arXiv research

Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of the Dirac operator under the chiral bag boundary condition. More precisely, we s…

Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.

Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…

Proves spacetime positive mass theorem for spin initial data sets with arbitrary ends.

Extends Llarull's theorem to noncompact manifolds with boundary.

New bounds for Dirac eigenvalue involving boundary capacity.

Study of Dirac operator with chiral boundary conditions on spin manifolds.

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}^{(2,2)}$ for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.

Develops a theorem for a 6D manifold with boundary.

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}_{6}^{(1,3)}$ for 6-dimensional spin manifolds with boundary and the gravity on boundary is derived by the noncommutative residue associated with Dirac operators.For 6-dimensional manifo…

The paper proves a 'long neck principle' for Riemannian spin manifolds with positive scalar curvature.

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.

Upper bound for total mean curvature of spin fill-ins is proven.

Proves rigidity for specific initial data sets under the dominant energy condition.

Real vector bundles are determined by their Dirac indices on specific spin manifolds.

In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem for spin manifolds without boundary. For $6$ dimensional spin manifolds with boundary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then we generalize this theorem to the general $n$ dimensional manifold. An equivariant Kastler-Kal…

Let M be an 8-manifold with a Spin(7)-structure. We first show that closed Cayley submanifolds of M form a smooth moduli space for a generic Spin(7)-structure. Then we study the deformations of a compact, connected Cayley submanifold X of M with non-empty boundary contained in a given submanifold W of M such that X and…

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 the gravitational action for boundary in the case of 4-dimensional manifolds with…

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.

Algorithm computes fundamental classes of spin components in moduli space.

Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.

The paper defines invariants for positive scalar curvature metrics on manifolds with boundary.

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

Abstract: Mapping 3-manifold bordisms to topological orders and domain walls.

Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne-Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected c…

We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …

Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.

Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

The paper calculates transformation operators and proves a theorem on 4D manifolds.

New rigidity theorems for spin fill-ins with non-negative scalar curvature.

Study cyclic group actions on spin 4-manifolds with boundary using Seiberg-Witten theory.

Sharp distance estimates for compact spin manifolds using Dirac operator.

In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on $Spin^c$ manifolds without boundary. The limiting case is then studied and an example is given.

The paper calculates a functional for a specific Dirac operator.

Paper extends index theorem to odd-dimensional manifolds with even-dimensional boundaries.

We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of $\mathbb{Z}_r$-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topolog…

New cubic forms linked to η-invariants and mod 2 indices.

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when …

Paper proves rigidity for spin bands with specific conditions.

We study the Yamabe flow on compact Riemannian manifolds of dimensions greater than two with minimal boundary. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin.

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

The paper proves positive energy-momentum theorems for charged AdS initial data sets.

In this paper, we define lower dimensional volumes of compact Riemannian manifolds with boundary. For five dimensional spin manifolds with boundary, we prove a Kastler-Kalau-Walze type theorem associated with one-form perturbations of Dirac operators in this case.

Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.