arXiv research

Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.

New inequalities generalize Li's theorem on mixed Hodge structures.

Proves Hodge-Riemann relations for mixed valuations and strengthens geometric inequalities.

In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \ci…

Disproves Fedotov's conjecture on higher-order Shephard inequalities.

This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact Kähler manifold $(M,h)$ of even complex dimension $2m$, we prove that $σ(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),Γ}^{p,q}(M)$ where $σ(M)$ is the signature of $M$ and $h_{(2),Γ}^…

We use topological methods to prove a semicontinuity property of the Hodge spectra for analytic germs defined on an isolated surface singularity. For this we introduce an analogue of the Seifert matrix (the fractured Seifert matrix), and of the Levine--Tristram signatures associated with it, defined for null-homologous…

A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.

New method generalizes eigenvalue inequality to surfaces with boundaries.

Proves $L^2$ Frölicher inequality on complex manifolds.

We examine the relationship between nonabelian Hodge theory for Riemann surfaces and the theory of vector valued modular forms. In particular, we explain how one might use this relationship to prove a conjectural three-term inequality on the weights of free bases of vector valued modular forms associated to complex, fi…

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

Study spectral and index properties of Hodge-Dirac operator on compact manifolds.

Short note proves Poincaré inequality for 4-manifold forms.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

Eigenvalue estimates for weighted manifolds with applications.

The paper proves inequalities for twisted differential forms on manifolds.

Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.

Extends extension formulas for Hodge numbers on complex manifolds.

We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…

Proves Kato inequalities for various conformal operators.

Derives integral formula for differential forms on compact spaces with applications.

We establish in this note some Cauchy-Schwarz-type inequalities on compact Kähler manifolds, which generalize the classical Khovanskii-Teissier inequalities to higher-dimensional cases. Our proof is to make full use of the mixed Hodge-Riemann bilinear relations due to Dinh and Nguy$\hat{\text{e}}$n. A proportionality p…

Study finds eigenvalue bounds for non-convex domains using cohomology.

Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.

Study harmonic representatives and cohomology of Oeljeklaus-Toma manifolds.

Reformulates elasticity complex with new differential and Hodge star operators.

Proves hard Lefschetz theorem and Hodge-Riemann relations for convex valuations.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

We prove the equivariant holomorphic Morse inequalities for a holomorphic torus action on a holomorphic vector bundle over a compact Kahler manifold when the fixed-point set is not necessarily discrete. Such inequalities bound the twisted Dolbeault cohomologies of the Kahler manifold in terms of those of the fixed-poin…

The paper defines signatures for Witt spaces with boundary and proves their equality.

The study calculates braid indices for two-bridge knots and proves inequalities.

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

Study of generalized knots and links, proving inequality involving crossing number and braid index.

We prove the existence of a knot whose braid index the Morton-Franks-Williams inequality fails to detect but a related inequality (KR-MFW inequality), which uses new information of Khovanov-Rozansky homology, detects. We also prove, by examples, that there exists infinitely many knots for which the KR-MFW inequality fa…

In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In part…

Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.

Short note on braid index and quasipositivity of certain pretzel knots.

Study links' arc index and Turaev genus, proving conjectures.

Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge…

The study improves inequalities for link diagrams and introduces weak rectangular diagrams.

The study explores discrete versions of Riemannian geometry structures on manifolds.

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of ri…

In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.

Let $(M^m,g)$ be a m-dimensional complete Riemannian manifold which satisfies the n-Sobolev inequality and on which the volume growth is comparable to the one of $\R^n$ for big balls; if the Hodge Laplacian on 1-forms is strongly positive and the Ricci tensor is in $L^{\frac{n}{2}\pm ε}$ for an $ε>0$, then we prove a G…

Study Dirac operators on incomplete cusp edge spaces, proving self-adjointness and Fredholm properties.

A kinematic method selects the deformation Laplacian for fluid dynamics on Riemannian manifolds.