Short proof given for a complex formula in spin manifold geometry.
problem Localization formula for loop space Chern character in spin manifolds.
method Using rescaled spinor bundle on tangent groupoid.
result A concise proof of the localization formula.
Paper solves long neck problem on odd-dimensional spin manifolds.
problem Long neck problem on odd-dimensional spin manifolds.
method Spectral flow of Callias operators.
result Complete answer to Gromov's long neck problem.
Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.
problem Understanding the space of metrics with positive Ricci curvature on spin manifolds.
method Analyzing spin manifolds of dimension 4k-1 for k ≥ 2, focusing on metrics with a specific property.
result The space of metrics has infinitely many path components.
Study proves a conjecture for certain spin manifolds.
problem Proving a conjecture about positive scalar curvature metrics.
method Connected sum construction and Gromov-Lawson area enlargement.
result Connected sum of spin manifolds admits no complete metric of positive scalar curvature.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
Rigidity of elliptic genera proven for non-spin manifolds with S1-action.
problem Rigidity of elliptic genera for non-spin manifolds with S1-action. method Analysis of universal covering spin condition and π2(M) for rigidity. result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.
New elliptic genera defined for spin manifolds.
problem Defining new elliptic genera for spin manifolds.
method Defined new two-variable elliptic genera using Jacobi forms and modular forms.
result Anomaly cancellation formulas derived from modular forms.
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
We study conformal Spin-subgeometry of submanifolds in a semi-Riemannian Spin-manifold, focusing on conformal Spin-manifolds (M,[h]) and their Poincaré-Einstein metrics (X,g+). Our approach is based on the spectral theory of Dirac operator in the ambient Spin-manifold, and associated spinor valued meromorp…
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
Revises Schwarzschild manifold rigidity proof for spin manifolds.
problem Rigidity of Schwarzschild manifold for spin manifolds.
method Spinorial proof approach.
result Generalizes and includes classical and recent black hole uniqueness theorems.
Proves Rudyak's conjecture for low-dimensional simply connected spin manifolds.
problem Rudyak's conjecture on the relationship between the Lusternik-Schnirelmann category of manifolds.
method Analyzes simply connected spin manifolds of dimensions up to 8.
result Proves the conjecture for n-dimensional simply connected spin manifolds for n≤8. Study uncoupled solutions to Dirac-Yang-Mills equations on spin manifolds.
problem Condition for vanishing Dirac current on harmonic spinors.
method Perturbation theory and index theorem.
result Existence of uncoupled solutions, classification of connection forms.
Paper proves a spinor inequality for magnetic fields on spin manifolds.
problem Proving a spinor inequality for magnetic fields on spin manifolds.
method Analyzing the zero mode equation and using the Yamabe constant.
result The inequality ∥dA∥n/2>Y(Mn,[g])/(4vn1/2) holds for non-trivial solutions. Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
Elliptic bouquets defined for spin manifolds with circular actions.
problem Defining integration in elliptic cohomology.
method Introducing elliptic bouquets of germs of holomorphic equivariant cohomology classes, integrating them as in K-theory.
result Witten's rigidity theorem follows from integration of elliptic bouquets.
Study shows strict inequality for Dirac-type equation on spin manifolds, leading to existence results.
problem Investigating strict inequality for a Dirac-type equation on compact spin manifolds.
method Analyzing a generalized conformally invariant equation involving the Dirac operator with a non-linear convolution term.
result Strict inequality holds, except for round sphere conformal cases, providing existence results for a ground state.
We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for A^-type genera.
New Witten rigidity theorems for elliptic genus in various dimensions.
problem Proving rigidity theorems for elliptic genus in different dimensions.
method Combining Liu's and Han-Yu's methods to prove Witten rigidity theorems for elliptic genus in even and odd dimensions.
result Several new Witten rigidity theorems for elliptic genus in even and odd dimensions have been established.
The round sphere is stable among spin manifolds with a specific scalar curvature bound.
problem Stability of the round sphere among spin manifolds with scalar curvature below a certain threshold.
method Showed that if the scalar curvature is bounded from below by n(n−1)−ε, the manifold is C0-close to a finite number of spheres outside a small bad set. result The spherical stability problem is completely solved.
The paper calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.
problem Characterizing Lorentzian Weyl spin manifolds with weighted parallel spinors.
method Analyzing holonomy algebras and introducing special coordinates.
result Local forms and examples of Lorentzian Weyl spin manifolds with weighted parallel spinors.
A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.
Study shows unbounded Pontryagin numbers on curved manifolds.
problem Understanding unbounded Pontryagin numbers on curved manifolds.
method Analyzing rational linear combinations of Pontryagin numbers and their relation to the universal elliptic genus.
result Proves existence of unbounded Pontryagin numbers on nonnegatively curved spin manifolds.
We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension 4k≥12, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…
We extend Witten's spinor proof of the positive mass theorem to large classes of complete asymptotically flat non-spin manifolds, including all manifolds of dimension less than or equal to 11 and all manifolds of dimension less than 26 which admit a codimension 3 immersion in Euclidean space.
