Searcharxiv⌕ Search

arXiv subjects

Daoqiang Liu

Publications and source records attributed to Daoqiang Liu.

9 recordsLinked to original sources
arXiv2026

Spectral Geroch conjecture and noncompact area enlargeable summands

We prove that the connected sum of a possibly noncompact area enlargeable manifold $M_1$ with an arbitrary spin manifold $M_2$ of the same dimension admits no complete Riemannian metric of uniformly positive scalar curvature. This extends a theorem of Wang--Zhang, where $M_1$ is assumed closed, to noncompact enlargeable summands; in this generality the uniform positivity hypothesis enters the argument in an essential way. We also prove a spectral analogue of the generalized Geroch conjecture in terms of the $γ$-spectral constant: for $γ>(\dim M_1-1)/(4\dim M_1)$, such a connected sum carries no complete metric with positive $γ$-spectral constant. The proofs are based on a covering connected sum construction together with the scalar-cowaist and spectral-cowaist inequalities.

math.DG↗
arXiv2026

A sharp inequality relating scalar curvature, bottom spectrum, and the relative $\widehat{A}$-cowaist on complete manifolds

We introduce a refined notion of relative $\widehat{A}$-cowaist, extending the framework of Cecchini--Zeidler to complete manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating this invariant to the scalar curvature and the bottom spectrum of the Laplacian. As a consequence, we partially answer a question raised by Shi, and we obtain, in all dimensions, sharp upper bounds for the bottom spectrum under scalar curvature lower bounds. In particular, this removes the bounded geometry assumption from a result of Wang--Zhu. Our approach is based on deformed Dirac operators.

math.DG↗
arXiv2026

Bottom spectrum, vertical $\widehat{A}$-cowaist and scalar curvature rigidity

We introduce the vertical \(\widehat{A}\)-cowaist, a codimension-one invariant for partitioned manifolds. It extends the concept of infinite vertical \(\widehat{A}\)-cowaist for bands to arbitrary partitioned manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating the scalar curvature, the bottom spectrum of the Laplacian, and this invariant. As an application, we obtain a high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate. We also prove a quantitative strengthening of Anghel's theorem together with a boundary version, as well as a Calabi-Yau type theorem that goes beyond the dimensional restrictions of the earlier \(μ\)-bubble method. Our approach is based on deformed Dirac operators.

math.DG↗
arXiv2026

Bottom spectrum and Llarull's theorem on complete noncompact manifolds

In this paper, we prove an extension of the noncompact version of Llarull's theorem due to Zhang and Li-Su-Wang-Zhang, giving an upper bound for the infimum of scalar curvature in terms of the bottom spectrum of the Laplacian. Moreover, we extend the theorem to manifolds with boundary, relaxing the strict positivity condition on the scalar curvature near the boundary that was required by Liu-Liu. Our approach is based on deformed Dirac operators.

math.DG↗
arXiv2026

Llarull's theorem on noncompact manifolds with boundary

Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary.

math.DG↗
arXiv2024

On distance estimates for complete manifolds with lower scalar curvature bounds

In this paper, we focus on the distance estimate problem on complete manifolds with compact boundary and with lower scalar curvature bounds. On these manifolds, relative to a background manifold with nonnegative curvature operator, we introduce a definition of the relative index of relative Gromov-Lawson pairs via a deformed Dirac operator trick in \cite{Zh20}. We prove that the relative index coincides with the index of associated Callias operators of the relative Gromov-Lawson pairs. As applications, we prove a short neck inequality with uniformly positive scalar curvature and the corresponding quantitative shielding result with nonnegative scalar curvature. Moreover, we generalize the concept of relative $\widehat{A}$-area in \cite{CZ24} to complete manifolds with compact boundary and then investigate width estimates of geodesic collar neighborhoods.

math.DG↗
arXiv2024

On the long neck principle and width estimates for initial data sets

In this paper, we prove the long neck principle, band width estimates, and width inequalities of the geodesic collar neighborhoods of the boundary in the setting of general initial data sets for the Einstein equations, subject to certain energy conditions corresponding to the lower bounds of scalar curvature on Riemannian manifolds. Our results are established via the spinorial Callias operator approach.

math.DG↗
arXiv2023

Tilted spacetime positive mass theorem with arbitrary ends

In this paper, we prove the spacetime positive mass theorem for asymptotically flat spin initial data sets with arbitrary ends and a non-compact boundary. Moreover, we demonstrate a quantitative shielding theorem, subject to the tilted boundary dominant energy condition. Our results are established by solving a mixed boundary value problem for the Dirac-Witten operator with a Callias potential.

gr-qc↗