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.
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.
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.
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.
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.
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.
Compact Riemannian manifolds with mostly positive curvature have finite fundamental groups.
problem Understanding conditions for finite fundamental groups in compact Riemannian manifolds.
method Using Bismut-Witten Laplacian and Ricci-Hessian inequalities.
result New conditions for finite fundamental groups in manifolds with mostly positive curvature.
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.
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.
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.
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.
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.
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.
The paper explores properties of Gauduchon curvature in Hermitian manifolds.
problem Investigating properties of Gauduchon curvature in Hermitian manifolds.
method Analyzing the Ricci curvature of Gauduchon connections and proving existence of metrics.
result Monotonicity theorem for Gauduchon holomorphic sectional curvature.
New metrics found on homogeneous spaces, preserving key geometric properties.
problem Finding Bismut Ricci flat metrics on compact homogeneous spaces.
method Exhibited and proved uniqueness of Bismut Ricci flat metrics on homogeneous spaces.
result Unique and non-homothetic metrics found on certain homogeneous spaces.
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.
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.
The paper proves stability of a Ricci flat metric on a product of Einstein homogeneous spaces.
problem Stability of Bismut Ricci flat metrics on product spaces.
method Generalized Ricci flow on aligned homogeneous spaces.
result The Bismut Ricci flat metric is asymptotically and globally stable under the generalized Ricci flow.
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…
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.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long…
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.
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 paper studies harmonic complex structures and special metrics on Sasakian manifolds.
problem Analyzing harmonic complex structures and special metrics on products of Sasakian manifolds.
method Examining 2-parameter families of Hermitian structures (Ja,b,ga,b) and their properties. result The complex structure Ja,b is harmonic with respect to ga,b, and conditions for various types of special metrics are determined. An SKT metric is a Hermitian metric on a complex manifold whose fundamental 2-form ω satisfies $\de\debarω=0$. Streets and Tian introduced in \cite{sttiPlur} a Ricci-type flow that preserves the SKT condition. This flow uses the Ricci form associated to the Bismut connection, the unique Hermitian connection with tota…
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.
Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
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 paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.
problem Proving Calderón-Zygmund inequalities on non-compact Riemannian manifolds without positive injectivity radius.
method Probabilistic tools, Hessian formulas, and Bismut type representations for heat semigroups.
result The paper proves the Calderón-Zygmund inequality for 1<p<2 under a lower Ricci curvature bound, and for p>2 under additional curvature conditions. New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
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…
The paper proves conditions for infinite lifetime of Brownian motion and regularity of heat flow.
problem Conditions for infinite lifetime of Brownian motion on Riemannian manifolds with bounded Ricci curvature.
method Derives a Bismut-Elworthy-Li derivative formula and proves the equivalence of lower bounded Ricci curvature to the existence of pathwise couplings.
result Conditions on Ricci curvature ensure infinite lifetime of Brownian motion and regularity of heat flow.
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 prove a completely new integral criterion for the existence and completeness of the wave operators W±(−Δh,−Δg,Ig,h) corresponding to the (unique self-adjoint realizations of) the Laplace-Beltrami operators −Δj, j=1,2, that are induced by two quasi-isometric complete Riemannian metrics g and h o…
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 derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
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.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
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.
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).
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.