Computes Weyl group of Kähler toric manifold isometries.
problem Computing the Weyl group of Kähler toric manifold isometries.
method Analyzes the group of holomorphic isometries of a Kähler toric manifold with real analytic Kähler metric.
result Computed the Weyl group of the group of holomorphic isometries.
We prove that the identity component of the holomorphic isometry group of a Sasaki-Einstein metric is the identity component of a maximal compact subgroup of its automorphism group.
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.
Study metrics on half plane with specific curvature properties.
problem Warped product metrics on half plane.
method Holomorphic isometries and sectional curvature analysis.
result Metrics with zero and unbounded negative curvature exist.
New representations of 3-manifold groups into complex hyperbolic space found.
problem Finding representations of 3-manifold groups into complex hyperbolic spaces.
method Using Lefschetz fibrations and orbifold fundamental groups of branched coverings of the projective plane.
result Infinitely many non-conjugate representations discovered.
The paper classifies Sasaki and Vaisman manifolds of unimodular Lie groups.
problem Understanding the structure of Sasaki and Vaisman manifolds on unimodular Lie groups.
method Basic structure theorem and complete classification for simply connected manifolds.
result A complete classification of simply connected Sasaki and Vaisman unimodular Lie groups.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.
It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manif…
Let Mn,d be the moduli space of semi-stable rank n, trace-free Higgs bundles with fixed determinant of degree d on a Riemann surface of genus at least 3. We determine the following automorphism groups of Mn,d: (i) the group of automorphisms as a complex analytic variety, (ii) the gro…
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
Holomorphic isometries from Poincaré disk to complex symmetric domains found.
problem Finding holomorphic isometries from the Poincaré disk to complex symmetric domains.
method Examined holomorphic isometries from the Poincaré disk into the product of the unit disk and complex unit n-ball, then extended to irreducible bounded symmetric domains of rank at least 2.
result Provided new examples of holomorphic isometries from the Poincaré disk into irreducible bounded symmetric domains of rank at least 2.
Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on …
Study limits of Kähler submanifolds and prove no holomorphic isometries.
problem Understanding limits of Kähler submanifolds and their isometries.
method Gromov-Hausdorff convergence and scalar curvature bounds.
result Holomorphic isometries cannot exist between certain Kähler manifolds and projective spaces.
Study Kähler-Einstein manifolds with holomorphic isometries into blow-ups of complex spaces.
problem Characterizing Kähler-Einstein manifolds with holomorphic isometries into specific blow-ups.
method Analyzing properties of Kähler-Einstein manifolds and their isometries into generalized Burns-Simanca and Eguchi-Hanson manifolds.
result Generalized Burns-Simanca and Eguchi-Hanson manifolds are not relatives to any homogeneous bounded domain.
No isometric holomorphic maps between Teichmüller spaces and bounded domains.
problem Holomorphic isometries between Teichmüller spaces and bounded symmetric domains.
method Analyzing intrinsic metrics and properties of Teichmüller spaces and bounded symmetric domains.
result No isometric holomorphic maps exist in dimensions two or more.
Paper proves disks are holomorphic in Teichmüller spaces.
problem Understanding holomorphic disks in Teichmüller spaces.
method Analyzes totally-geodesic isometries and disk-rigid domains.
result Every isometry is holomorphic or anti-holomorphic.
Local holomorphic maps preserving (p,p) forms are shown to be isometries.
problem Preserving (p,p) forms under holomorphic maps between Kähler manifolds.
method Analyzing local holomorphic maps between Kähler manifolds, proving isometries up to scalars.
result Holomorphic maps preserving (p,p) forms are isometries under certain conditions.
For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.
Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.
The paper proves rigidity theorems on holomorphic isometries into homogeneous domains.
problem Characterizing and comparing holomorphic isometries into homogeneous bounded domains.
method Two rigidity theorems on holomorphic isometries into homogeneous bounded domains.
result The flat (definite or indefinite) complex Euclidean space is not a relative of a homogeneous bounded domain.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by …
Classifies points in quaternionic hyperbolic spaces up to congruence.
problem Classifying points in quaternionic hyperbolic spaces up to congruence.
method Introduces geometric invariants and distance formulas to classify points.
result Congruence classes are described by quaternionic Cartan's angular invariants and distances.
The study examines holomorphic isometric embeddings of complex unit balls into bounded symmetric domains.
problem Holomorphic isometric embeddings of complex unit balls into bounded symmetric domains.
method Analyzes properties and structures of embeddings with non-minimal isometric constants and classifies images for specific cases.
result Complete classification of images for holomorphic isometries from (Bn,gBn) to (Ω,gΩ) for 1≤n≤n0(Ω)−1. The paper proves rigidity properties of holomorphic isometries into homogeneous Kähler manifolds.
problem Rigidity of holomorphic isometries into homogeneous Kähler manifolds.
method Analyzing Kähler-Ricci solitons, flat spaces, and homogeneous bounded domains.
result Strong extensions of rigidity results in previous studies.
