arXiv research

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we…

Explicit formula for Reidemeister torsion of two-bridge knots.

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space $\mathbb{H}^{4}$ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and …

Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcompl…

For a compact oriented 3-manifold with torus boundary the adjoint Reidemeister torsion is defined as a function on the $\mathrm{SL}_2(\mathbb{C})$-character variety depending on a choice of a boundary curve. Under reasonable assumptions, it is conjectured that the adjoint torsion satisfies a certain type of vanishing i…

The paper proves injectivity and vanishing theorems on compact Kahler manifolds.

Techniques of gauge theory are used to define and compute an invariant of certain diffeomorphisms of 4-manifolds. The invariant vanishes for any diffeomorphism which is smoothly isotopic to the identity. As an application, we give the first example of a diffeomorphism of a simply-connected 4-manifold which is homotopic…

We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a …

In this article, we first consider the $L^{2}$ \textit{Morse-Novikov cohomology} on a complete Riemannian manifold $M$ equipped with a parallel $1$-form which includes Vaisman manifold. Based on a vanishing theorem of $L^{2}$ \textit{Morse-Novikov cohomology}, we prove that the $L^{2}$-harmonic forms on $M$ are identic…

Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only sufficient, but also necessary for such a removability of singularities. This is the …

We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic f…

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…

We investigate complete minimal hypersurfaces in the Euclidean space $$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…

Improved CR Sobolev inequalities on CR sphere established.

We prove that the trace of the logarithmic term of the Toeplitz kernel on a contact manifold is a contact invariant, generalizing K. Hirachi's invariant for the Szego kernel on a CR manifold. When the base manifold is the three-sphere, this vanishes identically.

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where …

Study shows Futaki invariant vanishes on most Fano threefolds.

We present a formula for the trace of any symmetric power of a $n\times n$ matrix (with coefficients in a field) in terms of the ordinary powers of the matrix, an arbitrarily chosen linear function which vanishes on the identity matrix, and $n-2$ polynomial functions defined recursively.

It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.

We translate into the double forms formalism the basic identities of Greub and Greub-Vanstone that were obtained in the mixed exterior algebra. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure…

The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.

Analyzes $L^{2}$-harmonic forms on curved manifolds, proving integrability conditions.

Proves non-hyperbolicity of symplectic varieties with specific properties.

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conf…

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochne…

The paper develops $L^2$-Hodge theory on almost Kähler manifolds and proves the Hopf conjecture.

Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these de…

We give a geometric interpretation of weighted homogeneous solutions to the associativity equation in terms of the web theory and construct a massive Frobenius 3-fold germ via a singular 3-web germ satisfying the following conditions: 1) the web germ admits at least one infinitesimal symmetry, 2) the Chern connection f…

We show that a properly convex projective structure $\mathfrak{p}$ on a closed oriented surface of negative Euler characteristic arises from a Weyl connection if and only if $\mathfrak{p}$ is hyperbolic. We phrase the problem as a non-linear PDE for a Beltrami differential by using that $\mathfrak{p}$ admits a compatib…

Projective structures are mostly rigid at the boundary but some are not.

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of th…

New rigidity results for tensors on non-compact manifolds with curvature conditions.

The paper computes KV cochain differentials and their geometric implications.

We study power expansions of the characteristic function of a linear operator $A$ in a $p|q$-dimensional superspace $V$. We show that traces of exterior powers of $A$ satisfy universal recurrence relations of period $q$. `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. The…

We consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^…

We study the second order invariants of a Lorentzian surface in $\mathbb{R}^{2,2},$ and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrin…

We show that if a generator of a differential Gerstenhaber algebra satisfies certain Cartan-type identities, then the corresponding Lie bracket is formal. Geometric examples include the shifted de Rham complex of a Poisson manifold and the subcomplex of differential forms on a symplectic manifold vanishing on a Lagrang…

For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.

Study examines Weyl structures on Riemannian manifolds with vanishing Lee form.

We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…

We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the cla…

New tensors reveal full curvature structure from Riemann tensor.

A compact oriented 4-manifold is defined to be of ``superconformal simple type'' if certain polynomials in the basic classes (constructed using the Seiberg-Witten invariants) vanish identically. We show that all known 4-manifolds of $b_2^+>1$ are of superconformal simple type, and that the numerical invariants of 4-man…

Let $L$ be a closed, orientable, monotone Lagrangian 3-manifold of a symplectic manifold $(M, ω)$, for which there exists a local system such that the corresponding Lagrangian quantum homology vanishes. We show that its cohomology ring satisfies a certain dichotomy, which depends only on the parity of the first Betti n…