The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Study stability and instability of Ricci-flat metrics under generalized Ricci flow.
problem Stability and instability of Ricci-flat metrics under Ricci flow.
method Analysis of generalized Ricci flow for Ricci-flat metrics and vanishing 3-forms.
result Dynamical stability and instability results for Ricci-flat metrics and vanishing 3-forms.
The paper examines the limit of harmonic flow on flat vector bundles.
problem Understanding the limiting behavior of harmonic flow on flat complex vector bundles.
method Analyzes the harmonic flow and proves the limit is isomorphic to a graded flat complex vector bundle.
result The limit of the harmonic flow on flat complex vector bundles is isomorphic to a graded flat complex vector bundle.
We study Ricci flows on Rn, n≥3, that evolve from asymptotically flat initial data. Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is constant in time along the flow. We then specialize to the case of rotationally symmetr…
Study on the regularity of p-Gauss curvature flow near flat interfaces.
problem Regularity of p-Gauss curvature flow near flat interfaces. method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homoge…
New theorem using Ricci flow for Gromov almost flat manifolds.
problem Conditions for Gromov almost flat manifolds.
method Employing Ricci flow to derive a new theorem.
result New theorem with weaker condition than Gromov--Ruh Theorem.
Ancient solutions found for a specific flow on symplectic half-flat structures.
problem Existence of solutions for a particular geometric flow.
method Analyzes Type IIA flow on symplectic half-flat SU(3)-structures.
result Existence of ancient, immortal, and eternal solutions under suitable conditions.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when k=n/2.
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.
Study shows how flat flow solutions in 2D converge to disks.
problem Understanding the asymptotics of area-preserving mean curvature flow in 2D.
method Analyzes flat flow solutions starting from bounded sets of finite perimeter.
result Flat flow solutions converge to a union of equally sized disks with exponential rate.
In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on n-dimensional, n≥3, asymp…
Anomaly flow studied on flat and non-flat nilmanifolds.
problem Analyzing the Anomaly flow on nilmanifolds.
method Examined with respect to Hermitian connections, focusing on flat and non-flat cases.
result General solutions and qualitative behavior of the Anomaly flow on nilmanifolds.
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
We study relation of the Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds. In particular, we construct Ricci-flat cohomogeneity one metrics with respect to 3-dimensional Lie groups.
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension n≥3 has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists …
New theorem on flat tori stability using harmonic maps and Ricci flow.
problem Stability of flat tori under Ricci and scalar curvature bounds.
method Harmonic map heat flow, Ricci flow, and RCD theories.
result Gromov-Hausdorff stability theorem for flat 3-tori.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. As we have proved in [L], the geodesic flows associated with the flat metrics on T^2 minimize the polynomial entropy. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated to flat metrics are local strict minima for the polynomial entropy.…
Study on Gauduchon manifolds finds metrics for projectively flat bundles.
problem Existence of Hermitian-Poisson metrics on projectively flat bundles.
method Heat flow techniques and continuity methods.
result Established a correspondence between Hermitian-Poisson metrics and semi-simplicity.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
Study shows smooth convergence of round surfaces in flat space-time models.
problem Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces.
method Volume preserving mean curvature flow in asymptotically flat 3-manifolds.
result The flow converges smoothly to a stable CMC surface.
The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.
problem Characterizing parallel spinors on Ricci flat Lorentzian four-manifolds.
method Evolution flow defined by parallel spinors, proving preservation of constraints, solving left-invariant flows.
result Initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds.
In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We…
We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.
We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we s…
The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.
problem Understanding ancient solutions to curvature flows in bounded and unbounded regions.
method Constructing pancake-like and sausage-like ancient compact solutions.
result Ancient solutions to curvature flows in bounded and unbounded regions.
Identifies scales in Ricci-flat manifolds at infinity.
problem Identifying between distant scales in non-compact Ricci-flat manifolds.
method Gradient flow of an elliptic equation on a tangent cone at infinity.
result Natural identification map between scales at infinity.
In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under a…
Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
The study uses Ricci flow to prove flatness of certain Riemannian manifolds.
problem Proving the flatness of Riemannian manifolds with specific curvature properties.
method Ricci flow approach, quantitative existence theory, curvature estimates, and regularization.
result Manifolds with non-negative curvature and specific decay rates are necessarily flat.
The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.
problem Analyzing Ricci flow with Ricci curvature and volume constraints.
method Proving convergence and curvature bounds for Ricci flow with specific constraints.
result Ricci flow with specified constraints converges to a flat cone or static flow.
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
Noncompact Ricci-flat solutions have infinite unstable dimensions.
problem Understanding unstable dimensions of noncompact Ricci-flat solutions.
method Derived sufficient conditions for infinite-dimensional unstable manifolds.
result Noncompact Ricci-flat solutions have uncountably many unstable perturbations.
Ricci flow deforms metrics with positive curvature to include negative curvature.
problem Preserving positive sectional curvature under Ricci flow in dimension four.
method Evolved cohomogeneity one metrics on S4 and CP2 via Ricci flow. result Metrics with positive sectional curvature lose this property under Ricci flow.
New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
The paper examines conditions for Ricci solitons to be Ricci flat or Einstein.
problem Conditions for Ricci solitons to be Ricci flat or Einstein.
method Study of evolution equations of scalar and Ricci curvatures under Ricci flow.
result Conditions for Ricci solitons to be Ricci flat or Einstein.
The paper studies global Yamabe flow on AF manifolds, preserving ADM mass.
problem Existence and behavior of Yamabe flow on AF manifolds.
method New local existence theorem and maximum principle for parabolic equations.
result Global existence of Yamabe flow on AF manifolds with non-negative scalar curvature.
Paper uses flow to prove theorem on Higgs bundles.
problem Proving generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles.
method Affine Hermitian-Yang-Mills flow
result Generalized Donaldson-Uhlenbeck-Yau theorem proved.