Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.
problem Investigate positivity of Bismut curvature in non-Kähler manifolds.
method Analyze Bismut-Griffiths-positivity under Hermitian curvature flows.
result Identify HCFs that do not preserve Bismut-Griffiths-positivity.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
Characterizes Hermitian manifolds with Bismut parallel torsion.
problem Understanding curvature properties of Hermitian manifolds with specific torsion.
method Analyzes Bismut curvature tensor and parallel torsion conditions.
result Characterizes Bismut torsion parallel manifolds using Bismut curvature alone.
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
The study characterizes Hermitian manifolds with parallel Bismut-Strominger torsion.
problem Characterizing Hermitian manifolds with specific torsion properties.
method Analyzing the curvature tensor and properties of the Bismut-Strominger connection.
result A necessary and sufficient condition for Bismut torsion parallel manifolds.
Study on BAS manifolds with parallel torsion and curvature.
problem Characterizing and classifying BAS manifolds.
method Canonical reduction theorem, classification in homogeneous settings, construction of combined geometries.
result Classification of BAS manifolds in various settings.
Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on TM⊕ΛpT∗M is an example leading to a scalar curvature of the form R+H2 for a closed (p+2)-form H.
The paper confirms a conjecture for Bismut torsion parallel metrics.
problem The existence of metrics with constant holomorphic sectional curvature on non-Kähler manifolds.
method Investigation of Bismut torsion parallel metrics.
result The conjecture is confirmed for all non-balanced BTP manifolds.
The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on…
New insights on ACYT 6-manifolds reveal instanton conditions for curvature.
problem Understanding instanton conditions on ACYT 6-manifolds.
method Analyzing curvature and torsion properties of ACYT 6-manifolds.
result Curvature is an SU(3)-instanton if and only if torsion is parallel. We find Einstein metrics on homogeneous HKT manifolds.
problem Characterizing and finding Einstein metrics on homogeneous HKT manifolds.
method Characterization and construction of invariant HKT-Einstein metrics.
result Existence and uniqueness of invariant HKT-Einstein metrics.
The study characterizes compact homogeneous manifolds with Bismut parallel torsion.
problem Characterizing compact homogeneous manifolds with specific geometric properties.
method Investigating Hermitian manifolds with Bismut parallel torsion, focusing on locally homogeneous manifolds.
result Characterization of compact Chern flat BTP manifolds and properties of BTP compact Hermitian locally homogeneous manifolds.
Study Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
problem Characterize Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
method Analyze SKT condition and Bismut Kähler-like metrics, construct new examples, and study pluriclosed flow.
result Construct new examples of Hermitian manifolds satisfying Bismut Kähler-like condition.
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.
problem Prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds.
method Reduced to solving a semi-linear partial differential equation with exponential nonlinearity using super and sub-solution method.
result Existence of solution depends on the sign of a constant associated to Gauduchon degree.
The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.
problem Classifying Bismut Kähler-like manifolds in specific dimensions.
method Structural theorems and proving conjectures about BKL manifolds.
result Complete classifications of BKL manifolds in dimensions 4 and 5.
Estimates eigenvalue for Hermitian manifolds using curvature.
problem Estimating the first eigenvalue of Hermitian manifolds.
method Using holomorphic Ricci and sectional curvatures.
result Established estimates for the first eigenvalue.
Paper studies stability of generalized Ricci solitons with mathematical rigor.
problem Stability of generalized Ricci solitons under curvature assumptions.
method Computed second variation formula, proved stability conditions.
result Equivalence of dynamical and linear stability on steady gradient solitons.
Study SKT and CYT manifolds with parallel Bismut torsion.
problem Characterize and construct compact complex manifolds with specific geometric properties.
method Characterization via universal cover, construction using mapping torus, investigation of generalized Kaehler structures.
result Existence of non-Bismut flat examples and characterization of universal covers.
