Found first example of homogeneous gradient solitons for G2-Laplacian flow.
problem Existence of homogeneous gradient solitons for G2-Laplacian flow. method Provided the first known example of homogeneous gradient solitons.
result G2-Laplacian flow admits homogeneous gradient solitons on one-dimensional extensions. The paper connects complex Monge-Ampère equations to G2-structures on Calabi-Yau manifolds.
problem Establishing a relationship between complex Monge-Ampère equations and G2-structures. method Using a parabolic complex Monge-Ampère equation and Kähler metrics, the paper establishes the existence and convergence of G2-Laplacian and coflows. result The G2-Laplacian flow and coflow converge to G2-structures induced by Kähler Ricci-flat metrics. Study G2-flows reducing to complex geometry flows, focusing on G2-anomaly and G2-Laplacian coflow.
problem Investigate flows of G2-structures in relation to complex geometry. method Analyze G2-Laplacian coflow and G2-anomaly flow, compare their properties. result Compare G2-anomaly flow to G2-Laplacian coflow, investigate short-time existence and fixed points. We show the existence of expanding solitons of the G2-Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched G2-structure.
Nearly G2-structures are unstable under a modified G2-Laplacian co-flow.
problem Stability of nearly G2-structures under geometric flows. method Normalized modified G2-Laplacian co-flow. result Many nearly G2-structures are unstable, with the standard structure on the round 7-sphere being an unstable critical point. We prove the hypersymplectic flow of simple type on standard torus T4 exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one G2-Laplacian flow on a compact 7-manifold which exists for all time and…
We explicitly describe the solution of the G2-Laplacian flow starting from an extremally Ricci-pinched closed G2-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds m…
A flow from hypersymplectic to hyperkähler structures is described.
problem Flowing from hypersymplectic to hyperkähler structures on 4-manifolds.
method Positive triples, G2-Laplacian coflow, hypersymplectic flow. result The G2-Laplacian coflow descends to the hypersymplectic flow. We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space Rn,m, which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of…
A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive-definite subspace of Λ2 for the wedge product. This article is motivated by a conjecture of Donaldson: when X is compact ω can be deformed through cohomologous hype…
Geodesic concavity and hypersymplectic structures in G2-structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G2-structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G2 Laplacian flow decreases the length. New examples found for a type of geometric solitons.
problem Finding new shrinking Laplacian solitons.
method One-parameter family of examples and study of torsion forms.
result No closed eigenform for the Laplacian on the family.
Uniqueness proven for specific types of geometric structures.
problem Proving uniqueness of asymptotically conical gradient shrinking solitons.
method Extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons.
result G_2-structures are equivalent if asymptotically conical and asymptotic to the same closed G_2-cone.
The paper studies G2-Poisson equations on 7-spheres and classifies invariant solutions.
problem Existence and uniqueness of G-invariant solutions for G2-Laplacian on 3-forms. method Analyzes G-invariant solutions for G=SU(4),Spin(7),Sp(2)imesSp(1)/Z2 and discusses eigenvalue problem. result Classification of G-invariant solutions and determination of nearly parallel G2-structures. A short, elementary proof is given of the result: The number of components of a link arising from a medial graph M(G) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of the graph G.
Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.
problem Existence and uniqueness of geometric flows of G2-structures.
method Introduced geometric structures, geometric flows, and discussed qualitative features. Focused on Ricci flow and DeTurck trick, then extended to G2-structures.
result Clarified conditions for short-time existence and uniqueness of G2-Laplacian flow.
Rigidity of critical eigensections on spheres proven.
problem Rigidity of critical eigensections on spheres.
method Proved rigidity of critical eigensections through SO(3)-rotations.
result Minimal non-degenerate critical eigensections are deformation rigid.
Formula connects G2-structure geometry to Poisson equation.
problem Solvability conditions for G2-structures in a Poisson equation. method Developed a Gauss-Codazzi-like formula for G2-structures. result Necessary and sufficient conditions for solvability in cohomogeneity one.
Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
The H-type deviation measures how close step two Carnot groups are to H-type groups.
problem Quantifying how close step two Carnot groups are to H-type groups.
method Defined and analyzed the H-type deviation for step two Carnot groups.
result Explicitly computed H-type deviation for product of Heisenberg groups and verified the conjectural upper bound.
New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…
i-flow uses normalizing flows for high-dimensional integration and sampling.
problem High-dimensional integration in science and statistics.
method Normalizing flows for bijective mappings between distributions.
result i-flow outperforms other algorithms for high-dimensional correlated integrals.
The article calculates the F-convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-sense. Paper introduces Tensor Gauge Flow Models for better data encoding.
problem Lack of expressive flow dynamics in existing Generative Flow Models.
method Incorporates higher-order Tensor Gauge Fields into the Flow Equation.
result Tensor Gauge Flow Models achieve improved generative performance.
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in R4. method Analysis of ancient noncollapsed flows in R4. result Nonexistence of rotating ancient flows among ancient noncollapsed flows in R4. New flow preserves almost Hermitian metrics for manifold study.
problem Curvature flow for almost Hermitian manifolds.
method Introducing a new curvature flow matching Ricci flow and preserving almost Hermitian condition.
result Ricci flow can be used to study almost Hermitian manifolds.
Simplifies residual flows to make flow-based modeling more practical.
problem Extremely high computational cost of residual flows limits their applicability.
method Introduces Quasi-Autoregressive (QuAR) approach to residual flows.
result Significantly reduces compute time and memory requirements for flow-based modeling.
Existence of translating solutions shown for curve diffusion flow.
problem Existence of translating solutions for curve diffusion flow.
method Higher order curve shortening flow approach.
result Properly immersed translating solutions exist.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single…
The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally …
Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an en…
The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.
problem Analyzing mean curvature flow in a Ricci flow background.
method Computing variational properties and deriving evolution equations for mean curvature and second fundamental form.
result Established a Huisken's monotonicity-type formula for mean curvature solitons in an extended Ricci flow.
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
New derivation of Type IIA flow metrics.
problem Flow of metrics in Type IIA theory.
method Adapted to Laplacian flow, uses projected Levi-Civita connection.
result New derivation of flow equations.
Survey of geometric flows from unified string theories.
problem None explicitly stated, but related to understanding geometric flows in string theories.
method Survey of geometric flows in various geometries (complex, almost-complex, symplectic) motivated by string theories.
result Intermediate flows between Ricci and Kähler-Ricci flows, often coupled to additional fields.
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.