Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.
problem Investigate Cimes-families of flat connections with nilpotent Higgs fields. method Analyze families of flat connections including real twistor lines and conformal limits, deducing monodromy similarities.
result Traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.
New proof shows no flat embedding for Petersen family graphs.
problem Proving Petersen family graphs have no flat embeddings.
method Applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.
result Every Petersen family graph has no flat embedding.
Study on flat connections with controlled irregularity.
problem Boundedness of algebraic flat connections with limited irregularity.
method Analysis of families of algebraic flat connections and holonomic D-modules.
result Established boundedness of families of algebraic flat connections with controlled irregularity.
We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
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.
This work shows all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces are known families.
problem Characterize all conformally Kähler, Ricci-flat toric metrics on non-compact surfaces.
method Unified construction using axi-symmetric harmonic functions and methods from scalar-flat Kähler metrics.
result All such metrics are ALF and belong to known families.
Classifies toric dually flat manifolds into complex space forms.
problem Classifying 1D toric dually flat manifolds.
method Using complex space forms and exponential families.
result Toric dually flat manifolds are complex space forms.
Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.
problem Finding analytic Kähler potentials for Calabi-Yau manifolds.
method Numerically calculating Ricci-flat Kähler potentials via machine learning and fitting to Donaldson's Ansatz.
result Simple analytic expressions for approximately Ricci-flat Kähler potentials are found, including explicit dependence on complex structure parameter.
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
Study the spaces of flat connections for classical Lie groups using Chern-Weil theory.
problem Understanding the weak homotopy type of spaces of flat connections for classical Lie groups.
method Use Chern-Weil theory and relate to the functorial map involving continuous families of representations.
result Relate the spaces of flat connections to the weak homotopy type of the spaces of representations.
Constructs Einstein metrics on manifolds with specific orbits.
problem Finding Einstein metrics on manifolds with given orbits.
method Continuous families of metrics constructed using vector bundles and R4m+4. result Recovery of Spin(7) metrics A8 and B8. Researchers construct explicit bundles for ALF metrics, revealing rational patching matrices for gravitational instantons.
problem Constructing explicit toric Ricci-flat metrics and their associated bundles.
method Explicit construction of patching matrices for ALF metrics and gravitational instantons.
result Rational form of patching matrices for gravitational instantons in the Chen--Teo family.
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
Uniform estimates for complex Monge-Ampère equations on hermitian varieties.
problem Uniform boundedness of Chern-Ricci flat potentials in conifold transitions.
method Proving uniform a priori estimates for degenerate complex Monge-Ampère equations.
result Generalization of a theorem to hermitian contexts.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat …
Paper introduces new center of mass for flat manifolds.
problem Defining center of mass for asymptotically flat manifolds.
method Using double forms of Kulkarni and Labbi to prove existence and well-definedness.
result Existence and well-definedness of the Gauss-Bonnet-Chern center of mass.
New flat Minkowski planes created from convex functions.
problem Creating new geometric structures from convex functions.
method Constructing flat Minkowski planes using convex functions.
result Automorphism groups of these planes are at least 3-dimensional.
New flat surfaces found in 3D sphere space.
problem Constructing flat surfaces in 3D sphere.
method Using Ribaucour transformations and flat torus theory.
result Families of complete flat surfaces in S3 determined by parameters. We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if p:(X,B)→Y is a Kähler fiber space such that (Xy,B∣Xy) is generically klt, KX/Y+B is relatively trivial and p∗(m(KX/Y+B)) is Hermitian fla…
Introduces Legendre bundle for dually flat manifolds and quantum field theories.
problem Understanding duality in geometric structures and quantum field theories.
method Introduces Legendre bundle and para-Kähler structure.
result Exponential families and Hessian QFTs are realizations of the Legendre bundle.
Study cylindrical symmetric Finsler metrics that are projectively flat.
problem Characterize Finsler metrics that are projectively flat.
method Solve the system of differential equations for cylindrical symmetric Finsler metrics.
result Provide a family of solutions for the projectively flat Finsler metrics.