We study Hermitian metrics with a Gauduchon connection being "Kähler-like", namely, satisfying the same symmetries for curvature as the Levi-Civita and Chern connections. In particular, we investigate 6-dimensional solvmanifolds with invariant complex structures with trivial canonical bundle and with invariant Hermit…
Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.
problem Classifying specific types of manifolds with parallel Bismut torsion.
method Complete classification through mathematical analysis.
result Established a splitting theorem for certain manifolds.
Holonomy group of Bismut connection on Vaisman manifolds is studied.
problem Analyzing the holonomy group of Bismut connection on Vaisman manifolds.
method Proved and computed the holonomy group for Vaisman manifolds, including solvmanifolds and Hopf manifolds.
result Holonomy group of Bismut connection on Vaisman manifolds is contained in U(n-1).
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Study on balanced Hermitian threefolds with parallel Bismut torsion.
problem Characterizing compact, balanced BTP threefolds.
method Detailed description of all compact, balanced BTP threefolds.
result Characterization of all compact, balanced BTP threefolds.
New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
We show that there is a canonical construction of a zeta (Bismut-Quillen) connection on the determinant line bundle of a family of APS elliptic boundary problems and that it has curvature equal to the 2-form part of a relative eta form.
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
Study on Hermitian metrics on Lie algebras with specific ideals.
problem Classifying Hermitian metrics on Lie algebras with abelian ideals.
method Examined unimodular Lie algebras with abelian ideals of codimension two, classified metrics.
result Classification of Bismut Kähler-like and Bismut torsion-parallel metrics.
We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…
Novel analysis of generalized Ricci solitons leads to stability results and deformations.
problem Stability and classification of generalized Ricci solitons.
method Group of generalized gauge transformations, novel connection, second variation formula.
result All Bismut flat manifolds are linearly stable and admit nontrivial deformations.
Researchers classify invariant Hermitian structures on flag manifolds with parallel Bismut torsion.
problem Classifying invariant Hermitian structures with specific torsion properties on flag manifolds.
method Detailed analysis of invariant Hermitian structures on flag manifolds, proving conditions for parallel Bismut torsion.
result Conditions for the existence of parallel Bismut torsion on most flag manifolds.
In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. I…
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.
problem Extending variational formula for Bismut-Cheeger eta form without kernel bundle assumption.
method Twisting spinc Dirac operators by isomorphic vector bundles, proving Z2-graded additivity. result Analytic index in differential K-theory is a well-defined group homomorphism, and Riemann-Roch-Grothendieck theorem in R/Z K-theory. For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally …
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
Study of complex 3-folds with vanishing Bismut Ricci form via special Kähler geometry.
problem Characterizing non-Kähler BHE 3-folds with vanishing Bismut Ricci form.
method Dimensional reduction to special Kähler geometry, solving 6th order PDE, momentum map interpretation.
result Characterization of Samelson locally homogeneous BHE 3-folds and construction of new non-Kähler BHE structures.
In this paper we give a proof of an index theorem by Bismut. As a consequence we obtain another proof of the Grothendieck-Riemann-Roch theorem in differential cohomology.
We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…
The paper computes a residue density for a specific Laplacian on compact manifolds.
problem Computing a noncommutative residue density for a complex Laplacian.
method Explicit computation of the noncommutative residue density associated with equivariant twisted Bismut Laplacian with torsion.
result Proves equivariant twisted Kastler-Kalau-Walze type theorems with torsion on compact manifolds with boundary.
In this paper, we give a quantum interpretation of the Bismut-Chern character form (the loop space lifting of the Chern character form) as well as the Chern character form associated to a complex vector bundle with connection over a smooth manifold in the framework of supersymmetric quantum field theories developed by …
Derives an index formula for families of end-periodic Dirac operators.
problem Calculating the index of families of end-periodic Dirac operators.
method Using the renormalized Chern character and Fourier-Laplace transform of the Bismut superconnection.
result Establishes an index formula involving a new end-periodic eta form.
Wave trace singularity formula for fibre bundles generalizes Poisson summation.
problem Wave trace singularity formula for fibre bundles.
method Proved a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle.
result Generalizes Poisson summation formulae for families.
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.