arXiv research

The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.

Develops Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.

Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.

Defines curvature for spectral triples and applies to θ-deformations.

Positive scalar curvature metrics on cobordisms yield only reducible Seiberg-Witten solutions.

Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.

Study shows upper limit for torical band width with spectral curvature bounds.

Researchers prove unique connection and curvature for Podleś quantum sphere.

Define mass invariant for asymptotically hyperbolic 3-manifolds with toroidal infinity.

Theorem proves minimal hypersurfaces in nonnegative scalar curvature manifolds are smooth.

Study metrics with specific spectral properties to compute Bartnik and Bartnik-Bray masses efficiently.

Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.

Study shows how certain metrics can be split into warped products.

Paper solves long neck problem on odd-dimensional spin manifolds.

Theorem proves spectral rigidity of warped product metrics.

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obt…

Paper bounds the A-hat genus using curvature and isoperimetric constants.

Study isotopy and concordance for metrics on manifolds with boundary.

Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.

Study complex structures and curvature equations on compact manifolds.

Study eta invariant on non-compact manifolds with positive scalar curvature.

For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

New proof of Llarull's rigidity theorem in odd dimensions via spectral flow.

Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume…

Sharp distance estimates for compact spin manifolds using Dirac operator.

Study spectral functionals on manifolds with torsion.

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a h…

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.

New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.

The paper extends spin geometry to weighted manifolds and defines a new mass for Ricci flow.

The paper stratifies positive scalar curvature metrics on manifolds.

A map between manifolds is an isometry if it's Lipschitz and scalar curvature bounded.

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the s…

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed pro…

We give a local expression for the {\it scalar curvature} of the noncommutative two torus $A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $ζ_a(s): …

Constructs equivariant spectral flow for Dirac-type operators on manifolds.

We had previously defined the rho invariant $ρ_{spin}(Y,E,H, g)$ for the twisted Dirac operator $\not\partial^E_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree differential form on $Y…

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set o…

Formula for spectral flow connects manifold properties to index theorem.

Extends Dirac operator results to foliations with invariant measures.

Proves effective positive mass theorem for AF manifolds and singular spaces.

We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and…

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable spa…

Quantizes symplectic fibrations to analyze vector bundles and metrics.

Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.

In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the La…

The paper studies Dirac operators on large spectral three-manifolds.