We prove a perturbative result concerning the uniqueness of Kerr-Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons, if it agrees with a non-extremal Kerr-Newman space-time asymptotically flat at infinity and it is sufficiently close to the Kerr-Newman family, then the space-t…
The aim of this paper is to calculate the eta invariants and the dimensions of the spaces of harmonic spinors of an infinite family of closed flat manifolds. It consits of some flat manifolds M with cyclic holonomy groups.
Determinants remain constant along specific families of differential operators.
problem Local constancy of regularized determinants for differential operators.
method Analyzing families of operators Dτ=[δτ,d∇], showing flat-regularized determinant's constancy. result The flat-regularized determinant is constant in τ when restricted to im(δτ) under suitable assumptions. Exponential families and mixture families are parametric probability models that can be geometrically studied as smooth statistical manifolds with respect to any statistical divergence like the Kullback-Leibler (KL) divergence or the Hellinger divergence. When equipping a statistical manifold with the KL divergence, th…
In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in CP3 as the image of the composition of the Hopf map H:S7→CP3 and a map ψ:R3→S7 with certain conditions.
We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family RBM of real Bott manifolds and the family GHW of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection GHW∩RBM is not empty. We also …
We prove that any conformally flat submanifold with flat normal bundle in a conformally flat Riemannian manifold is locally holonomic, that is, admits a principal coordinate system. As one of the consequences of this fact, it is shown that the Ribaucour transformation can be used to construct an associated large family…
New instantons show Einstein-Maxwell fields are more complex.
problem Disprove Euclidean Einstein-Maxwell black hole uniqueness.
method Explicit construction of new three-parameter family of asymptotically flat instantons.
result Demonstrate subtle properties of coupled gravitational and electromagnetic fields.
Study families of Lie algebroids on complex spaces, introducing unfoldings.
problem Investigate singular holomorphic Lie algebroids on complex analytic spaces.
method Introduce and study unfoldings of Lie algebroids, showing a correspondence with holomorphic flat connections.
result Existence of a one-to-one correspondence between transversal unfoldings and holomorphic flat connections.
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
Study integrable discretizations of cyclic systems with circular coordinate lines.
problem Integrable discretizations of 3D cyclic systems with circular coordinate lines.
method Investigate circle congruences and flat connections in the context of discrete cyclic systems.
result Characterization of circle congruences and existence of certain flat connections.
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…
Geodesic descent optimizes likelihood in dually flat spaces.
problem Maximum likelihood estimation in exponential families.
method m-geodesic and e-geodesic updates on dually flat spaces.
result Geodesic updates can reach maximum likelihood estimator in one step.
Study connects G2-structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of G2-structures and flat connections. method Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. result Equivalence between moduli spaces of G2-structures and flat connections on compact 3-manifolds. Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental g…
We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…
Develops a new method for minimal Lagrangian surfaces in complex quadrics.
problem Finding minimal Lagrangian surfaces in complex quadrics.
method Loop group representation and flat connections.
result Equivalence of minimality and flatness of connections, explicit families of examples.
We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over CP2, HP2 and OP2 with respective principal orbits G/K the Wallach spaces SU(3)/T2, Sp(3)/(Sp(1)Sp(1)Sp(1)) and F4/spin(8). Almost all the …
Lin-Lu-Yau introduced an interesting notion of Ricci curvature for graphs and obtained a complete characterization for all Ricci-flat graphs with girth at least five [1]. In this paper, we propose a concrete approach to construct an infinite family of distinct Ricci-flat graphs of girth four with edge-disjoint 4-cycles…
Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.
problem Understanding the relationships between different types of surfaces and their integrable systems.
method Unified view through families of flat connections and parallel sections.
result Complete description of links between different surface types and their dressing transformations.
We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold K, we produce the generalized connected sum along K. On this third manifold we produ…
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…
The paper studies scalar flat Kähler metrics on line bundles and proves their properties.
problem Understanding scalar flat Kähler metrics on line bundles.
method Analyzes two families of scalar flat Kähler metrics on Cn+1 and O(−k), proving existence of asymptotic expansions and approximations. result Characterizes the Burns-Simanca metric as the only projectively induced scalar flat metric on O(−k) with a vanishing second coefficient in its asymptotic expansion.