Constructs modular forms and proves divisibility results for odd-dimensional manifolds.
problem Constructing modular forms over specific groups and proving divisibility results.
method SL(2, Z) modular forms and Witten genus in odd dimensions.
result Obtained divisibility results of index of Toeplitz operators on spin and spin^c manifolds.
Paper proves existence of minimum energy solutions in 5D contact spin manifolds.
problem Finding minimum energy solutions for CR Yamabe equation in 5D contact spin manifolds.
method Spinorial approach based on a positive mass theorem.
result Existence of minimum energy solutions in 5D contact spin manifolds.
We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we giv…
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}.
New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.
problem Existence of parallel spinors on Ricci-flat manifolds.
method Generalization of existing results to manifolds with non-vanishing Rosenberg index.
result Every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.
problem Proving a theorem for a specific type of Dirac operator on various manifolds.
method Extending previous results to even-dimensional almost product Riemannian spin manifolds.
result Established the general Kastler-Kalau-Walze type theorem for even-dimensional manifolds.
Study shows certain spin manifolds can't meet DEC condition.
problem Non-existence of spin fill-ins meeting DEC condition.
method Analyzes spin Riemannian manifolds and generalized mean curvature functions.
result Closed spin manifolds cannot satisfy DEC if curvature is large.
Dirac operator invertibility proven for specific manifolds.
problem Invertibility of twisted Dirac operator on manifolds.
method Closed connected spin manifold with non-negative scalar curvature, flat Hilbert module bundle.
result Dirac operator is invertible under given conditions.
In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume 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.
In this paper we investigate the properties of a semi-linear problem on a spin manifold involving the Dirac operator, through the construction of Rabinowitz-Floer homology groups. We give several existence results for sub-critical and critical non-linearities as application of the computation of the different homologie…
Extends Llarull's theorem to noncompact manifolds with boundary.
problem Generalizing Llarull's theorem to noncompact manifolds with boundary.
method Extends previous results to include compact boundaries.
result Generalized theorem to noncompact manifolds with compact boundaries.
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…
We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…
Paper shows zero Rosenberg index for certain foliated manifolds.
problem Finding conditions for zero Rosenberg index.
method Examined foliated manifolds with leafwise positive scalar curvature.
result Rosenberg index is zero for such manifolds.
Researchers prove an inequality linking spin manifold fill-ins to hyperspherical radius.
problem Proving an inequality relating the infimum of mean curvature to hyperspherical radius for spin manifolds.
method Combining Hijazi-Montiel-Roldán inequality and Bär's recent theorem; providing an alternative proof based on Bär-Ballmann work.
result Proved inequality linking spin manifold fill-ins to hyperspherical radius.
The paper proves a quantitative positive mass theorem for spin manifolds with distance estimates.
problem Proving a positive mass theorem for spin manifolds with arbitrary ends.
method Analyzing the scalar curvature and using distance estimates.
result Quantitative answer to Schoen and Yau's question on the positive mass theorem.
Since the first work of Thomas Friedrich showing that isometric immersions of Riemann surfaces are related to spinors and the Dirac equation, various works appeared generalizing this approach to more general Spin-manifolds, in particular the case of submanifolds of Spin-manifolds of constant curvature. In the present w…
Sharp distance estimates for compact spin manifolds using Dirac operator.
problem Metric inequalities for compact spin manifolds with lower bounds on scalar and mean curvatures.
method Using the Dirac operator technique with spectral estimates and local boundary conditions.
result Optimal estimates for Riemannian bands and long neck problem solutions.
The paper derives spectral (0,4)-tensor functionals using the noncommutative residue.
problem Deriving spectral (0,4)-tensor functionals on compact spin manifolds.
method Using four one-forms and the Dirac operator, the noncommutative residue is applied to even-dimensional compact spin manifolds.
result Spectral (0,4)-tensor functionals are extended to a general spectral triple.
Study on spaces of metrics with intermediate curvature bounds.
problem Understanding spaces of metrics with lower bounds on intermediate curvatures.
method Analyzing spaces of Riemannian metrics with specific curvature bounds on high-dimensional Spin-manifolds.
result Spaces of metrics with positive p-curvature and k-positive Ricci curvature have non-trivial homotopy groups.
The paper proves a mass theorem for non-spin manifolds with low regularity curvature.
problem Establishing a mass theorem for non-spin manifolds with low regularity curvature.
method Smooth approximations of the metric, Sobolev version of Friedrichs' Lemma, comparison theory of RCD-spaces, rigidity theorem for compact manifolds.
result Asymptotically flat manifolds with nonnegative distributional scalar curvature have nonnegative ADM mass.
The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.
problem Proving the Rosenberg index does not vanish for certain manifolds.
method Analyzing isometric immersions of wide Riemannian bands and cube-like domains on spin manifolds.
result Closed spin manifolds with infinite KO-width have non-vanishing Rosenberg index.