Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. In this note, we describe the Hermitian metrics that leave the total Monge-Ampere volume invariant. In particular, we give several characterizations of the Hermitian metrics which satisfy the comparison principle for the complex Monge-Ampere operator
The paper introduces a volume invariant for Hermitian-symplectic metrics and proves its critical points are Kähler.
problem Investigating volume invariants for Hermitian-symplectic metrics.
method Introducing a functional acting on metrics in Aeppli cohomology classes and proving critical points are Kähler.
result The volume invariant generalises the volume of a Kähler class and vanishing is a necessary condition for the existence of a Kähler metric.
This study calculates the average number of common zeros of holomorphic functions on complex manifolds.
problem Calculating the average number of common zeros of holomorphic functions.
method Defined a Hermitian mixed volume for a mix of non-negative Hermitian forms and proved the average number of common zeros equals this mixed volume.
result The average number of common zeros of holomorphic functions equals the mixed volume of the manifold.
Study of Monge-Ampère volumes on hermitian manifolds, focusing on plurisigned metrics.
problem Behavior of Monge-Ampère volumes under hermitian variations.
method Analysis of hermitian forms and cohomology, studying boundedness and positivity.
result Existence and non-existence of plurisigned hermitian metrics on specific manifolds.
We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant r, a, b.
In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian manifold of compact type into its Cartesian products.
The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.
problem Analyzing semilinear elliptic equations and inequalities on pseudo-Hermitian manifolds.
method Using a generalized Jerison-Lee's formula and volume estimates.
result Established Liouville theorems and nonexistence results for specific equations and inequalities.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
This paper divides into two parts. Let (X,ω) be a compact Hermitian manifold. Firstly, if the Hermitian metric ω satisfies the assumption that ∂∂ωk=0 for all k, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is…
The paper extends Pappus-Guldin theorems to 3D-Heisenberg group surfaces.
problem Extending classical theorems to a new geometric setting.
method Deriving formulas for p-areas and volumes in the Heisenberg group.
result Pappus-Guldin theorems hold for surfaces in the Heisenberg group.
A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …
Study volumes of Bott-Chern classes on complex manifolds.
problem Understanding volumes of transcendental Bott-Chern classes.
method Extending non-pluripolar products to quasi-positive currents, establishing quasi-monotonicity of Monge-Ampère masses, and solving degenerate complex Monge-Ampère equations.
result Positive answer to Demailly-Păun-Boucksom conjecture regarding bounded mass property.
We compute the measure with multiplicity of the set of complex planes intersecting a compact domain in a complex space form. The result is given in terms of the so-called hermitian intrinsic volumes. Moreover, we obtain two different versions for the Gauss-Bonnet-Chern formula in complex space forms. One of them gives …
In this article we show that any finite cover of the moduli space of closed Riemann surfaces of g genus with g≥2 does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…
New method solves complex Monge-Ampère equations on hermitian manifolds.
problem Solving degenerate complex Monge-Ampère equations on hermitian manifolds.
method New approach using compactness and envelopes properties of quasi-plurisubharmonic functions.
result New relative a priori estimates and existence results for degenerate complex Monge-Ampère equations.
Calabi-Yau theorem extended to Vaisman manifolds.
problem Uniqueness of Vaisman metrics and their characterization.
method Analyzing the Lee form and Lee class properties.
result Vaisman metrics uniquely determined by volume and Lee class.
Based on uniform CR Sobolev inequality and Moser iteration, this paper investigates the convergence of closed pseudo-Hermitian manifolds. In terms of the subelliptic inequality, the set of closed normalized pseudo-Einstein manifolds with some uniform geometric conditions is compact. Moreover, the set of closed normaliz…
We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…
Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of Z-positive and Z-critical metrics leading to bundle stability. result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
Study classifies gravitational instantons based on their asymptotic geometry.
problem Classifying gravitational instantons based on their asymptotic properties.
method Investigation of asymptotic geometry of Hermitian non-Kähler Ricci-flat metrics.
result All Hermitian non-Kähler gravitational instantons can be compactified to log del Pezzo surfaces.
Researchers classify and decompose valuations on convex functions.
problem Classifying valuations on convex functions.
method Geometric decomposition of valuations, using properties of special subspaces and Monge-Ampère-type operators.
result Valuations decompose into subspaces defined by vanishing properties.
The paper studies volumes of direct images for high tensor powers of ample bundles.
problem Understanding asymptotics of Monge-Ampère volumes for high tensor powers of ample line bundles.
method Analyzes the leading term of asymptotics and classifies bundles saturating a topological bound.
result Provides a characterization of bundles admitting projectively flat Hermitian structures in the case of high symmetric powers of ample vector bundles.
Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…
We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalu…
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. The paper finds explicit instantons on a specific 6-manifold.
problem Finding explicit solutions to the Hermitian Yang-Mills equations.
method Dimensional reduction and reformulation of the equations on quotient spaces.
result Explicit abelian instantons and deformed Hermitian Yang-Mills connections are found.
We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …
In this paper we calculate the Lagrangian Floer homology HF(L0,L1:Z2) of a pair of real forms (L0,L1) in a monotone Hermitian symmetric space M of compact type in the case where L0 is not necessarily congruent to L1. In particular, we have a generalization of the Arnold-Givental inequality…
We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\θ$ on the CR sphere S 2n+1 ⊂ C n+1 , achieves its maximum when $\θ$ is the standard contact form.
Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
A discussion of torsion of Riemannian G-structures leads to a survey of contributions of Alfred Gray and others on almost Hermitian manifolds, G_2-manifolds, curvature identities, volume expansions, plotting geodesics, and the geometry of nilmanifolds. The paper concludes with a new example of a compact 8-manifold with…
Study bounds on Monge-Ampère volumes for degenerate complex equations.
problem Bounds on volumes of Monge-Ampère measures for degenerate complex equations.
method Fine use of quasi-plurisubharmonic envelopes.
result Established a transcendental version of the Grauert-Riemenschneider conjecture.
Study cohomology of ball quotients and their compactifications.
problem Cohomology of symmetric power of cotangent bundles of ball quotients and their compactifications.
method Hodge theory for complete hermitian manifolds, Green's operator, extension of results.
result Established existence of Hodge decomposition and Green's operator for ball quotients and their compactifications.
The Fubini-Study metric minimizes a volume-normalized holomorphic systole in CPn.
problem Finding metrics with minimal holomorphic systoles in complex projective spaces.
method Introduced holomorphic k-systole and used Gauduchon metrics to establish minimization. result The Fubini-Study metric locally minimizes the volume-normalized holomorphic (n−1)-systole. Let (M,I) be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold is called nearly Kaehler if it adm…
Abstract: Study of metrics on line bundles over complex varieties.
problem Chern-Weil and Hilbert-Samuel formulae for singular metrics.
method Theory of b-divisors and multiplier ideal volume function.
result Generalization of a result for elliptic curves to higher degrees.
The paper analyzes systoles of complex projective spaces under various metrics.
problem Behavior of systoles in complex projective spaces for different metrics.
method Integral geometric techniques and careful analysis of systole functional.
result Balanced metrics locally minimize the systole on volume-normalized metrics.
Paper solves complex equations on noncompact manifolds.
problem Establishing estimates and existence for fully non-linear equations.
method General class of fully non-linear equations on Kähler and Hermitian manifolds.
result Constructs complete Kähler metrics with prescribed volume forms.
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.