Related papers: The Weitzenb\"ock formula for the Fueter-Dirac ope…

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce…

It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the…

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the…

We consider the Dirac operator on asymptotically static Lorentzian manifolds with an odd-dimensional compact Cauchy surface. We prove that if Atiyah-Patodi-Singer boundary conditions are imposed at infinite times then the Dirac operator is…

In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of…

We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for $3,4$-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in…

Motivated by our conjecture of an earlier work predicting the degeneration at the second page of the Fr\"olicher spectral sequence of any compact complex manifold supporting an SKT metric $\omega$ (i.e. such that…

We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature…

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma > 0$, the…

We give some rigidity theorems for an n$(\geq4)$-dimensional compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $\sigma_2$. Moreover, when $n=4,$ we prove that a 4-dimensional compact…

We establish an index theorem for Toeplitz operators on odd dimensional spin manifolds with boundary. It may be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with…

The Dirac operator enters into zero curvature representation for the cubic nonlinear Schr\"{o}dinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The…

The time evolution of several interacting Ginzburg-Landau vortices according to an equation of Schroedinger type is approximated by motion on a finite-dimensional manifold. That manifold is defined as an unstable manifold of an auxiliary…

We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular…

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension…

We develop the deformation theory of instantons on asymptotically conical $G_2$-manifolds, where an asymptotic connection at infinity is fixed. A spinorial approach is adopted to relate the space of deformations to the kernel of a twisted…

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in…

Let X be a compact manifold with boundary, and suppose that the boundary is the total space of a fibration with base Y and fibre Z. Let D be a generalized Dirac operator associated to a Phi-metric g on X. Under the assumption that D is…

In this paper we establish new Bochner-Kodaira formulas with quadratic curvature terms on compact K\"ahler manifolds: for any $\eta\in \Omega^{p,q}(M)$, $$ \left\langle\Delta_{\overline \partial} \eta,\eta\right\rangle =\left\langle…

We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a "geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the…