Mathematical proof of S-duality and universal isometries in q-map spaces.
problem Understanding the isometries of q-map spaces and their relation to S-duality.
method Purely mathematical proof of the existence of an SL(2,R)-action on q-map spaces, describing interactions with existing isometries.
result Presentation of a (3n+6)-dimensional group of universal isometries, enlarging previous results.
We prove that Kobayashi isometries between strongly convex domains are holomorphic or anti-holomorphic. More precisely, let n1,n2 be positive integers and let $Ω_i \subset \C^{n_i}, \ i=1,2$, be bounded C3 strongly convex domains. If φ:(Ω1,dΩ1K)→(Ω2,dΩ2K) is an isometry, i.e. $ d^K_…
Study shows Dolbeault cohomology is unchanged by complex Lie group actions.
problem Understanding how complex Lie group actions affect Dolbeault cohomology.
method Analyzes Dolbeault cohomology of compact complex manifolds with group actions.
result Induced action on Dolbeault cohomology is trivial.
A k-reflection of the n-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in U(n,1) with negative type eigenvalue λ, ∣λ∣=1, of multiplicity k+1 and positive type eigenvalue 1 of multiplicity n−k. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most fou…
We show that holomorphic riemannian metrics on compact complex threefolds are locally homogeneous (the pseudogroup of local isometries acts transitively on the manifold).
Study on special Hermitian metrics on cohomogeneity one manifolds.
problem Characterizing and constructing Hermitian metrics on cohomogeneity one manifolds.
method Investigation of geometry of Hermitian manifolds with compact Lie group action by holomorphic isometries.
result Construction of new examples of cohomogeneity one Hermitian metrics solving specific equations.
New symmetries and line bundles on hyperkähler spaces explored.
problem Exploring new symmetries on hyperkähler spaces.
method Introducing new continuous symmetries and their geometric implications.
result Found new types of symmetries and their associated line bundles.
We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kähler mom…
Researchers describe a new dynamical zeta function for certain Hermitian manifolds.
problem Analyzing holomorphic torsion for specific Hermitian locally symmetric manifolds.
method Developed a Weil-type zeta function involving geometry of universal cover.
result Valid for real rank one groups and some rank two groups, but not for all.
We obtain explicitly all solutions of the SU(infinity) Toda field equation with the property that the associated Einstein-Weyl space admits a 2-sphere of divergence-free shear-free geodesic congruences. The solutions depend on an arbitrary holomorphic function and give rise to new hyperKahler and selfdual Einstein metr…
We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.
The paper classifies all left invariant metrics on complex hyperbolic space.
problem Classifying left invariant Riemannian metrics on complex hyperbolic space.
method Analyzing the structure of the Lie group and using properties of constant curvature metrics.
result All metrics are of constant negative scalar curvature, with only one Einstein.
The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.
problem Classifying Heintze groups up to isometry and quasi-isometry in low dimensions.
method Analyzing quasi-isometries and isometries of Heintze groups, applying existing tools to groups of dimension 4 and 5.
result Complete classification of simply connected solvable groups in dimension 4 and groups of polynomial growth in dimension 5 up to isometry.
Study of multi-moment map for nearly Kähler S³ × S³.
problem Investigating the multi-moment map for nearly Kähler S³ × S³.
method Analyzing the multi-moment map associated with an almost Hermitian manifold with a torus action.
result The multi-moment map behaves similarly to the moment map of a toric manifold in the nearly Kähler S³ × S³ case.
Entire minimal graphs in Heisenberg space have negative Gauss curvature.
problem Characterizing entire minimal graphs in Heisenberg space.
method Defining holomorphic quadratic differentials and using them to describe entire graphs.
result Entire minimal graphs in Heisenberg space have negative Gauss curvature.
Study finds all isometries for specific Lie groups.
problem Identifying isometry groups in nonunimodular Lie groups.
method Examined left-invariant Riemannian metrics on Lie groups of dimension four.
result Determined full group of isometries for each metric.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
Classifies conjugation orbits of loxodromic pairs in SU(n,1).
problem Classifying conjugation orbits of loxodromic pairs in SU(n,1).
method Classification based on fixed points and conjugation.
result Classification of SU(n,1) conjugation orbits of pairs of loxodromic elements.
Study describes isometry groups of specific Lie groups.
problem Understanding isometry groups of compact Lie groups.
method Analyzing bi-invariant metrics on SO(4) and U(n).
result Full groups of isometries identified for specific Lie groups.
Study reveals structure of isometry group for specific manifolds.
problem Understanding the isometry group of non-compact, homogeneous manifolds.
method Analyzes non-compact, homogeneous manifolds with immortal Ricci flows.
result Establishes structure result for isometry group.
We show that a compact complex surface which admits a conformally Kähler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g beco…
Quantum isometry groups exist for certain metric spaces.
problem Existence of quantum isometry groups for specific metric spaces.
method Proved existence for geodesic metrics and uniformly distributed measure spaces.
result Quantum isometry groups are classical (commutative) for Riemannian manifolds.
Explicit isometry groups found for nearly Kähler manifolds.
problem Understanding the symmetries of nearly Kähler manifolds.
method Alternative, less algebraic approach to find isometry groups.
result Explicit expression for isometry groups of six-dimensional nearly Kähler manifolds.