In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…
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48 results for “harmonic flows”
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
The study derives formulas for functionals on surface with boundary under harmonic Ricci flow.
problem Understanding functionals along harmonic Ricci flow on surfaces with boundaries.
method Derivation of formulas for functionals under harmonic Ricci flow.
result Established formulas for functionals on surface with boundary.
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In add…
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
Study sesqui-harmonic map flow from Riemannian surfaces
problem Investigate sesqui-harmonic map flow from Riemannian surfaces
method L2-gradient flow of an energy functional
result Generalizes Struwe's regularity result for harmonic maps
New rigidity estimate derived via harmonic map flow.
problem Rigidity of maps from to .
method Harmonic map flow approach.
result Rigidity estimate derived.
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for -harmonic maps.
result Regularized -conformal heat flow does not develop finite time singularities.
Study proves existence of non-trivial harmonic map flows to hemispheres.
problem Existence of non-trivial harmonic map flows to hemispheres.
method Construction of infinitely many weak solutions to harmonic map flow starting from non-minimizing but stationary maps.
result Proves existence of non-trivial self-expanding harmonic map flows to hemispheres.
Study proves short-term existence for harmonic maps under evolving metrics.
problem Analyzing harmonic maps under time-dependent metrics.
method Proves short-term existence for harmonic map heat flow coupled with a smooth family of complete metrics.
result Generalizes short-term existence results for harmonic map heat flow.
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.
The paper connects Ricci flow and harmonic spinors, proving new inequalities.
problem Understanding the behavior of harmonic spinors under Ricci flow.
method Introduced a weighted monopole equations and used Perelman's entropy.
result Ricci flow is the gradient flow of energy related to harmonic spinors.
New approach to heat flow for half-harmonic maps, related to minimal surfaces.
problem Heat flow for half-harmonic maps from to closed target manifolds.
method Classical approach using Dirichlet-to-Neumann operator for the Laplace equation.
result Analogous results to 1985 harmonic map heat flow, valid for finite-energy data.
Paper studies heat flow for VT harmonic maps on compact manifolds.
problem Existence of VT harmonic maps and geodesics on compact manifolds.
method Heat flow method to solve Dirichlet problem and existence of geodesics.
result Existence of VT harmonic maps and geodesics under certain conditions.
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvaturesmath.DG
The paper examines how parabolic frequency behaves under Ricci flow and Ricci-harmonic flow on manifolds.
problem Understanding the behavior of parabolic frequency under Ricci flow and Ricci-harmonic flow.
method Investigates the monotonicity of parabolic frequency for solutions of linear and heat equations with bounded curvatures.
result Establishes monotonicity results for parabolic frequency under specific curvature conditions.
Harmonic flow of -structuresmath.DG
Study harmonic flow of Spin(7)-structures on compact 8-manifolds.
problem Isometric flow of Spin(7)-structures on compact 8-manifolds.
method Establishing Shi-type estimates, self-similar solutions, monotonicity formula, compactness theorems, and Bryant-type description.
result Conditions for long-time existence and characterisation of singularities.
Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometrymath.DG
Study Liouville theorems for harmonic maps along ancient super Ricci flows.
problem Proving Liouville theorems for harmonic maps under specific geometric conditions.
method Using Perelman's reduced geometric viewpoint, derive Liouville theorems with controlled growth.
result Sharp growth conditions and new Liouville theorems for both non-positively and positively curved target spaces.
Paper shows no finite time singularities for smooth conformal heat flow of harmonic maps.
problem Smoothness of conformal heat flow of harmonic maps.
method Combines harmonic map flow with metric evolution in conformal direction.
result No finite time singularity occurs for the flow, and under certain conditions, maps converge to a point.
Paper solves Minkowski problem for p-harmonic measures.
problem Solving the Minkowski problem for p-harmonic measures on convex domains.
method Using the Gauss curvature flow method.
result Existence of smooth solution to the Minkowski problem for p-harmonic measures.
Strict type-II blowup in harmonic map flow is proven to have Hölder continuous body map.
problem Finite-time singularity of harmonic map flow.
method Analysis of outer energy scale and Hölder continuity proof.
result Strictly type-II blowup body map is Hölder continuous.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
New flow for G2-structures helps find torsion-free structures.
problem Finding torsion-free G2-structures on compact manifolds.
method Ricci-harmonic flow of G2-structures, analyzing Taylor series expansion.
result Stationary points of the flow are torsion-free G2-structures.
Study of quaternion-Kähler structures via harmonic flow on 8-manifolds.
problem Geometric flow of quaternion-Kähler structures on 8-manifolds.
method Formulated gradient Dirichlet flow, analyzed harmonicity, proved almost-monotonicity.
result Proved long-time existence and constructed harmonic solitons.
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtain…
Paper studies Laplace operator estimates in harmonic map heat flows.
problem Estimating Laplace operator in harmonic map heat flows outside singularities.
method Investigates estimates using spherical coordinates for and boundary conditions.
result Provides higher-order estimates for the Ericksen--Leslie system.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
Study of Type IIA flow on symplectic Lie algebras for geometric structures.
problem Detecting geometric structures like Lagrangian torus fibrations and harmonic almost complex structures.
method Investigate -harmonic forms and the long-time behavior of the Type IIA flow.
result The Type IIA flow helps in detecting desired geometric structures.
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
Static spacetimes are stable attractors in a flow equation.
problem Stability of static spacetimes with negative cosmological constant.
method New expander entropy for Ricci-harmonic flow.
result Static metrics are stable if and only if a positive mass theorem holds.
We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology,…
Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.
problem Establishing Liouville theorems for ancient solutions to V-harmonic map heat flows.
method Refined gradient estimates and exponential growth conditions.
result Better Liouville theorems for ancient solutions to V-harmonic map heat flows.
Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…
In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where maps from a fixed closed surface with metric to a general target manif…
We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes th…
Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure . This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure has small energy (depending on the norm ), then the flow ex…
New heat flow for harmonic maps avoids singularities but not bubbles.
problem Finite time singularities in harmonic maps.
method Introduces a conformal heat flow for harmonic maps defined by an evolution equation.
result Global weak solution exists, smooth except at most finitely many points.
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 …
The paper studies harmonic map heat flow stability and decay rates.
problem Analyzing stability and decay rates of harmonic map heat flow solutions.
method Use of homogeneous Besov space for small initial data and self-similar decay assumption.
result Decay rates for solutions of the harmonic map flow of the form and self-similar decay under stronger initial conditions.
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…
A new method estimates marginal likelihood using normalizing flows.
problem Estimating marginal likelihood in Bayesian model selection.
method Learned harmonic mean estimator using normalizing flows.
result Normalizing flows avoid the exploding variance problem.