We establish global existence of smooth solutions to heat flow for Yang-Mills-Higgs functional on Kahler fibrations. As an application, we give a new proof of the key inequality for Mundet's Hitchin-Kobayashi correspondence theorem using the heat flow technique.
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We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the -space produces the same evolution as the gradient flow of the relative entropy in the -Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…
The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.
problem Analyzing mass concentration and nodal domains of Laplace eigenfunctions.
method Heat diffusion technique to study eigenfunctions and their nodal sets.
result Discovers new insights into the decay and behavior of Laplace eigenfunctions.
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.
Asymptotic Spectral Flowmath.DG
Study the spectral flow of Dirac operators on spinor bundles.
problem Understanding the asymptotic behavior of spectral flow for Dirac operators.
method Variation of eta invariant and local index theory technique.
result Established a uniform estimate of the eta invariant for large parameter values.
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 on biharmonic map heat flow with monotonicity formula.
problem Properties of biharmonic heat kernel and extrinsic biharmonic map heat flow.
method Derived an entropy type quantity exhibiting monotonicity behaviors.
result Monotonicity formula for extrinsic biharmonic map heat flow.
New method learns latent group structures without clustering, using heat flow dynamics.
problem Learning with latent group sparsity in machine learning problems.
method Heat flow dynamics on network structure to incorporate group structure.
result Effective performance and provable bounds on sample complexity.
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.
Paper studies smoothness of bi-conformal heat flow on 4-manifolds.
problem Smoothness of bi-conformal heat flow on 4-manifolds.
method Introduces bi-conformal heat flow (bi-CHF) and proves global smoothness without finite time singularities.
result Global smoothness and no finite time singularity for bi-conformal heat flow.
In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.
Heat flow fails to preserve concavity in curved spaces.
problem Non-preservation of concavity properties in curved spaces.
method Analysis of Dirichlet heat flow on Riemannian manifolds.
result No concavity properties are preserved unless curvature is zero.
Bounds on Hessian of heat equation coupled with Ricci flow.
problem Estimating the Hessian of a solution to the conjugate heat equation coupled with Ricci flow.
method Obtained upper bounds for the Hessian.
result Local and global upper bounds for the Hessian of a positive solution.
Extends heat kernel estimates for super Ricci flow.
problem Heat kernel estimates for super Ricci flow.
method Generalizes Bamler-Zhang's geometric analysis to super Ricci flow.
result Obtains Gaussian heat kernel estimates for super Ricci flow.
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.
The paper proves a Liouville theorem for heat flows on manifolds with specific curvature conditions.
problem Investigating heat flows on manifolds with specific curvature conditions.
method Gradient estimate and Liouville type theorem for ancient solutions.
result Established a Liouville theorem for -harmonic heat flows.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Proves upper bounds for heat kernels evolving on manifolds.
problem Bounding heat kernels on evolving manifolds.
method Logarithmic Sobolev inequalities and ultracontractivity estimates.
result Gaussian upper bounds for heat kernels are derived.
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.
Calculates spinor heat flow using Gaussian-Grassmann integrals.
problem Computing the spinor heat flow symbol.
method Getzler calculus and Gaussian-Grassmann integrals.
result Computed the spinor heat flow symbol.
Differential Harnack inequalities for nonlinear heat equations with potentials under the Ricci flowmath.DG
We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the -Ricci flow on a closed surface. These new Harnac…
We present two initial graphs over the entire , for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. …
Study heat flow on changing surfaces, proving existence and uniqueness.
problem Existence and uniqueness of heat flow on time-varying manifolds.
method Establishes estimates for heat flow under minimal assumptions, focusing on logarithmic derivative of volume measure.
result Proves estimates hold for Ricci flow with scalar curvature bounded below, dependent only on initial data.
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.
Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.
problem Establishing estimates for heat and conjugate heat equations under Ricci flow.
method Proving matrix Li-Yau-Hamilton estimates for positive solutions to the heat and conjugate heat equations coupled with Ricci flow.
result Monotonicity of parabolic frequencies established up to correction factors.
The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.
problem Analyzing heat flow and constants on graphs.
method Introducing concepts, recalling graph theory, and proposing new discrete Morse flows.
result Weak discrete Morse flows for heat flow on finite graphs under suitable assumptions.
Extends gradient estimates for heat equation under Finsler geometric flows.
problem Global gradient estimates for positive solutions to heat equation.
method General compact Finsler geometric flow.
result Derives Harnack inequality for positive solutions.
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.
Proves estimates for Kähler-Ricci flow solutions.
problem Positive solutions to Kähler-Ricci flow.
method Matrix Li-Yau-Hamilton estimates coupled with flow.
result Monotonicity formula derived.
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.
The paper improves heat equation estimates under weaker Ricci curvature conditions.
problem Improving heat equation estimates under weaker Ricci curvature conditions.
method Establishing Li-Yau-type and Hamilton-type estimates for positive solutions of the heat equation under generalized Ricci flow.
result Deriving Harnack-type inequalities and monotonicity of parabolic frequency.
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.
Researchers find stable solutions for heat map flow in higher dimensions.
problem Stability of shrinkers for harmonic map heat flow in higher dimensions.
method Construction of specific target manifolds allowing for stable shrinkers.
result Existence of corotational self-similar shrinkers representing stable blowup mechanisms.
Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flowmath.DG
In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's …
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.
In this paper, we consider the heat flow for p-pseudoharmonic maps from a closed Sasakian manifold M into a compact Riemannian manifold N. We prove global existence and asymptotic convergence of the solution for the p-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold N is nonpos…
Study shows global oscillatory solutions for Yang-Mills heat flow in 4D space.
problem Investigating long-time dynamics of Yang-Mills heat flow with specific initial data.
method Analysis of -equivariant Yang-Mills heat flow with group in 4D space.
result Global solutions can exhibit oscillatory behavior at time infinity.
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…
Study heat content in sub-Riemannian manifolds, obtaining asymptotic expansion.
problem Heat content in sub-Riemannian manifolds with non-characteristic domains.
method Fourth-order asymptotic expansion, combining rough boundary temperature and stochastic completeness.
result Obtained a fourth-order asymptotic expansion for relative heat content.
Sharp estimates for heat flow on nonconvex domains.
problem Quantitative estimates for heat flow on nonconvex domains.
method Sharp gradient and transport estimates with novel dependence on time.
result Equivalent characterization of lower bound on second fundamental form.
The study characterizes heat flow and concentration on directed graphs with a lower Ricci curvature bound.
problem Understanding heat flow and concentration on directed graphs with a specific curvature bound.
method Characterization via gradient estimate and transportation inequality for the heat semigroup.
result Concentration of measure inequality for directed graphs with positive Ricci curvature.
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 differential 1-forms, adapted flow construction.
result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
Proves heat kernel superconvexity in hyperbolic space.
problem Heat kernel superconvexity in hyperbolic space.
method Proves conjecture by Bernstein in all dimensions.
result Analog of Huisken's monotonicity formula for mean curvature flow.
Study on uniquely determining thermal properties from boundary temperature and heat flux measurements.
problem Determine thermal conductivity and volumetric heat capacity from boundary measurements.
method Uniqueness proof for isotropic and anisotropic media under thermal diffusivity assumption.
result Uniqueness of thermal properties in all dimensions and up to a gauge in two dimensions.
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…
We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…