of manifolds.
Study on manifolds with controlled Ricci curvature, focusing on volume and eigenvalue estimates.
problem Analyzing geometric properties of manifolds with specific curvature conditions.
method Use Euclidean volume growth, Sobolev inequalities, and spectral conditions on Ricci curvature.
result Eigenvalue and heat kernel estimates for closed manifolds with controlled Ricci curvature.
The compact curves of an intermediate Kato surface S form a basis of H2(S,Q). We present a way to compute the associated rational coefficients of the first Chern class c1(S). We get in particular a simple geometric obstruction for c1(S) to be an integral class, or equivalently index(S)=1. We also f…
We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is based on the r…
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Generalizes complex manifolds to manifolds with corners and generalized corners.
problem Tackles the extension of complex structures to manifolds with corners and generalized corners.
method Uses complex structures on the b-tangent bundle and proves a formal Newlander-Nirenberg type theorem.
result Proves that along each corner stratum, the b-complex structure agrees with a standard model to infinite order.
The paper studies eigen-solutions on Riemannian manifolds and their growth as distance increases.
problem Analyzing the growth of eigen-solutions on Riemannian manifolds.
method Developed Kato's methods on manifolds to study the asymptotic behaviors of eigen-solutions.
result Established the growth of eigen-solutions as distance increases.
Witten deformation connects manifold spectra to Morse functions.
problem Understanding spectral properties of Riemannian manifolds.
method Rellich-Kato theorem applied to Witten deformation.
result Relates spectral package to Morse complex and harmonic oscillators.
New framework for higher-order singular-value derivatives of rectangular matrices.
problem Challenging to derive higher-order Fréchet derivatives of singular values in real rectangular matrices.
method Using Kato's analytic perturbation theory for self-adjoint operators and embedding rectangular matrices into block self-adjoint operators.
result Closed-form expressions for the n-th order spectral variations of singular values. This paper derives a new semi closed-form approximation formula for pricing an up-and-out barrier option under a certain type of stochastic volatility model including SABR model by applying a rigorous asymptotic expansion method developed by Kato, Takahashi and Yamada (2012). We also demonstrate the validity of our app…
This short note shows how the Novikov conjecture for mapping class groups follows from a theorem of Kato and a result theorem of Hamenstadt.
Study proves NN matching is equivalent to Riesz regression for debiased machine learning.
problem Addressing bias in machine learning models.
method Interprets NN matching as Riesz regression and derives it from LSIF.
result NN matching is shown to be equivalent to Riesz regression.
Study shows Dehn twists on certain 4-manifolds cannot be realized by finite order diffeomorphisms.
problem Realization of Dehn twists as finite order diffeomorphisms on spin 4-manifolds.
method Use Y. Kato's 10/8-type inequality for involutions and its refinement.
result Dehn twists about specific spheres in certain 4-manifolds are not homotopic to finite order diffeomorphisms.