arXiv research

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

Study of special Kato manifolds derived from toric geometry.

Proves Kato inequalities for various conformal operators.

This paper presents a new asymptotic expansion method for pricing continuously monitoring barrier options. In particular, we develops a semi-group expansion scheme for the Cauchy-Dirichlet problem in the second-order parabolic partial differential equations (PDEs) arising in barrier option pricing. As an application, w…

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

In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…

Proves Kato manifolds satisfy Hodge decomposition.

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

New criterion for wave operators on Kato-Ricci manifolds.

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.

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

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

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

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

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.

New inequalities for spectral zeta kernels on spheres and manifolds.

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…

We describe some relations between coefficients of irreducible components of the first Chern class [FP15] and birational germs introduced by Dloussky {Dl16] for intermediate Kato surfaces.

It is shown that if the Kato constant of the negative part of the Ricci curvature below a positive level is small, then the volume of the corresponding manifold can be bounded above in terms of the Kato constant and the total Ricci curvature. Together with the results from [5] and [6], this yields a generalization of t…

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…

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…

Some properties of non-orientable 3-manifolds are shown. The semi-group of cobordism of immersions of surfaces in such manifolds is computed and proven actually to be a group. Explicit invariants are provided.

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…

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…

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 $p$.

Study infinitesimal characters on semi-groups to prove interior properties and apply to Teichmüller spaces.

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.

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…

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…

Study of foliations' geometric and topological structures.

Generalizes complex manifolds to manifolds with corners and generalized corners.

Consider a sample of $n$ points taken i.i.d from a submanifold $Σ$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $Σ$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carré du Champ for diffusion semi-groups, the theory of Empirical process…

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We argue that the obtained formal s…

Study of stochastic differential equations on non-compact manifolds, solving open problem on strong completeness.

Witten deformation connects manifold spectra to Morse functions.

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

Study large deviations and speed of random walks in hyperbolic spaces.